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UBC Theses and Dissertations

Topics in black hole evaporation Leahy, Denis Alan 1980

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TOPICS IN BLACK HOLE EVft.POfiAT.IOH t l . S c . , The U n i v e r s i t y o f B r i t i s h C olumbia, 1.976 A THESIS SUBMITTED IN PftBTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s ) We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 1980 © Denis A l a n Leahy, 1980 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of P U ^ C ( ( ^ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date r p o r n . i ^ o ABSTRACT Two major a s p e c t s of p a r t i c l e c r e a t i o n by g r a v i t a t i o n a l f i e l d s of b l a c k h o l e s a re s t u d i e d ; the n e u t r i n o e m i s s i o n from r o t a t i n g b l a c k h o l e s ; and i n t e r a c t i o n s between s c a l a r p a r t i c l e s e m i t t e d by a b l a c k h o l e . The n e u t r i n o e m i s s i o n i s i n v e s t i g a t e d under t h r e e t o p i c s . The asymmetry o f the a n g u l a r dependence o f n e u t r i n o e m i s s i o n from r o t a t i n g b l a c k h o l e s i s c a l c u l a t e d f i r s t . A low fre q u e n c y a n a l y t i c a p p r o x i m a t i o n demonstrates the p r e f e r e n t i a l e m i s s i o n of n e u t r i n o s ( a n t i n e u t r i n o s ) a n t i p a r a l l e l < p a r a l l e l ) t o the d i r e c t i o n o f the b l a c k h o l e ' s a n g u l a r momentum v e c t o r . N u m e r i c a l c a l c u l a t i o n s are performed which r e v e a l the dependence of t h e n e u t r i n o e m i s s i o n on p o l a r a n g l e , n e u t r i n o energy, and b l a c k hole a n g u l a r momentum and mass. Next we c o n s i d e r t h e p r o d u c t i o n of a l o c a l m a t t e r e x c e s s by r o t a t i n g b l a c k h o l e s i n a baryon symmetric u n i v e r s e . B l a c k h o l e s form a t e a r l y c o s m o l o g i c a l t i m e s w i t h t h e i r r o t a t i o n axes a l i g n e d over t h e same s c a l e s i z e as the a n g u l a r momentum i n t h e u n i v e r s e . The e v a p o r a t i o n of these b l a c k h o l e s produces l a r g e s c a l e n e u t r i n o c u r r e n t s , whose e f f e c t i v e n e s s i n s e p a r a t i n g baryons from a n t i b a r y o n s d u r i n g t h e hadron e r a o f the e a r l y u n i v e r s e i s e s t i m a t e d . The l o c a l b a r y o n t o photon r a t i o o v e r a g a l a c t i c s i z e s c a l e depends on the i i i subsequent evolution of the re s u l t i n g matter and antimatter regions, but i s found to have an upper l i m i t of 13-* 6. This i s much less than the present observed value of about 10 -*. We then study cosaological magnetic f i e l d generation by neutrinos from evaporating black holes. During the radiation era the neutrinos scatter o f f protons and alectrons, producing a net charge current. This current generates magnetic f i e l d s . I f present in large enough numbers, rotating black holes could account for the present observed magnetic f i e l d i n our galaxy. F i n a l l y we study the e f f e c t s of interactions on the black hole evaporation process. Perturbation theory i s used, to second order, to calculate the e f f e c t s of a a<|> 4 s e l f -i n t e r a c t i o n f o r a scalar f i e l d <J> i n the 2 dimensional black hole spacetime. a mass renormalization was found to be i n s u f f i c i e n t to remove a l l divergences that occur i n the calc u l a t i o n s . However, the inte r a c t i o n appears to destroy the thermal character of the emission from a black hole evaporating i n a vacuum. W TABLE OF CONTESTS TITLE PAGE 1 ABSTRACT i i LIST OF TABLES ix LIST OF FIGURES . X ACKNOWLEDGEHENTS . . . . . . x i i l 1. INTBODOCTION 1 Quantum F i e l d Theory In Curved Spacetime ......... 2 Black Holes And Thermodynamics ................... 6 P a r t i c l e Creation By Gra v i t a t i o n a l F i e l d s -H e uristic Discussion .......................... 9 Overall Features Of Black Hole Evaporation ....... 13 P a r t i c l e Creation In Curved Spacetime- A Short F i e l d Theoretic Derivation «. 17 Black Holes Formed By S t e l l a r Collapse And The P a r t i c l e Creation Process ..................... 24 Thesis Outline And Summary ....................... 28 2. NEUTRINO EMISSION FROM ROTATING BLACK HOLES. ........ 36 I. Introduction ..................................... 36 I I . The Heutrino F i e l d In The Kerr Hetric ........... 39 I I I . Quantization Of The Neutrino F i e l d ; And .48 IV. The Choice Of Vacuum State ...................... 52 V. The Vacuum Expectation Values Of J * And T^y ...... 57 V VI. Summary Of Quantities For Calculation 64 VII. Low Freguency analytic approximation ........... 67 VIII. Numerical Calculations ........................ 73 1. angular Eigenfunctions ........................ 73 2. The Transmission C o e f f i c i e n t s ................. 74 3. The Emission Bates .......................... ..76 IX. Results and Discussion .......................... 77 X. Summary ......................83 3. HATTER ANTIHATTER SEPARATION IN THE EARLY UNIVERSE BI ROTATING BLACK HOLES 87 I. Introduction ..................................... 87 Baryon Symmetric Cosmologies ..................... 87 Hodel For The Early Universe ..................... 92 Black Boles In The Early Oniverse ................ 96 Chapter Outline • 100 I I . Calculation Of The Baryon To Photon Ratio ....... 101 The Neutrino Current From Rotating Black Holes ...103 The Baryon Current Produced By Neutrino Scattering 111 The Baryon Number Density And Baryon To Photon Ratio • ......113 I I I . Case A. Small Black Hole Density ...............117 IV. Friedmann Expansion With Black Hole Formation ...123 V. Case B. Opper Limits On The Baryon To Photon Ratio For p y « ^ / H ..,,.,...126 t V*l VI. D i s c u s s i o n ....131 Table 1. Expansion With Black Hole Formation-yQ>yWz/M 134 4. COSMOLOGICAL MAGNETIC FIELD GENERATION BY ROTATING BLACK HOLES . 135 I. I n t r o d u c t i o n ......135 I I . C a l c u l a t i o n Of The Charge Current ..140 The Neutrino Flux 142 The Proton And E l e c t r o n L i f e t i m e s ................ 148 Dominance Of The Proton C u r r e n t .................. 152 I I I . Generation Of The Magnetic F i e l d ...............155 Magnetic F i e l d Generated Over A P r o t o g a l a c t i c S c a l e ..159 IV. D i s c u s s i o n ............•.•••.••...••....•..••..••161 V. Summary .166 5 . Jhfr* SCALAR FIELD IN THE 2-D BLACK HOLE METRIC .......168 I. I n t r o d u c t i o n .......168 Chapter C u t l i n e 176 I I . The Free S c a l a r F i e l d ....178 I I I . C a l c u l a t i o n Of The Out St a t e With The *4>* I n t e r a c t i o n ....••••....••...•....••.•.•.••..•••••187 IV. E x p r e s s i o n s For The F l u x Of Outgoing P a r t i c l e s , And Zero Order C a l c u l a t i o n ....................... 189 V. The Black Hole Flux As A Thermal Sum over S t a t e s ,194 VI. Diagrams For Processes C o n t r i b u t i n g To The * V I I Outgoing F l u x . 198 V I I . D e t a i l e d Balance F o r P u r e l y O u t g o i n g I n t e r a c t i o n s .......•.•........•••.....••...•••••.207 V I I I . Comparison Of The B a d i a l Dependence Of B l a c k Hole And F l a t Space P a r t i c l e P r o d u c t i o n .....210 I X . I n t e g r a t i o n Over The I n t e r a c t i o n Region, And Sum Over Thermal S t a t e s ..............................215 Sum Over Thermal S t a t e s ........ .................. 221 X. E v a l u a t i o n Of Diagrams W i t h I n g o i n g P a r t i c l e s . ...222 No Decays To I n g o i n g P a r t i c l e s F o r The Case Of Women turn C o n s e r v a t i o n 223 I n c l u s i o n Of A Thermal I n g o i n g F l u x 224 E v a l u a t i o n Of Diagram No. 1 ...................... 225 R e s u l t Of E v a l u a t i o n Of Graphs 1 To 9 And 11 To 14 , 233 The Nonzero C o n t r i b u t i o n To The F l u x .............238 Zero Temperature L i m i t ........................... 239 E v a l u a t i o n Of The Loop Graph .....................241 X I . E e n o r m a l i z a t i o n Of The D i v e r g e n c e s To Second Order 242 Mass B e n o r m a l i z a t i o n • 244 I n s u f f i c i e n c y Of B e n o r m a l i z a t i o n In 2 D F o r M a s s l e s s 34* 247 S c a l i n g Of a To O b t a i n F i n i t e E e s u l t s .....250 ft • m v n i XII. D i s c u s s i o n .........253 Results 253 The I n f r a r e d Problem ............................. 257 Belated Work 260 BIELIOGRAPHY , .......263 References. Chapter 1. ...........................263 References. Chapter 2. ......264 References. Chapter 3. ........................... 266 References. Chapter 4. 267 References. Chapter 5. ........................... 268 References. Appendices ........................... 270 APPENDIX A. FIGURES ..271 APPENDIX B. DETAILS OF CALCULATIONS FOR CHAPTER 2. .....303 1. Angular Functions ....303 2. Transmission C o e f f i c i e n t s .304 APPENDIX C. NATURAL (PLANCK) UNITS .....309 Table 2. Conversion From N a t u r a l To Standard U n i t s ..........309 APPENDIX D. THE RATIO OF NEUTRINO BARYON TO NEUTRINO ANT IB ARYON CROSS SECTIONS AT HIGH ENERGIES. 310 APPENDIX E. THERMALIZATION OF BLACK HOLE EMISSION IN THE EARLY UNIVERSE 315 i x LIST OF TABLES Tabl e 1. Expansion With Black Hole Formation-^>y 1 / 2/H Table 2. Conversion From Natu r a l To Standard U n i t s 3 0 ^ X LIST OF FIGUBES A l l f i g u r e s are l o c a t e d i n Appendix A. Page 2 7 4 F i g l a . Diagrams of c o l l a p s e t o form a black hole: Opper- Penrose diagram, Lower- Eddington F i n k e l s t e i n diagram with i n g o i n g n u l l rays a t 45 degrees. 275 r i g . 1b. Kerr metric topology along symmetry a x i s . N u l l l i n e s (constant u,U or v,V) are a t plus or minus 45 degrees. 276 F i g . 2a. Fe r m i - D i r a c f a c t o r and t r a n s m i s s i o n p r o b a b i l i t y | B _ | z f o r r o t a t i o n parameter «*=. 8 and angular mode (l,m) = (3/2,3/2) vs. frequency Mw. 277 F i g . 2b. S i n g l e mode (l,m)= (3/2,3/2) n e u t r i n o p l u s a n t i n e u t r i n o number l o s s r a t e spectrum f o r *>(,=.8. 278 p i g . 3a-e. Neutrino p l u s a n t i n e u t r i n o number l o s s r a t e spectrum f o r the dominant angular modes for<X=.3, i e (l,m) = (1/2,1/2), (1/2,-1/2), (3/2,3/2), (3/2,1/2) and (5/2,5/2). 280 F i g . , 4a-f. Neutrino p l u s a n t i n e u t r i n o number l o s s r a t e spectrum f o r the dominant angular modes f o r f<=.999, i e (l,m) = (1/2,1/2), (3/2,3/2), (5/2,5/2), (7/2,7/2), (9/2,9/2) and (11/2,11/2). 283 F i g . 5a. Neutrino plus a n t i n e u t r i n o number l o s s r a t e spectrum (summed over angular modes) f o r r o t a t i n g b l a c k h o l e s with vi= 0, .3, .5, .7, .8, .9, .99 and .999. 284 F i g . 5b. Power spectrum (energy l o s s from a r o t a t i n g b l a c k hole v i a n e u t r i n o s and a n t i n e u t r i n o s v s . frequency Mw) f o r r o t a t i o n parameters oC= 0, .8, and .999. 285 F i g . 6a-b. The angular e i g e n f u n c t i o n S,(l,m,aw,6) vs. p o l a r angle & f o r 1=3/2, m=*3/2, +1/2 and aw=0, .4, 1. The second e i g e n f u n c t i o n i s given by S i(l,m,aw,&)= ( - 1 ) * — S , (1, m, aw ,-n-e) . 287 F i g . 6c-e. The angular e i g e n f u n c t i o n S,(l,m,aw,&) vs. p o l a r angle & f o r 1=5/2, m=i5/2, £3/2, ±.1/2 and aw values i n d i c a t e d . 290 F i g . 7. Net number l o s s r a t e ( n e u t r i n o s minus a n t i n e u t r i n o s ) vs. p o l a r angle © f o r r o t a t i o n parameters <X=. 1, .3, .8 and .999. x i P A S E 2 9 1 F i g . 8a. Net number l o s s r a t e ( i n t e g r a t e d over frequency flw) vs. p o l a r angle & f o r each of the dominant (l,m) modes f o r *=.1. 2 9 2 F i g . 8b. Net number l o s s r a t e ( i n t e g r a t e d over frequency Hw) vs. p o l a r angle & f o r each of the dominant (l,m) modes f o r c6=.5. 293 F i g . 8c. Net number l o s s r a t e ( i n t e g r a t e d over frequency Hv) vs. p o l a r angle &• f o r each of the dominant (l,m) modes f o r *=.999. 294 F i g . 9. Neutrino plus a n t i n e u t r i n o number l o s s r a t e ( i n t e g r a t e d over frequency Mw and summed over angular modes) vs. p o l a r angle 9 f o r r o t a t i o n parameters i n d i c a t e d . 295 F i g . 10. Emitted power i n n e u t r i n o s p l u s a n t i n e u t r i n o s vs. p o l a r angle 6 f o r r o t a t i o n parameters <x =0, .8, .99 and . 999. 2 9 6 F i g . 11. Angular momentum l o s s r a t e v i a n e u t r i n o and a n t i n e u t r i n o e m i s s i o n vs. p o l a r angle & f o r r o t a t i o n parameters t>i=.3, .8 and .999. 2 9 7 F i g . 12. Asymmetry i n n e u t r i n o emission ( i e the d i f f e r e n c e d i v i d e d by the sum, of n e u t r i n o and a n t i n e u t r i n o number l o s s rates) vs. p o l a r angle & f o r r o t a t i o n parameters <x=. 1, .3 and . 7. 298 F i g . 13. Schematic i l l u s t r a t i o n of the bl a c k hole e v a p o r a t i o n process f o r n e u t r i n o s , with the Fermi-Dirac f a c t o r l i s t e d at top. 299 F i g . 14. Large s c a l e baryon and antibaryon c u r r e n t s caused by n e u t r i n o s from r o t a t i n g black h o l e s . The long arrows r e p r e s e n t the n e u t r i n o , a n t i n e u t r i n o , baryon, and an t i b a r y o n c u r r e n t s , whereas the s h o r t arrows represent the r o t a t i o n axes of the evap o r a t i n g black h o l e s . 300 F i g . 15a. Time dependence o f 8, T, n, /0r, yo b K, and f o r formation of black h o l e s a t time t 0=M with number d e n s i t y p/W3. (Logarithmic s c a l e s ) . Case ^ <yV«/ n. 3 0 1 F i g . 15b. Time dependence of B, T, n, fir, , and ^ t f m f o r formation of black h o l e s a t time t^=M with number d e n s i t y n toK=^/fl3- (Logarithmic s c a l e s ) . Case yS>yl/2/M. 302 F i g . 16. Boundaries (u,v=*L) on which the s c a l a r f i e l d i s r e q u i r e d to be p e r i o d i c , i n order to r e g u l a t e the i n f r a r e d divergences i n the a 4 * p a r t i c l e number c a l c u l a t i o n . The i n t e r a c t i o n r e g i o n i s shaded. x i i i ACKNOWLEDGEMENTS I wish to thank my s u p e r v i s o r Dr. w. G. Unruh f o r a s s i s t a n c e throughout the course of t h i s work, p a r t i c u l a r l y on the m a t e r i a l o f chapter 5. I would a l s o l i k e to thank Dr. D. Page f o r p r o v i d i n g a t a b l e o f c o e f f i c i e n t s f o r c a l c u l a t i n g the e i g e n v a l u e s t o the angular e i g e n f u n c t i o n s and a l s o making a v a i l a b l e a l i s t of h i s d e t a i l e d r e s u l t s f o r e m i s s i o n r a t e s ( i n t e g r a t e d over s o l i d angle) of n e u t r i n o s from r o t a t i n g black h o l e s , which served as a check on my own c a l c u l a t i o n s . I thank Dr. A. V i l e n k i n f o r h e l p f u l d i s c u s s i o n s on the m a t e r i a l of chapt e r s 3 and 4, and Dr. D. Beder f o r h e l p f u l d i s c u s s i o n s on chapter 5. F i n a l l y , I acknowledge support from the N a t i o n a l Research C o u n c i l o f Canada i n the form of a Scie n c e S c h o l a r s h i p , d u r i n g the pe r i o d Sept.,1975 t o Aug. 1979. 1 1. IHTBODUCTIOH The emission by black holes of quanta of the neutrino f i e l d and of the scalar f i e l d i s investigated i n t h i s t h e s i s . The quantum mechanical process of p a r t i c l e emission by black holes i s known as black hole evaporation. When the p a r t i c l e s are noninteracting the emission i s thermal with c h a r a c t e r i s t i c temperature T. This implies: i ) the energy spectrum i s proportional to (exp (E/kT) • 1 ) - 1 , with • for emitted fermions, - f o r emitted bosons; i i ) separate emitted p a r t i c l e s are uncorrelated; i i i ) the black hole would be i n thermal equilibrium (obeys the p r i n c i p l e of detailed balance between emission and absorption) i f immersed i n a heat bath of the same temperature T as the black hole. Black hole evaporation r e s u l t s from the ap p l i c a t i o n of quantum f i e l d theory to p a r t i c l e f i e l d s (e.g., neutrinos, photons) propagating i n the background g r a v i t a t i o n a l f i e l d of a black hole. Quantum mechanics has not yet been applied successfully to gravity i t s e l f . Studying black hole evaporation i s one way of seeing the e f f e c t s of quantum mechanics on matter when gravity i s present , without having to deal with the numerous problems which plague quantum gravity. The e f f e c t s on neutrinos of the parity unsymmetric g r a v i t a t i o n a l f i e l d of a rotating black hole are considered 2 f i r s t . In c h a p t e r 2 we c a l c u l a t e the d e t a i l s of n e u t r i n o emission from the vacuum by r o t a t i n g black holes. I n c h a p t e r s 3 and 4, c o s m o l o g i c a l consequences of the n e u t r i n o e m i s s i o n are s t u d i e d . Chapter 5 i s an i n v e s t i g a t i o n o f the e f f e c t s of i n c l u d i n g i n t e r a c t i o n s on t h e e m i s s i o n from a b l a c k h o l e . N a t u r a l u n i t s are used throughout the t h e s i s , except where otherwise noted. In the n a t u r a l u n i t system, a l l q u a n t i t i e s are d i m e n s i o n l e s s . U n i t s of mass, l e n g t h , time, e t c . , are chosen such t h a t the fundmental c o n s t a n t s , i . e . G, £ , c, and k, a l l have the value u n i t y . Due t o the l a r g e value o f the speed of l i g h t , the s m a l l value of Planck's c o n s t a n t , e t c . , the n a t u r a l u n i t s of mass, l e n g t h , time, e t c . , are unwieldy f o r use i n the l a b o r a t o r y . The advantage o f n a t u r a l u n i t s l i e s i n the s i m p l i f i c a t i o n which r e s u l t s from r e p l a c i n g the v a r i o u s powers and combinations of the fundamental c o n s t a n t s by u n i t y i n t h e o r e t i c a l c a l c u l a t i o n s . Appendix C presents c o n v e r s i o n f a c t o r s between c o n v e n t i o n a l u n i t s and n a t u r a l u n i t s . The r e a r l e a f of Misner, Thorne, and Hheeler ( 1970) l i s t s a s i m i l a r s e t of c o n v e r s i o n f a c t o r s . Quantum F i e l d Theory In Curved Spacetime The term 'quantum f i e l d u s u a l l y reserved f o r s t u d i e s s c a l a r , s p i n o r , or vector) theory i n curved spacetime' i s i n which the matter ( i . e . f i e l d s are q u a n t i z e d but the 3 g r a v i t a t i o n a l f i e l d i s not. Parker (1976) and D e B i t t (1975) review quantum f i e l d theory i n curved spacetime. The matter f i e l d s r e a c t back on the g r a v i t a t i o n a l f i e l d through t h e i r c o n t r i b u t i o n to the energy momentum t e n s o r T*y appearing on the r i g h t hand s i d e of E i n s t e i n ' s e q u a t i o n s : < ^ = 8TT T^ V (1) The E i n s t e i n t e n s o r appears on the l e f t hand s i d e and r e p r e s e n t s the g r a v i t a t i o n a l f i e l d . . I t i s e q u a l t o the ( B i c c i ) c u r v a t u r e t e n s o r p l u s a term t o make i t c o n s e r v e d 1 . A problem of i n t e r p r e t a t i o n a r i s e s s i n c e E i n s t e i n ' s equations have the E i n s t e i n t e n s o r , a c l a s s i c a l f i e l d , on t h e l e f t and the energy-momentum t e n s o r T M V , an opera t o r - v a l u e d e x p r e s s i o n f o r the case that the matter f i e l d s are q u a n t i z e d , on the r i g h t . A d e f i n i t e although u n p h y s i c a l meaning can be g i v e n t o the E i n s t e i n equations by t a k i n g the e x p e c t a t i o n value of T M* i n a s u i t a b l e guantum s t a t e . T h i s l i m i t s one to s i t u a t i o n s i n which the quantum nature of the f i e l d i n some sense produces only s m a l l v a r i a t i o n s i n T^ v. Then c l a s s i c a l f i e l d s are present on both s i d e s o f the E i n s t e i n e quations and the 1 i . e . G/*v;v=0, where ; v means c o v a r i a n t d e r i v a t i v e with r e s p e c t t o x w. I t i s then c o n s i s t e n t with the conserved energy momentum t e n s o r . 4 problem reduces t o s o l v i n g a s e t of n o n - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . T h i s i s q u i t e a d i f f i c u l t problem i n i t s e l f z. A f u r t h e r problem now comes i n the s e l e c t i o n of the s t a t e of the quantum f i e l d . The quantum s t a t e can be e i t h e r pure or mixed. Examples of each a r e : a vacuum s t a t e and a thermal s t a t e , r e s p e c t i v e l y . The s t a t e depends, i n some yet unknown way, on the g r a v i t a t i o n a l f i e l d v i a the m e t r i c t e n s o r . The m e t r i c tensor i s not determined u n t i l a f t e r the s o l u t i o n . of E i n s t e i n ' s e q u a t i o n s , which i n t u r n depend d i r e c t l y on the s t a t e chosen t o f i x the e x p e c t a t i o n value of the energy momentum t e n s o r . Besides t h i s problem of t r y i n g t o f i n d a s e l f - c o n s i s t e n t s t a t e , the d i v e r g e n c e s i n the e x p e c t a t i o n value of T^„ must be removed, and i n a way t h a t r e s p e c t s the f a c t t h a t , i n g e n e r a l r e l a t i v i t y , an a d d i t i v e c o n s t a n t to the energy has r e a l p h y s i c a l s i g n i f i c a n c e 3 . The q u e s t i o n of what c o n s t i t u t e s a s u i t a b l e quantum s t a t e s t i l l l a c k s a s a t i s f a c t o r y answer. The problem of developing a theory i n which both the matter and g r a v i t y are quantized i s much more ambitious. I n t h i s , the metric f u n c t i o n s of the g r a v i t a t i o n a l f i e l d are 2 Only a s m a l l number of e x a c t s o l u t i o n s t o the E i n s t e i n eguations with c l a s s i c a l (unquantized) matter are known. These are a l l based on imposing some symmetries a t the o u t s e t , e.g. the Friedmann cosmologies which are s p a t i a l l y homogeneous. 3 T h i s f o l l o w s from E i n s t e i n s eqns (1) above. 5 usually treated as the f i e l d s to be quantized. The construction of a consistent theory has many d i f f i c u l t i e s , but promising approaches have been put f o r t h . An excellent summary of work up to 1975 i s 'Quantum Gravity 1975- an Oxford Symposium'. We return to consider quantum f i e l d theory i n curved spacetime. The problem of choosing a suitable quantum state i s usually avoided by assuming that the back reaction of the quantum matter f i e l d s on the g r a v i t a t i o n a l f i e l d i s ne g l i g i b l e . In t h i s case, the purely c l a s s i c a l expression for the energy-momentum tensor (e.g. that for a perfect fluid) i s used. This represents some background (non-quantized) matter, and allows one to solve the Ei n s t e i n g r a v i t a t i o n a l f i e l d equations. For the case of evaporating black holes, the assumption of no back reaction i s equivalent to the assumption that the curvature due to the quantum f i e l d s i s much l e s s than that already present. The theory predicts that the average energy of p a r t i c l e s emitted by quantum processes i s 1/(8irH) i n natural units, where M i s the black hole mass. This r e s u l t s i n the requirement that 1/(8irH) be much l e s s than H or that the black hole mass be much greater than the Planck mass; 10~ s gram. 6 Black Holes And Thermodynamics The concept of t h e black hole i s not new. L a p l a c e was the f i r s t t o c o n s i d e r strong g r a v i t a t i o n a l f i e l d s , using Newtonian g r a v i t y , as f o l l o w s : The escape v e l o c i t y from r a d i u s r away from t h e ce n t r e o f a s p h e r i c a l l y symmetric body (of mass M and r a d i u s E<r) to i n f i n i t y i s : v e, f c 2/2=GH/r B e g u i r i n g the body to tr a p l i g h t ( i . e . , y ^ c ) g i v e s the c r i t i c a l r a d i u s f o r a Newtonian black hole of: B c„ T=2GH/c 2 =2M i n n a t u r a l u n i t s T h i s value i s v a l i d a l s o f o r a black h o l e i n the t h e o r y of ge n e r a l r e l a t i v i t y . T h i s i s a very s m a l l r a d i u s : the e a r t h ' s mass of 6 x 1 0 2 7 g would have t o be compressed i n t o a sphere of r a d i u s 1 ce n t i m e t e r , and the sun's mass of 2 x 1 0 3 3 g would have t o be compressed i n t o a sphere of r a d i u s 3 k i l o m e t e r s , i n order t o form black h o l e s . Since l i g h t cannot escape from a black hole, nothing can. The s u r f a c e d e s c r i b e d by R=2GM/c2 s e p a r a t e s the region from which escape i s p o s s i b l e (r>B), from the r e g i o n from which escape i s not p o s s i b l e (r<B). T h i s s u r f a c e i s the event 7 h o r i z o n of the b l a c k h o l e . In the r e g i o n r<R, g r a v i t y dominates a l l other f o r c e s and r e s u l t s i n a l l matter being crushed t o a s i n g u l a r i t y ( i n present theory) a t t h e c e n t e r . A l l f u t u r e d i r e c t e d paths (of massive or a a s s l e s s p a r t i c l e s ) n e c e s s a r i l y h i t the s i n g u l a r i t y very s h o r t l y a f t e r they c r o s s the event h o r i z o n (within a m i l l i s e c o n d f o r a s o l a r mass black hole) . See i l l u s t r a t i o n below: An important development i n g r a v i t a t i o n theory i s the c l o s e connection between quantum f i e l d t h e o r y , event h o r i z o n s and thermodynamics*. S t u d i e s of the e n e r g e t i c s of c l a s s i c a l p rocesses f o r b l a c k h o l e s l e d to the i d e a that black holes were thermodynamic systems. More s p e c i f i c a l l y i t led to the f o r m u l a t i o n of •The Four Laws of Black Hole Mechanics* by Bardeen,Carter, and Hawking (1973): The Zeroth Law- the s u r f a c e g r a v i t y of a s t a t i o n a r y black hole i s constant over the event h o r i z o n . The F i r s t Law- any two neighboring s t a t i o n a r y axisymmetric black h o l e s t a t e s are r e l a t e d by: • For a recent review see Davies,,1978 8 d H= (K/8TT) d A* ndJ (2) H and J are the mass and angular momentum of the black h o l e , K and A are the surface g r a v i t y and area of the event h o r i z o n , and SL i s the angular v e l o c i t y of dragging of i n e r t i a l frames a t the h o r i z o n . The Second Law- the area A o f the event h o r i z o n of each b l a c k hole i s a nondecreasing f u n c t i o n of time. Also i f two black holes c o a l e s c e the area of the r e s u l t i n g event h o r i z o n i s g r e a t e r than the sum of the i n i t i a l two areas. The T h i r d Law- i t i s i m p o s s i b l e t o reduce K t o z e r o by any f i n i t e seguence of o p e r a t i o n s . These laws are analogues t o the corresponding laws of thermodynamics. Bekenstein (1973,1974,1975) suggested some m u l t i p l e of the area o f the event h o r i z o n of a b l a c k h o l e r e p r e s e n t e d entropy and some m u l t i p l e of the su r f a c e g r a v i t y represented temperature. These i d e a s were i n c o n f l i c t with the c l a s s i c a l concept t h a t black holes were p e r f e c t a bsorbers o f r a d i a t i o n but not e m i t t e r s and thus e q u i v a l e n t t o a black body at 0 degrees K e l v i n . Hawking's (1974,1975) d i s c o v e r y of the quantum mechanical process of p a r t i c l e emission from black holes provided the mis s i n g l i n k . A black hole emits p a r t i c l e s t h e r m a l l y and thus can be i n e q u i l i b r i u m with a heat bath. The temperature i s low: T=K/2W=1/9TTM= 1-2X10* *K/(M/1gm) (3) f o r a n o n r o t a t i n g , uncharged b l a c k hole of mass H. E. q.f a s o l a r mass black h o l e has T=6x10~ e K e l v i n . The p a r t i c l e e m i ssion process w i l l next be d e s c r i b e d i n a q u a l i t a t i v e way, then i n more q u a n t i t a t i v e f a s h i o n . P a r t i c l e C r e a t i o n By G r a v i t a t i o n a l F i e l d s - H e u r i s t i c D i s c u s s i o n The f i r s t q u e s t i o n t o ask i s : why would one expect a g r a v i t a t i o n a l f i e l d to c r e a t e p a r t i c l e s from the vacuum? P a r t i c l e c r e a t i o n i n another context comes t o mind- the c r e a t i o n of e l e c t r o n p o s i t r o n p a i r s by e l e c t r o m a g n e t i c f i e l d s . T h i s can occur by c o n v e r s i o n of a photon t o an e + - e _ p a i r i n the Coulomb f i e l d of a nucleus, by n u c l e a r de-e x c i t a t i o n with r e s u l t a n t e * - e - emission, or even by an i n t e n s e e l e c t r i c f i e l d between the p l a t e s of a c a p a c i t o r . The l a t t e r case has the c l o s e s t analogy with p a r t i c l e p r o d u c t i o n from the vacuum by the g r a v i t a t i o n a l f i e l d . For t h i s l a t t e r case, c o n s e r v a t i o n of energy r e q u i r e s the c r e a t e d p a i r t o be separated by a d i s t a n c e d, such t h a t eEd=2mc 2, where m i s the e l e c t r o n mass, e the e l e c t r o n charge and E the e l e c t r i c f i e l d . C r e a t i o n of a p a i r occurs at a p o i n t and thus, c l a s s i c a l l y , e + - e _ c r e a t i o n by the e l e c t r i c f i e l d cannot 10 occur. However i n quantum mechanics, a v i r t u a l p a i r can e x i s t f o r a time, i n accord with the Heisenberg u n c e r t a i n t y p r i n c i p l e , o f At=Jl/(2mc 2) , without c o n s e r v i n g energy. Vacuum f l u c t u a t i o n s i s a term o f t e n g i v e n t o v i r t u a l p a r t i c l e s i n the vacuum ( v i r t u a l p a r t i c l e s occur i n many other processes which i n v o l v e sources) . T h i s v i r t u a l p a i r has a c e r t a i n p r o b a b i l i t y of t u n n e l l i n g through the r e g i o n where 2mc 2-eEd>0, which i s a p o t e n t i a l b a r r i e r , t o become a r e a l p a i r . The p r o b a b i l i t y , and thus the c r e a t i o n r a t e per u n i t volume between the c a p a c i t o r p l a t e s , i s p r o p o r t i o n a l to the t r a n s m i s s i o n f a c t o r f o r the p o t e n t i a l b a r r i e r 5 : exp ( —irm 2c VeEif) The black hole e v a p o r a t i o n process can a l s o be viewed as r e s u l t i n g from the presence of vacuum f l u c t u a t i o n s . T h i s d e s c r i p t i o n i s s p e c u l a t i v e s i n c e no a c t u a l c a l c u l a t i o n l i n k s the p i c t u r e presented here with the p h y s i c s of the process. The vacuum f l u c t u a t i o n s o c c u r as f o l l o w s . Throughout spacetime, p a i r s of v i r t u a l p a r t i c l e s with opposite quantum numbers, i . e . p a r t i c l e a n t i p a r t i c l e p a i r s , are c r e a t e d and subsequently destroyed (see i l l u s t r a t i o n below). s See e.g. Frolov,1976, who a l s o d i s c u s s e s the analogy of e l e c t r o m a g n e t i c and g r a v i t a t i o n a l p a r t i c l e p r o d u c t i o n V i r t u a l P a i r 11 Cresrf: ion. — > c.f a c e The a n t i p a r t i c l e i s l a b e l l e d {in accord with standard convention) as a p a r t i c l e which t r a v e l s backwards i n time. In accord with the Heisenberg u n c e r t a i n t y r e l a t i o n (4EAt>21/2) , these p a i r s e x i s t f o r only a very s h o r t time. The vacuum f l u c t u a t i o n s r e s u l t i n measurable e f f e c t s , e.g. the C a s i m i r e f f e c t 6 . E i t h e r r a p i d time v a r i a b i l i t y or strong s p a t i a l g r a d i e n t s of the g r a v i t a t i o n a l f i e l d can supply the energy to make a v i r t u a l p a i r i n t o a r e a l created p a i r , i n s i m i l a r way t o the p a i r c r e a t i o n process f o r the e l e c t r o m a g n e t i c f i e l d . The expansion of the universe a t very e a r l y times p r o v i d e s an environment with r a p i d time v a r i a b i l i t y where p a r t i c l e c r e a t i o n i s important., Parker (1976) reviews p a r t i c l e c r e a t i o n by g r a v i t a t i o n a l f i e l d s with a p a r t i c u l a r emphasis on c o s m o l o g i c a l p a r t i c l e c r e a t i o n . The p a r t i c l e c r e a t i o n has a f u r t h e r i m p l i c a t i o n when event h o r i z o n s a re present, as f o r a black h o l e . T h i s i s the *The C a s i m i r e f f e c t i s the presence of a weak a t t r a c t i v e f o r c e between two p a r a l l e l f l a t e l e c t r i c a l l y n e u t r a l conducting p l a t e s i n a vacuum environment. The f o r c e i s caused by m o d i f i c a t i o n by the conducting p l a t e s of the vacuum f l u c t u a t i o n s of the e l e c t r o m a g n e t i c f i e l d . The C a s i m i r e f f e c t has been e x p e r i m e n t a l l y v e r i f i e d . Ford (1974) DeWitt(1975) and Parker(1976), f o r example, d e s c r i b e the u n d e r l y i n g p h y s i c s o f the C a s i m i r e f f e c t . 12 l o s s of i n f o r m a t i o n a c r o s s the event h o r i z o n , which means t h a t the quantum mechanical s i t u a t i o n must be d e s c r i b e d by a d e n s i t y matrix r a t h e r than a pure s t a t e (e.g. Hawking, 1976a, 1976b). although l o c a l l y an event h o r i z o n has no s p e c i a l s i g n i f i c a n c e , no p a r t i c l e t h a t has c r o s s e d the event h o r i z o n can escape to the o u t s i d e a g a i n . When a p a r t i c l e c r e a t i o n occurs j u s t o u t s i d e the h o r i z o n , one p a r t i c l e of the p a i r can escape while the a n t i p a r t i c l e i s trapped and e n t e r s the black h o l e . That the observed p a r t i c l e s are d e s c r i b e d by a d e n s i t y matrix, i s one reason why the emission i s thermal i n nature. From the p o i n t of view of an observer f a r from the black ' h o l e , the e m i s s i o n can be d e s c r i b e d as f o l l o w s . P a r t i c l e s are emitted with a thermal spectrum. A l l p a r t i c l e s p e c i e s are emitted. The emission of s p e c i e s with r e s t mass g r e a t e r than t h e temperature i s suppressed due to the thermal exp(-E/kT) f a c t o r , where B i s the p a r t i c l e energy i n c l u d i n g r e s t mass. Due to the l o s s of energy, the black hole decreases i n mass. The i l l u s t r a t i o n below summarizes t h i s ( q u a l i t a t i v e ) p i c t u r e f o r p a r t i c l e e mission from a black hole. 13 r = 0 O v e r a l l F e a t u r e s Of Black Hole Evaporation Be now very crudely c o n s i d e r the e n e r g e t i c s of the c r e a t i o n process. Viewed l o c a l l y , both c r e a t e d p a r t i c l e s have p o s i t i v e energy. The energy f o r the c r e a t i o n of the p a r t i c l e p a i r comes from the g r a v i t a t i o n a l f i e l d . The v i r t u a l p a i r can become r e a l i f enough energy can be s u p p l i e d l o c a l l y during the u n c e r t a i n t y p r i n c i p l e l i f e t i m e of 1/w of the p a i r , where w i s the ( l o c a l ) energy of each member of the p a i r . Here we use n a t u r a l u n i t s , i n which £=G=c=k=1, see appendix C. The t i d a l f o r c e F a t d i s t a n c e r from the black h o l e , between two p a r t i c l e s of energy w and a d i s t a n c e 1 a p a r t , i s approximately: F=flwl/r 3 (4) 14 where H i s t h e black hole mass. The energy s u p p l i e d by the g r a v i t a t i o n a l f o r c e to a v i r t u a l p a i r , which can t r a v e l a d i s t a n c e 1/w i n the time 1/w, i s roughly: i/w F d l = M/wr* (5) o For t h i s t o be l a r g e r than t h e energy r e q u i r e d t o make the p a i r r e a l , one r e q u i r e s w 2<H/r 3. Since the minimum v a l u e o f r a p p l i c a b l e i s near the event h o r i z o n a t r=2H, s i g n i f i c a n t p a r t i c l e c r e a t i o n w i l l occur o n l y f o r p a r t i c l e e n e r g i e s s a t i s f y i n g w<1/H. , The p a r t i c l e emission by t h e black hole, which r e s u l t s from quantum processes, can be shown to be thermal i n c h a r a c t e r 7 . The immediately p r e c e d i n g a n a l y s i s gave a c u t o f f i n frequency: w< 1/H, r a t h e r than the smooth e x p o n e n t i a l dependence of a thermal spectrum. T h i s i s due t o the o v e r s i m p l i c a t i o n s - the f u l l quantum mechanical c a l c u l a t i o n r e s u l t s i n the thermal spectrum, c r u d e l y analogous t o the e x p o n e n t i a l dependence f o r the b a r r i e r p e n e t r a t i o n f a c t o r o c c u r i n g f o r the e+-e~ p a i r c r e a t i o n by an e l e c t r i c f i e l d . F or the black h o l e e m i s s i o n , s e p a r a t e outgoing p a r t i c l e s are u n c o r r e l a t e d with each other. The i n t e n s i t y o f p a i r c r e a t i o n i s g r e a t e s t near the h o r i z o n , where the g r a v i t a t i o n a l f i e l d 7 T h i s i s done i n chapter 2. 4 15 g r a d i e n t (the curvature) i s l a r g e s t , and i s c o n c e n t r a t e d near the h o r i z o n a l s o because the p r o b a b i l i t y of one member of a c r e a t e d p a i r e n t e r i n g the black hole i s g r e a t e s t t h e r e . Taken together with the g r a v i t a t i o n a l r e d s h i f t e f f e c t 8 , the p r o b a b i l i t y f o r p a i r c r e a t i o n r e s u l t s i n the d i s t a n t observer s e e i n g a thermal f l u x of p a r t i c l e s from the black h o l e . The f l u x i s d i l u t e d by a 1/r 2 g e o m e t r i c a l f a c t o r f o r a b l a c k hole i n a spacetime with 3 s p a t i a l d i m e n s i o n s 9 . The c h a r a c t e r i s t i c temperature o f the e m i s s i o n i s 1/8-irM, as mentioned above, and i n c r e a s e s f o r s m a l l e r mass black holes due t o the g r e a t e r c u r v a t u r e ( i . e . s u r f a c e g r a v i t y ) at t h e i r event h o r i z o n s . The l u m i n o s i t y of the b l a c k h o l e can be e s t i m a t e d , as w e l l as the l i f e t i m e to when the black hole has completely evaporated by e m i t t i n g p a r t i c l e s . Here we assume t h a t nothing prevents the black hole from ev a p o r a t i n g c o m p l e t e l y . For thermal emission a t temperature T=1/8trM from very near the h o r i z o n , with area A=4-nr2 (r=2M), the emitted power L i s r o u g h l y : L=fAT»=10-»f/M2 = fL o(2x10«gm/H) 2 (6) 8 For a c l a s s i c a l l i g h t source f a l l i n g i n t o a black hole, the r e d s h i f t (decrease i n freguency of a l i g h t wave due to escaping the g r a v i t y p o t e n t i a l well) i n c r e a s e s e x p o n e n t i a l l y as the source approaches t h e event h o r i z o n . 9 T h i s f a c t o r i s absent i n the case of one s p a t i a l and one time dimension, which i s used i n chapter 5. 16 where L Q i s the s o l a r l u m i n o s i t y . The f a c t o r f i s roughly p r o p o r t i o n a l t o the number of p a r t i c l e s p e c i e s e m i t t e d , and i s i n c l u d e d s i n c e each s p e c i e s w i l l c o n t r i b u t e a b l a c k body l u m i n o s i t y . f a l s o i n c l u d e s t h e e f f e c t t h a t the b l a c k hole does not behave as a p e r f e c t black body r a d i a t o r . T h i s happens s i n c e the surrounding g r a v i t a t i o n a l f i e l d w i l l s c a t t e r some of the emitted p a r t i c l e s back i n t o the black h o l e , r e d u c i n g i t s e m i s s i v i t y . The l u m i n o s i t y L i n c r e a s e s a s the b l a c k hole mass decreases. S e t t i n g dM/dt egua l t o L and s o l v i n g f o r the time t h a t M reaches z e r o y i e l d s the black hole l i f e t i m e *C: t=3x103HVf =10» 7sec (H/2x 10» *gm) V f (7) One sees the s t r o n g dependence of l i f e t i m e on black hole mass. 10* 7 sec i s roughly t h e age o f the u n i v e r s e , so t h a t any black h o l e s formed i n the e a r l y universe with masses l e s s than 2x10 l * f 1 / 3 gm s h o u l d have evaporated by now. The magnitude of the quantum eva p o r a t i o n e f f e c t s i s s t r o n g l y dependent on the b l a c k hole mass, being g r e a t e r f o r s m a l l e r H. T h i s can be seen by examining the p r e v i o u s r e l a t i o n s f o r Z, L and T. 17 P a r t i c l e C r e a t i o n In Curved Spacetime- & Short F i e l d T h e o r e t i c D e r i v a t i o n above we d i s c u s s e d the p a r t i c l e p r o d u c t i o n by a black h o l e from a q u a l i t a t i v e p o i n t of view. Here a more q u a n t i t a t i v e d e s c r i p t i o n i s given f o r the p r o d u c t i o n o f elementary p a r t i c l e s by g r a v i t a t i o n a l f i e l d s 1 " , we r e s t r i c t the d i s c u s s i o n to spacetimes (or spacetime regions) which can be f o l i a t e d i n t o a s e t of s p a c e l i k e s u r f a c e s . T h i s i s not very r e s t r i c t i v e , s i n c e only p a t h o l o g i c a l spacetimes, such a s the Godel universe with i t s c l o s e d t i m e l i k e l i n e s (Hawking and E l l i s , 1 9 7 3 ) , v i o l a t e t h i s c o n d i t i o n . Very b r i e f l y , the p a r t i c l e c r e a t i o n occurs f o r the f o l l o w i n g r easons: The d e f i n i t i o n of a p a r t i c l e depends on the proper time of an observer. The quantum f i e l d o f the p a r t i c l e propagates v i a the f i e l d equations, which depend on the spacetime but not on the w o r l d l i n e of any o b s e r v e r . Thus d i f f e r e n t o b s e r v e r s , or the same observer a t d i f f e r e n t times, w i l l have d i f f e r e n t p a r t i c l e i n t e r p r e t a t i o n s of the same quantum f i e l d . The c r e a t i o n p rocess i s d e r i v e d here f o r s c a l a r p a r t i c l e s . The s c a l a r f i e l d e quation possesses s e t s o f mode s o l u t i o n s t h a t form complete s e t s f o r expanding any s o l u t i o n . 1 0 For a more d e t a i l e d d e s c r i p t i o n see, f o r example, Parker, 1976. 18 An analogy i s the use of plane waves: f*(x,t)=exp(i(k.x-wt)) /J"27w' with w2=k*+m2 (8) f o r expanding any s o l u t i o n o f t h e s c a l a r f i e l d e g u ation (such as a l o c a l i z e d wave packet) i n f l a t space. w and k are the energy and wave v e c t o r of the mode and m i s the mass o f the s c a l a r p a r t i c l e s . He w r i t e the i n n e r product of two s o l u t i o n s F,G as an i n t e g r a l over a s p a c e l i k e s u r f a c e (x° i s a t i m e l i k e c oordinate) : (F,G)= (-i/2) I dfxF^ g«^[ ( ^ F*) G-F* (V*) 3 (9) where g^ v i s the spacetime m e t r i c and g i s i t s determinant. Next, the s e t of s o l u t i o n s i s separated i n t o p o s i t i v e and n e g a t i v e frequency modes. In curved space the s o l u t i o n s are more complicated i n form than eqn (8) , and we w i l l w r i t e the p o s i t i v e frequency modes as fj , the negative freguency modes as f^ *. Here we use a d i s c r e t e mode index, j , f o r convenience. Then o r t h o n o r m a l i t y of the modes i s expressed by: The s e p a r a t i o n i n t o p o s i t i v e and negative freguency modes i s 19 done s i n c e n e g a t i v e frequency sodes have negative n o r m s 1 1 . T h i s presents a problem r e g a r d i n g the p r o b a b i l i t y i n t e r p r e t a t i o n of the wave f u n c t i o n , which i s standard t o quantum mechanics. The problem i s r e s o l v e d by t a k i n g o n l y the p o s i t i v e frequency modes to represent p a r t i c l e s t a t e s . How does one decide the d i f f e r e n c e between p o s i t i v e and negative freguency? One p o s s i b i l i t y i s to f o l l o w Unruh(1976) (now f a i r l y s t a n d a r d ) . One d e f i n e s p o s i t i v e frequency as meaning a time dependence of the form e x p ( - i w t ) , w>0, where t i s the proper time along some observer's w o r l d l i n e . T h i s d e f i n i t i o n corresponds t o p a r t i c l e s which a d e t e c t o r , moving along the w o r l d l i n e with the observer, would see (Onruh,1976). One i s thus f a c e d with having a d e f i n i t i o n of p a r t i c l e s which i s s t r o n g l y observer dependent, an a l t e r n a t e d e f i n i t i o n i s i n terms of e v o l u t i o n of the modes o f f a s p a c e l i k e s u r f a c e : &f/dxo=-iwf w>0 (11) T h i s has the disadvantages t h a t i t depends on the s u r f a c e s chosen (independent o f the observer) and does not seem to correspond to measurable p a r t i c l e s i n any way. The f i e l d o p e r a t o r <J> i s i n t r o d u c e d next. I t i s expanded 1 1 See egn (10). T h i s can be v e r i f i e d with the modes of eqn (8) i n the i n n e r product, eqn (9 ) . 20 i n t e r n s of the mode s o l u t i o n s : The a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s , a and a + , destroy and c r e a t e p a r t i c l e s with wavefunctions f^ . The vacuum s t a t e |0> i s def i n e d as the s t a t e with no f type p a r t i c l e s : a^|0>=0 f o r a l l j (13) The s e p a r a t i o n of modes i n t o p o s i t i v e and n e g a t i v e freguency p a r t s determines what the vacuum s t a t e i s . Since the wavefunctions evolve from s p a c e l i k e s u r f a c e t o s p a c e l i k e s u r f a c e , the d e f i n i t i o n of p o s i t i v e frequency t h a t a p p l i e s i s th a t of eqn (11)» In f l a t space the s e p a r a t i o n i s unambiguous: w>0 i s p o s i t i v e frequency and w<0 i s negative frequency f o r the modes of eqn ( 8 ) . The d e f i n i t i o n (11) and the exp(-iwt) d e f i n i t i o n o f p o s i t i v e frequency are e q u i v a l e n t . In the l a t t e r case, Lorentz boosts of the observer from proper time t t o proper time t * do not cause exp (-iwt) , w>0, to have exp(-iw*t*),w*<0, components, so a l l i n e r t i a l o b s e r vers i n f l a t space agree on the s e p a r a t i o n . In curved space, the d e f i n i t i o n of p o s i t i v e frequency i s not i n v a r i a n t . The s p l i t t i n g of modes i n t o p o s i t i v e and n e g a t i v e freguency a c c o r d i n g to (11) a t one time w i l l not, i n 21 g e n e r a l , agree with the s p l i t t i n g at a l a t e r time. T h i s i s because the c u r v a t u r e can cause a mode f- , of p o s i t i v e freguency as d e f i n e d a t c o o r d i n a t e time x°, t o be e q u a l to a l i n e a r combination o f modes f' and f^*» where f^ and f^** are p o s i t i v e and ne g a t i v e freguency as d e f i n e d at time x ° ' . A simple example of the e f f e c t s of mixing of p o s i t i v e and negative frequency modes i s given here. Say f j , d e f i n e d above, are p o s i t i v e freguency modes as d e f i n e d by (11) at the s p a c e l i k e s u r f a c e d e s c r i b e d by x°=constant. A l s o , take f ^ ' as p o s i t i v e freguency modes as d e f i n e d by (11) at the s u r f a c e x 0 , = c o n s t a n t . Since the f- and f. * together form a complete s e t f o r expanding any s o l u t i o n to the f i e l d e q u a t i o n , we expand any given primed mode as: The l a t t e r c o n d i t i o n ensures t h a t the modes f ' are k orthonormal i f the f- are (see egn (10)). The v«s are a measure of the amount of n e g a t i v e frequency f * modes i n a p o s i t i v e frequency f• mode. The f i e l d o perator (see eqn (12)) can a l s o be expanded i n terms of the f • , f ^ 1 * modes and the a k f , a^'* o p e r a t o r s : <P= £(Vak«*V*afc'+) (15) k 22 The r e l a t i o n between primed and unpritned o p e r a t o r s i s given by: He wish t o determine the number of x°• type p a r t i c l e s g iven that there are no x° type p a r t i c l e s present. The s t a t e v e c t o r of the system i s then the vacuum |0> f o r s c a l a r p a r t i c l e s of type f j . The number ope r a t o r f o r p a r t i c l e s of type f • i s given by: The t o t a l number of p a r t i c l e s of type f i n the s t a t e j0> i s the sum over a l l modes of the vacuum e x p e c t a t i o n value of (16) ( 1 7 ) N«= X<0ja, ,' t"a L • J0> (18) k = ^ |V w. J 2 Thus the f (or x°) vacuum c o n t a i n s f» (or x»') type p a r t i c l e s . To get a p h y s i c a l i n t e r p r e t a t i o n of what i s o c c u r i n g , we 23 switch t o the observer dependent exp(-iwt) d e f i n i t i o n of p a r t i c l e s . A p a r t i c l e d e t e c t o r responds to p a r t i c l e s t a t e s of the form exp(-iwt) where t i s the proper time along the d e t e c t o r ' s path ( t h i s i s d e s c r i b e d i n d e t a i l i n Unruh,1976). He can f o l i a t e the spacetime i n t o s p a c e l i k e s u r f a c e s such t h a t they are l a b e l l e d by t , the proper time of the observer (or d e t e c t o r ) . Only f u n c t i o n s which have exp(-iwt) components f o r a l l t , are t r u l y p o s i t i v e frequency. In g e n e r a l these f u n c t i o n s are not s o l u t i o n s to the f i e l d equations f o r the p a r t i c l e s . The modes f j , which are s o l u t i o n s , appear to be p o s i t i v e frequency near time t=x°, but t u r n out t o have n e g a t i v e frequency components near time t=x 0', where the f^' modes appear t o be p o s i t i v e frequency. The c u r v a t u r e o f spacetime has i n some sense c r e a t e d p a r t i c l e s from the vacuum. I t must be emphasized t h a t the exp(-iwt) d e f i n i t i o n o f p o s i t i v e freguency depends on the w o r l d l i n e o f the observer, s i n c e t i s the proper time f o r the observer. One would l i k e t o d e f i n e p a r t i c l e s everywhere. The s e t of mode s o l u t i o n s extends over the whole spacetime. T h i s pr o v i d e s one method of extending the d e f i n i t i o n of p o s i t i v e frequency over a l l of spacetime, i . e . , by keeping the same se p a r a t i o n of p o s i t i v e and negative freguency modes everywhere as def i n e d on the s u r f a c e l a b e l l e d x °. However, i n g e n e r a l , the corresponding p a r t i c l e s would only be p h y s i c a l p a r t i c l e s f o r o b s e r v e r s with proper time the same as the c o o r d i n a t e time x° near the s u r f a c e , and only near the time that the observers pass through the s u r f a c e . Another approach i s t o choose a spacetime f i l l i n g s e t of w o r l d l i n e s ( i . e . , set o f observers) to guarantee that the d e f i n i t i o n of p o s i t i v e freguency corresponds to p h y s i c a l p a r t i c l e s at any given point, i n spacetime i n a known way. The s e p a r a t i o n of any set o f modes i n t o p o s i t i v e and n e g a t i v e freguency s e t s w i l l then, i n g e n e r a l , be d i f f e r e n t at each p o i n t i n spacetime. A problem with t h i s l a t t e r approach i s t h a t there i s no way i n g e n e r a l to d e f i n e a guantum s t a t e (e.g. the vacuum) c o n s i s t e n t l y over a complete s p a c e l i k e s u r f a c e . Always when d i s c u s s i n g p a r t i c l e c r e a t i o n i n curved spacetime, one must consider who's point of view i s most important to the problem a t hand. For example, i n f l a t space, normally observers at r e s t are more u s e f u l to c o n s i d e r than a s e t of h i g h l y a c c e l e r a t e d o b s e r v e r s . The above d e s c r i p t i o n o f p a r t i c l e c r e a t i o n a p p l i e s to the guantum e v a p o r a t i o n process f o r black h o l e s and a l s o t o c o s m o l o g i c a l p a r t i c l e p r o d u c t i o n (not d i s c u s s e d f u r t h e r i n t h i s t h e s i s ) . In both cases, s e t s of p r e f e r r e d observers e x i s t f o r which one can base d e f i n i t i o n s of p o s i t i v e freguency. For example, i n the Friedmann cosmologies, o b s e r v e r s at r e s t with respect to the matter are chosen. Since the Friedmann u n i v e r s e s are homogeneous i n space, these 25 o b s e r v e r s a l l have the same d e f i n i t i o n of p o s i t i v e frequency. Black Holes Formed By S t e l l a r C o l l a p s e A n d T h e P a r t i c l e C r e a t i o n Process Here we d i s c u s s the a p p l i c a t i o n of the above p a r t i c l e c r e a t i o n c a l c u l a t i o n s t o a black hole formed by s t e l l a r c o l l a p s e 1 2 . The abrupt formation of the event h o r i z o n causes the quantum f i e l d to change. T h i s change i s such t h a t an observer to the f u t u r e o f the c o l l a p s e sees a thermal spectrum of p a r t i c l e s i n the guantum s t a t e which i s the vacuum s t a t e f o r an observer to the past of the c o l l a p s e . F i g . l a (in Appendix A) shows the c o l l a p s e of a s p h e r i c a l body (star) t o form a black hole, using two d i f f e r e n t c o o r d i n a t e systems. The lower diagram, i n the h o r i z o n t a l plane, shows two s p a t i a l dimensions (r and <p of s p h e r i c a l p o lar c o o r d i n a t e s ) o f the three ( r , © , and tf) - The Schwarzschild c o o r d i n a t e t i s a t i m e l i k e c o o r d i n a t e o u t s i d e the event h o r i z o n . However, i n s i d e the event h o r i z o n , changes i n t , at c o n s t a n t r and <f>, are s p a c e l i k e i n t e r v a l s . The c o o r d i n a t e t ' has been chosen so t h a t i n g o i n g s p h e r i c a l l i g h t wave f r o n t s a r e cones with 45 degree apex angles. t ' i s r e l a t e d to Schwarzschild time t 1 2 Hawking (1975), f o r example, g i v e s a more complete d i s c u s s i o n . 26 b y : t»=t+2li l n ( r / 2 M - 1 ) , o u t s i d e the c o l l a p s i n g body. t * and r a r e the Eddington- F i n k e l s t e i n c o o r d i n a t e s f o e the Schwarzchild m e t r i c 1 3 . The s t a r c o l l a p s e s i n roughly a f r e e f a l l t i m e 1 * , ending i n a s i n g u l a r i t y 1 5 . The event h o r i z o n forms j u s t b e f o r e the s i n g u l a r i t y , and i s the boundary from w i t h i n which nothing can escape to i n f i n i t y . Incoming l i g h t r a y s (to the f u t u r e of the one l a b e l l e d 2) c r o s s the h o r i z o n and are t r a p p e d . The l i g h t ray, l a b e l l e d 2, i s delayed a p p r e c i a b l y i n i t s outward progress from the black h o l e . As seen from f a r away, a t r a i n o f l i g h t rays coming i n at a f i x e d frequency ( i . e . , e q u a l l y spaced) would undergo a d r a s t i c r e d u c t i o n i n freguency, i . e . a d r a s t i c r e d s h i f t , at the time of c o l l a p s e of the black hol e . The upper diagram i n f i g u r e 1a i s a Penrose diagram o f the same c o l l a p s e . A Penrose diagram i s c o n s t r u c t e d so that a l l n u l l rays ( l i g h t paths) are represented by l i n e s a t * or - 45 degrees. I n f i n i t y i s transformed t o a f i n i t e c o o r d i n a t e d i s t a n c e from the o r i g i n . By p r e s e r v i n g the l i g h t l i k e s t r u c t u r e of a spacetime, Penrose diagrams allow an easy 1 3 For a more complete d e s c r i p t i o n , see Hawking and E l l i s , 1973. The Schwarzschild m e t r i c d e s c r i b e s the g r a v i t a t i o n a l f i e l d o u t s i d e any i s o l a t e d , s p h e r i c a l l y symmetric body. 1 4 For i n i t i a l r a d i u s R and mass H, the f r e e f a l l time i s (B 3/2M) 1 5 T h i s i g n o r e s quantum g r a v i t y e f f e c t s which are important f o r s i z e s of order the Planck l e n g t h , 1.6x10 - 3 3 cm. 27 v i s u a l i z a t i o n of the behaviour of l i g h t r a y s and a l s o of the l i m i t s on t i m e l i k e c u rves ( p o s s i b l e w o r l d l i n e s of observers) , whose tangent v e c t o r s must l i e w i t h i n the f u t u r e d i r e c t e d n u l l cone. Hawking and E l l i s (1976) d i s c u s s Penrose diagrams more f u l l y and make e x t e n s i v e use of them. Along past n u l l i n f i n i t y , t eguals minus i n f i n i t y and r equals i n f i n i t y , and along f u t u r e n u l l i n f i n i t y , ^ , t eguals p l u s i n f i n i t y and r eguals i n f i n i t y . Far away from the black hole t and r are the normal f l a t space c o o r d i n a t e s . Only one s p a t i a l dimension i s d i s p l a y e d here (& and ^ are suppressed). u and v are n u l l c o o r d i n a t e s , along which l i g h t r a y s t r a v e l . Because of the e x p o n e n t i a l r e d s h i f t t h a t occurs near the event h o r i z o n , waves which are p o s i t i v e frequency as d e f i n e d by an observer near propagate to waves which are not pure p o s i t i v e frequency as d e f i n e d by an observer to the f u t u r e of t h e c o l l a p s e (near $•) • T h i s i s p r e c i s e l y the requirement f o r p a r t i c l e c r e a t i o n . The vacuum s t a t e s f o r observers hear § _ and are d i f f e r e n t , s i n c e a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s f o r p a r t i c l e s near and are d i f f e r e n t . They a r e r e l a t e d by egn (16). Egn (14) expresses the r e l a t i o n between f» modes, which we take as p o s i t i v e freguency near and f modes, which we take as p o s i t i v e frequency near Qr. The v c o e f f i c i e n t s can be c a l c u l a t e d , so t h a t the p a r t i c l e p r o d u c t i o n i s given by egn (18). The v»s are such t h a t the black hole emits a thermal spectrum of p a r t i c l e s when the 28 e n t i r e system i s i n the vacuum s t a t e f o r an observer near For c a l c u l a t i o n s of quantum mechanical processes around a black hole, i t i s p o s s i b l e to dispense with the spacetime i n v o l v i n g c o l l a p s e to form the black hole. T h i s i s done i n order t o a v o i d the messy problems a s s o c i a t e d with the d e t a i l s o f the c o l l a p s e . For quantum processes a f f e c t i n g an observer f a r to the f u t u r e of the c o l l a p s e , the messy d e t a i l s are no lon g e r important. Instead of the c o l l a p s e spacetime, one uses the s t a t i o n a r y black hole spacetime of f i g u r e 1 b 1 6 . T h i s diagram i n c l u d e s the a n a l y t i c a l l y extended r e g i o n of the black hole ( r e g i o n I I ) , but not the s i n g u l a r i t y a t r=0. Since waves from can no lo n g e r propagate through r=0 (see f i g . 1a), waves which reach f u t u r e n u l l i n f i n i t y (§*)# must a l l o r i g i n a t e from the past h o r i z o n . I t t u r n s out t h a t d e f i n i n g p o s i t i v e frequency, near the past h o r i z o n , i n terms of the proper time c o o r d i n a t e of an observer f r e e f a l l i n g i n t o the black hole ( i . e . the c o o r d i n a t e V of f i g . 1b), r e s u l t s i n the same black h o l e e v a p o r a t i o n * 7 that occurs i n the c o l l a p s e spacetime of f i g . l a . *' although t h i s i s drawn f o r the symmetry a x i s of a r o t a t i n g black h o l e , i t a p p l i e s t o a n o n r o t a t i n g black h o l e f o r a l l value s of 6> and * 7 Onruh, 1976, f i r s t d e r i v e d t h i s r e s u l t , as a consequence the vacuum s t a t e , based on the V c o o r d i n a t e f o r outgoing p a r t i c l e s and the u c o o r d i n a t e f o r i n g o i n g p a r t i c l e s , i s known as the Onruh vacuum. 29 T h e s i s O u t l i n e And Summary The f o l l o w i n g paragraphs d e s c r i b e the work o f t h i s t h e s i s . In chapter 2, the thermal p a r t i c l e e m i s s i o n process i s examined f o r the quantized n e u t r i n o f i e l d i n the g r a v i t y f i e l d of a r o t a t i n g black h o l e . The angular dependence and energy dependence of the n e u t r i n o emission i s c a l c u l a t e d i n d e t a i l . P r e v i o u s work concentrated on the c a l c u l a t i o n o f em i s s i o n r a t e s f o r massless p a r t i c l e s ( n e u t r i n o s , photons and g r a v i t o n s ) from a no n r o t a t i n g b l a c k hole (Page,1976a) or the c a l c u l a t i o n o f the e v o l u t i o n of r o t a t i n g black h o l e s due t o emi s s i o n of massless p a r t i c l e s (Page, 1976b) . Thus ch a p t e r 2 i s an e x t e n s i o n of Page's work. However the emphasis here i s on the asymmetry and d e t a i l e d form of the neutrino emission and not on the e f f e c t s of the emission on the e v o l u t i o n of the black h o l e . Apart from the d e t a i l s , the main r e s u l t demonstrated here i s the predominance of n e u t r i n o emission a n t i p a r a l l e l t o the black h o l e ' s r o t a t i o n a x i s and of a n t i n e u t r i n o emission p a r a l l e l t o the a x i s . T h i s r e s u l t can be e a s i l y understood. The r o t a t i n g black hole l o s e s angular momentum i n a d d i t i o n t o mass (energy) i n the ev a p o r a t i o n process. The asymmetry i n n e u t r i n o emission i s then due to the f a c t t h a t a l l n e u t r i n o s have t h e i r s p i n (angular momentum) a n t i p a r a l l e l to t h e i r momentum ( i . e . , d i r e c t i o n of emission from the black hole) 30 whereas a n t i n e u t r i n o s have t h e i r s p i n p a r a l l e l t o t h e i r momentum. T h i s i s the p a r i t y v i o l a t i o n f o r n e u t r i n o s (the p a r i t y transformed n e u t r i n o , which would have s p i n and •omentum p a r a l l e l , does not e x i s t ) . 0nruh(1973) f i r s t pointed out t h e e x i s t e n c e of the asymmetry i n the n e u t r i n o emission, and V i l e n k i n (1978a) has a l s o r e c e n t l y c o n s i d e r e d i t . Chapters 3 and 4 examine two important c o s m o l o g i c a l e f f e c t s : the s e p a r a t i o n of matter from a n t i m a t t e r i n the e a r l y u n i v e r s e ; and the g e n e r a t i o n of a p r i m o r d i a l magnetic f i e l d . R o t a t i n g black holes which evaporate very e a r l y i n the h i s t o r y of the u n i v e r s e could cause these e f f e c t s by asymmetric em i s s i o n of n e u t r i n o s . The black holes would form from l a r g e eddies and thus be r o t a t i n g i n p a r a l l e l to produce l a r g e s c a l e n e u t r i n o c u r r e n t s . The f i r s t e f f e c t might e x p l a i n the overwhelming dominance of matter over a n t i m a t t e r i n space around us (at l e a s t i n a r e g i o n as l a r g e as our galaxy (Steigman: 1976) ) . The dominance of matter c o n t r a s t s with the symmetry of p h y s i c a l theory under p a r t i c l e - a n t i p a r t i c l e i nterchange. The s c e n a r i o f o r matter a n t i m a t t e r s e p a r a t i o n i s as f o l l o w s . The universe i s assumed t o i n i t i a l l y have e g u a l amounts of matter and a n t i m a t t e r . These can c o e x i s t u n t i l the temperature i n the u n i v e r s e drops below the p a r t i c l e r e s t mass. Neutrino c u r r e n t s from black holes push matter and a n t i m a t t e r apart while the temperature i s s t i l l high. However, c a l c u l a t i o n s 31 here i n d i c a t e t h a t the e f f e c t i s t o o s m a l l t o e x p l a i n the presence of matter around us. T h i s i s b a s i c a l l y because the temperature becomes low enough f o r matter and a n t i m a t t e r t o a n n i h i l a t e before t h e r e i s enough time to s e p a r a t e them s u f f i c i e n t l y . The second e f f e c t c o u l d e x p l a i n the presence of l a r g e s c a l e magnetic f i e l d s i n our galaxy (of order 1 0 - 6 Gauss(Heiles:1976)) and p o s s i b l y i n i n t e r g a l a c t i c space. These magnetic f i e l d s must have played an important r o l e i n the formation of g a l a x i e s from the p r e g a l a c t i c medium (e.g. Harrison,1973). Only magnetic f i e l d s can reasonably provide t h e c o r r e c t s i z e s c a l e s f o r galaxy f o r m a t i o n . . without magnetic f i e l d s , the Jeans mass 1 8 i s of order 1 0 1 7 s o l a r masses j u s t before and 10 6 s o l a r masses j u s t a f t e r the time o f r e c o m b i n a t i o n 1 9 (e.g. Weinberg, 1972) , whereas a g a l a c t i c mass i s of order 1 0 1 1 s o l a r masses. Furthermore, the gas, which possesses l a r g e v e l o c i t y i n order to account f o r the present g a l a c t i c angular momentum, i s thrown i n t o a shocked s t a t e at the time of recombination, and subsequently i n t o s m a l l c o l l a p s e d fragements, i f no magnetic f i e l d s are 1 8 The Jeans mass i s the minimum mass f o r a body f o r which the a t t r a c t i v e g r a v i t a t i o n a l f o r c e can overcome the s u p p o r t i v e pressure f o r c e s which prevent a body from c o l l a p s i n g . 1 9 Recombination i s name of the b r i e f p e r i o d i n the h i s t o r y o f t h e expansion of the u n i v e r s e when the temperature has j u s t dropped low enough f o r e l e c t r o n s and protons to combine t o form hydrogen. Z2 p r e s e n t . The c a l c u l a t i o n s of chapter 1 i n d i c a t e t h a t r o t a t i n g black holes could have generated the g a l a c t i c magnetic f i e l d . The f i e l d i s generated by proton and e l e c t r o n c u r r e n t s , i n t u r n caused by s c a t t e r i n g from the n e u t r i n o s emitted from the black h o l e s . T h i s f i e l d i s generated before the time of recombination and has s i g n i f i c a n t i m p l i c a t i o n s f o r galaxy f o r m a t i o n as w e l l as p r o v i d i n g an e x p l a n a t i o n f o r p r e s e n t l y observed magnetic f i e l d s . A hi g h number d e n s i t y of black emission "tVonx "tWt h o l e s i s r e g u i r e d t o generate the f i e l d . The^black h o l e s can d i s t o r t the 3°K background r a d i a t i o n . For the emission to t h e r m a l i z e , the r a t i o of the d e n s i t y i n the universe to the c r i t i c a l d e n s i t y cannot be much l e s s than 1. In the next paragraph, a r e c e n t advance i n the theory of galaxy f o r m a t i o n , which depends on the presence of magnetic f i e l d s , i s b r i e f l y d e s c r i b e d . The c o n v e n t i o n a l approach to cosmology t r e a t s the problems of galaxy formation, the angular momenta of g a l a x i e s , and g a l a c t i c magnetic f i e l d s s e p a r a t e l y . However Wasserraan(1978) has shown th a t , given an a p p r o p r i a t e magnetic f i e l d at the time of recombination, the e x i s t e n c e of g a l a x i e s , t h s i r angular momenta, and t h e i r magnetic f i e l d s can a l l be accounted f o r . Wasserman a p p l i e s the e q u a t i o n s of magnetohydrodynamics to the matter i n the universe j u s t a f t e r recombination has occured. The e f f e c t s o f r a d i a t i v e pressure 33 and r a d i a t i v e v i s c o s i t y on the matter are d r a s t i c a l l y reduced a f t e r recombination compared with b e f o r e , a l l o w i n g the Lorent2 f o r c e to d r i v e compressional and r o t a t i o n a l p e r t u r b a t i o n s i n t o p r o t o g a l a x i e s . These subsequently become g a l a x i e s a f t e r a f u r t h e r n o n l i n e a r phase of c o n t r a c t i o n . For a galaxy mass of 1.4X10*1 s o l a r masses, such as our own galaxy i s thought t o have, Wasserman's values of angular momentum and magnetic f i e l d f o r the galaxy agree with the o b s e r v a t i o n a l l y determined values, an i n t e r g a l a c t i c f i e l d o f 10- 9 Gauss i s a l s o p r e d i c t e d by Wasserman, c o n s i s t e n t with present o b s e r v a t i o n a l upper l i m i t s of about 10 - aGauss. S i m i l a r t o p i c s to those of c h a p t e r s 3 and 4 have been c o n s i d e r e d r e c e n t l y a l s o by V i l e n k i n i n another c o n t e x t . V i l e n k i n (1978b) has shown t h a t n e u t r i n o c u r r e n t s e x i s t i n r o t a t i n g thermal r a d i a t i o n . With t h i s type of n e u t r i n o c u r r e n t present i n high temperature medium of the e a r l y u n i v e r s e and with assumed r o t a t i o n , V i l e n k i n (1979a) c o n s i d e r s the s e p a r a t i o n of matter and ant i m a t t e r , with n e g a t i v e r e s u l t s . V i l e n k i n (1979b) a l s o c o n s i d e r s the generat i o n o f cosmic magnetic f i e l d s , with the c o n c l u s i o n t h a t the present g a l a c t i c f i e l d c o uld have had i t s o r i g i n i n such n e u t r i n o c u r r e n t s . We next c o n s i d e r the q u e s t i o n of whether b l a c k hole emission remains thermal when p a r t i c l e i n t e r a c t i o n s are taken i n t o account. Before t h i s can be answered one must be a b l e t o 34 handle p a r t i c l e i n t e r a c t i o n s when a g r a v i t a t i o n a l f i e l d i s present. Techniques f o r c o n s t r u c t i n g i n t e r a c t i n g quantum f i e l d t h e o r i e s i n curved spacetime are under development. In g e n e r a l though, r e l a t i v e l y l i t t l e work has been done on t h i s t o p i c . In chapter 5, we c o n s i d e r the a<£* i n t e r a c t i o n i n the b l a c k hole spacetime. C a l c u l a t i o n s of p h y s i c a l e f f e c t s of the A 4 4 i n t e r a c t i o n i n s p e c i f i c spacetimes have so f a r been l i m i t e d t o the Robertson Walker cosmologies. B i r r e l l and Davies (1979) i n v e s t i g a t e conformal symmetry breaking t o demonstrate that p a r t i c l e c r e a t i o n occurs f o r the ty+ i n t e r a c t i o n . B i r r e l l and Ford (1979) c a l c u l a t e the c o n t r i b u t i o n by the i n t e r a c t i o n to the t o t a l energy and pressure i n a Robertson Walker u n i v e r s e . Here, the e f f e c t of p a r t i c l e i n t e r a c t i o n s i s c o n s i d e r e d f o r a simple model: ^<j>* s e l f - i n t e r a c t i o n f o r the massless s c a l a r f i e l d i n the 2 dimensional black hole m e t r i c . A two dimensional model was chosen i n order to keep the problem s i m p l e - the mode s o l u t i o n s f o r the f r e e s c a l a r f i e l d are known e x a c t l y and are of simple form so that an e x p l i c i t p e r t u r b a t i o n c a l c u l a t i o n can be c a r r i e d out to i n v e s t i g a t e the e f f e c t s of the i n t e r a c t i o n s . A comparison i s made with 4^>4 f o r f l a t space i n which an outward going thermal f l u x i s present. The use o f a massless s c a l a r f i e l d i n t r o d u c e s low freguency ( i n f r a r e d ) d i v e r g e n c e s . In f a c t , massless 9<f>* i n 2 dimensions i s known to be nonrenormalizable (Wightman, 1967), 35 §o t h a t r e n o r m a l i z a t i o n w i l l not n e c e s s a r i l y g i v e f i n i t e r e s u l t s . A mass r e n o r m a l i z a t i o n was c a r r i e d out but was i n s u f f i c i e n t t o remove a l l o f the i n f r a r e d d i v e r g e n c e s . We c a l c u l a t e the changes i n the number of p a r t i c l e s observed at i n f i n i t y . To f i r s t order, the change i s zero while t o second order a number of s i g n i f i c a n t r e s u l t s are found. The e f f e c t of the i n t e r a c t i o n on the p a r t i c l e spectrum goes t o zero as one approaches the event h o r i z o n of the black h o l e , suggesting that near the h o r i z o n t h e r e are no p a r t i c l e s t o i n t e r a c t , t h a t the p a r t i c l e c r e a t i o n i n some sense takes p l a c e o u t s i d e the h o r i z o n . The change i n the emitte d spectrum comes about o n l y because of the c r e a t i o n of i n g o i n g p a r t i c l e s e i t h e r by i n t e r a c t i o n s between p a r t i c l e s of the outgoing thermal f l u x or because of decays of outgoing p a r t i c l e s . Furthermore, t h i s c r e a t i o n i s p o s s i b l e f o r a massless f i e l d only because the g r a v i t a t i o n a l f i e l d of the black hole allows momentum nonconserving i n t e r a c t i o n s to take p l a c e . Although these r e s u l t s are t e n t a t i v e because they are i n f i n i t e , we would expect these c o n c l u s i o n s to remain v a l i d i f the problems of the i n f i n i t i e s could be s o l v e d . 36 2. NEOTBINO EMISSION FROM ROTATING BLACK HOLES. I . INTRODUCTION In t h i s c h a p t e r , we study the ne u t r i n o f i e l d i n the g r a v i t a t i o n a l f i e l d o f a r o t a t i n g black h o l e . N e u t r i n o s are l e f t handed, and a r o t a t i n g black hole has a d e f i n i t e handedness. Thus t h i s t o p i c i s w e l l s u i t e d f o r s t u d y i n g the e f f e c t s of p a r i t y n o n - i n v a r i a n c e * and of s p i n - g r a v i t y c o u p l i n g . Quantum theory p r e d i c t s t h a t black h o l e s emit a thermal spectrum of p a r t i c l e s i n t o the surrounding vacuum (Hawking: 1974, 1975) . Kerr ( r o t a t i n g ) black holes l o s e angular momentum and mass as a r e s u l t of the p a r t i c l e e m i s s i o n process (Page : 1976a) . A consequence of the angular momentum l o s s i s : p a r t i c l e s with p o s i t i v e h e l i c i t y 2 are emitted p r e f e r e n t i a l l y along the r o t a t i o n a x i s , i n the d i r e c t i o n of the angular momentum vector of the black hole, and n e g a t i v e h e l i c i t y p a r t i c l e s are emitted i n the opposite d i r e c t i o n . Most p a r t i c l e s occur i n nature with both s i g n s of » The p a r i t y operator exchanges l e f t and r i g h t handedness. 2 H e l i c i t y i s the s c a l a r product of a p a r t i c l e ' s i n t r i n s i c spin fcr) with i t s momentum (p): <y»p/Jp2 * a z 37 h e l i c i t y s i n c e they are i n v a r i a n t under the p a r i t y o p e r a t o r . However, n e u t r i n o s v i o l a t e p a r i t y maximally, o c c u r r i n g only with negative h e l i c i t y 3 . Since n e u t r i n o s are CP i n v a r i a n t , the a n t i n e u t r i n o has a p o s i t i v e h e l i c i t y . The r e s u l t i s that n e u t r i n o s are p r e f e r e n t i a l l y emitted from the ' s o u t h e r n 1 hemisphere and a n t i n e u t r i n o s from the 'northern' hemisphere d u r i n g the e v a p o r a t i o n of a r o t a t i n g black h o l e . The importance of p a r i t y v i o l a t i o n by n e u t r i n o s i n the presence of a Kerr b l a c k hole was pointed out by Unruh (1973) and has been independently i n v e s t i g a t e d by A. , V i l e n k i n ( 1 978) . An o u t l i n e of t h i s c h a p t e r i s presented here. The n e u t r i n o f i e l d equations are d i s c u s s e d i n s e c t i o n I I . The eguations are .separated f o r a p a r t i c u l a r choice of the D i r a c m a t r i c e s and asymptotic s o l u t i o n s are expressed t o g e t h e r with some symmetries of the s o l u t i o n s . In s e c t i o n I I I the n e u t r i n o f i e l d i s g u a n t i z e d and e x p r e s s i o n s f o r and T^ v i n terms o f the mode s o l u t i o n s are found. The c h o i c e of an a p p r o p r i a t e vacuum s t a t e f o r t h e n e u t r i n o f i e l d i s d i s c u s s e d i n s e c t i o n IV. Two d i f f e r e n t vacuum s t a t e s are proposed- the Boulware vacuum and the Unruh vacuum. The l a t t e r corresponds* to the 3 P o s i t i v e h e l i c i t y n e u t r i n o s could e x i s t and thus be produced i n black hole e v a p o r a t i o n . As they would not i n t e r a c t with matter except g r a v i t a t i o n a l l y , we e x c l u d e them from f u r t h e r c o n s i d e r a t i o n . 38 vacuum used by Hawking i n h i s d e r i v a t i o n o f the b l a c k hole e v a p o r a t i o n process. In s e c t i o n V, e x p r e s s i o n s f o r the e x p e c t a t i o n v a l u e s o f the c u r r e n t J** , and the energy momentum t e n s o r T M V are given f o r the two vacuum s t a t e s . These are r e l a t e d to the p h y s i c a l o b s e r v a b l e s f a r from the black hole, i . e . the r a d i a l f l u x e s o f n e u t r i n o number, energy and angular momentum. The f l u x e s are given as a f u n c t i o n o f angle, and a l s o as i n t e g r a t e d over a l a r g e 2-sphere s u r r o u n d i n g the black h o l e . I n s e c t i o n VI, a l l of the eguations r e l e v a n t t o c a l c u l a t i n g the ob s e r v a b l e s a r e put i n dim e n s i o n l e s s form s u i t a b l e f o r c a l c u l a t i o n . T h i s a l s o s e r v e s t o summarize the e s s e n t i a l parameters o f the n e u t r i n o e v a p o r a t i o n process. Only the q u a n t i t i e s d e r i v e d with the Unruh vacuum, r e l e v a n t to the Hawking process of black hole e v a p o r a t i o n , are c a l c u l a t e d subsequently. In s e c t i o n VII, a low frequency a n a l y t i c approximation i s made to v e r i f y the asymmetry i n n e u t r i n o e m i s s i o n . I n s e c t i o n VIII a b r i e f survey of the method of c a l c u l a t i o n i s given. S e c t i o n IX presents and d i s c u s s e s the major r e s u l t s of the c a l c u l a t i o n s with the a i d of the f i g u r e s i n Appendix A 5. These d i s p l a y the d e t a i l e d dependence of the e m i s s i o n on p o l a r angle (fi) , neutrino energy (w) , and black hole angular * They correspond i n the sense t h a t f a r i n the f u t u r e of the c o l l a p s e ( f o r Hawking's d e r i v a t i o n , see f i g . 1a) or near f u t u r e n u l l i n f i n i t y ( for Unruh's d e r i v a t i o n , see f i g . 1b) , the two vacua r e s u l t i n t h e same e f f e c t s , i . e . thermal emission by the b l a c k h o l e . 5 Page (1976b) has c a l c u l a t e d r a t e s i n t e g r a t e d over s o l i d angle i n h i s d i s c u s s i o n of e v o l u t i o n of r o t a t i n g b lack holes. 39 moment um (J) and mass ( H ) . N a t u r a l (Planck) u n i t s , i n which -n=c=G=k=1, are used throughout, except as noted. Por the c o n v e r s i o n t o more common u n i t s see Appendix C or the l a s t (red) page of Misner et a l . (1970) . I I . THE NEOTBINO FIELD IN THE KERB MET SIC The Kerr metric d e s c r i b e s a r o t a t i n g black h o l e o f mass M and angular momentum J . In B o y e r - L i n d g u i s t c o o r d i n a t e s ( B o y e r , L i n d g u i s t : 1967) (here w r i t t e n as x M= ( t , © , <P ,r) y*=0,1,2,3 ) the m e t r i c takes the form (Onruh: 1973) : dsz=(A/ /o 2) (dt-asin2ed<p) 2 -/>2d$2 - (sin2©/^ ,2) (adt-(r2+a2) d^) 2 - (f 2 / A ) d r 2 (2.1) with: &=r2+a2-2Mr yo2=r2«-a2cos2 a=J/M (2.2) The g M component o f t h e me t r i c , pz/Ar d i v e r g e s f o r A approaches 0, i . e . a t : r ± =M (1* (l-oC.2) »^2) with <* = a/M (2.3) T h i s g i v e s the r a d i a l d i s t a n c e s , r + and r _ , o f the o u t e r and 40 i n n e r event h o r i z o n s 6 * 7 . The r e g i o n of spacetime of concern here i s t h a t e x t e r i o r to the out e r event h o r i z o n at r + . The n e u t r i n o f i e l d i s d e s c r i b e d by a wave f u n c t i o n t h a t i s a s o l u t i o n to the g e n e r a l l y c o v a r i a n t massless D i r a c e q u a t i o n and i s of negative h e l i c i t y : aMVMY=0 ( U i y s ) t = 0 (2.4) i s the 4x4 matrix d e f i n e d by ^ ^ o ^ s ^ Y ^ ^ ' 4 1 * i s the permutation symbol: =*1 f o r an even permutation of i n d i c e s ; =-1 f o r an odd permutation of i n d i c e s ; =0 i f two i n d i c e s are the same. i s obtained from € ^ f i s i by lowering i n d i c e s with the metric g M V . In the r e p r e s e n t a t i o n f o r the gamma mat r i c e s of eqns (2.7) and (2.8) below, v» has the form: (2.5) The second eguation of eqns (2.4) expresses the requirement * C a r t e r (1966, 1968) d i s c u s s e s the g l o b a l p r o p e r t i e s of the Kerr metric. 7 An event h o r i z o n (in an a s y m p t o t i c a l l y f l a t spacetime) i s a n u l l ( l i g h t l i k e ) s u r f a c e from w i t h i n which nothing can escape to i n f i n i t y . Hawking and E l l i s (1973) g i v e more p r e c i s e d e f i n i t i o n s . 41 t h a t n e u t r i n o s ( a n t i n e u t r i n o s ) be l e f t ( r i g h t ) handed, i e the s p i n , <li/2, i s a n t i p a r a l l e l ( p a r a l l e l ) t o the momentum. T h i s i s the p a r i t y v i o l a t i n g f a c t o r which i s c r u c i a l to the asymmetry f o r the emission of n e u t r i n o s from r o t a t i n g black h o l e s . As e x p l a i n e d by B r i l l and I h e e l e r (1957), the s p i n o r c o v a r i a n t d e r i v a t i v e i s giv e n b y 8 : (2.6) F o l l o w i n g Unruh ( 1974) , the matrices fiM are chosen 9 t o take the form: ^o=(r2+a2)'jo/Ai« • a s i n d f l 2 ^ (2.7) In the above, v M are the f l a t - s p a c e gamma matrices: (2.8) 8 i^ttxpl i s the C h r i s t o f f e l symbol of the f i r s t k i n d . i s d e f i n e d by irf-C*^*. , ^ » Any set r e l a t e d by a s i m i l a r i t y t r a n s f o r m a t i o n ^ j j M = s / s - 1 ) w i l l do. The ^ must s a t i s f y yT+ffi^g*". CT t a r e the 2x2 P a u l i matrices (e.g., these are l i s t e d i n the appendix of Bjorken and D r e l l , 1965). 0nrah(1973) was the f i r s t to demonstrate s e p a r a t i o n o f v a r i a b l e s f o r t h e n e u t r i n o equations i n the Kerr m e t r i c . The n e u t r i n o wavefunction i s w r i t t e n a s 1 0 : </*=exp (-iwt)exp(+im^) fo^ /fi,(r) Sx (&)\ (2.9) (Asin2£ ( r + i a c o s 5 ) 2 ) v i / ^ \ B - 2 J R ) si}&h The r a d i a l and ang u l a r f u n c t i o n s , B and S, s a t i s f y : [ d/dr-i / A(w (r2+a2)-ma) ]B, (r) = (k/^ 1 / 2) Rx (r) (2. 10) [d/dr + i/£(w (r 2+a 2)-ma) ]B a ( r ) = ( k / ^ ) B , (r) £d/d&+(wasinfe-m/sin©) ]S , (*) =kS1<(&) (2.11) [ d/d&- (wasinft-m/sinft) ]S t(&)=-kS, (6) As a consequence of the boundary c o n d i t i o n s f o r c o n t i n u i t y a £=0 and 6-=Tr# t h e equations above f o r S( and S x r e s u l t i n an eigenv a l u e problem with k as eiq e n v a l u e . A s e t of k values and S, , S z s o l u t i o n s to eqn (2.11) e x i s t s f o r each p a i r of values (w,m). Because of t h i s , i t i s convenient t o l a b e l the mode s o l u t i o n s '/'(from eqn (2.9)) of the separated equations *° m here i s the neqative o f Unruh's m. 43 by the s e p a r a t i o n c o n s t a n t s f o r t h a t node: w, a, and k, i . e . , Y^jJ1*)» R^(«#o,k #r) r Sj (w,m,k,fr) . The angular f u n c t i o n s Si and S z are taken t o be r e a l and t o be normalized by: TT f S, 2(v,m,k,&)dft = Js2^ 2(¥,m,k,6-) d$=1/(4-n;) (2. 12) S, and S^ are o r t h o g o n a l : "IT j S . l K . i . k ^ J S . (w,*,!^) d© = ( V 4 - i r ) ^ ^ i,j=1,2 (2.13) o They possess the symmetries: S, (-»,-m,k , A ) \ =±/sl (B,m,k,ft)\ (2.14) fs, (w,m,-k,&)\ =±/s, (w,m,k,&)\ Is fM,m,-k,6)/ V"S j /S, (w,m,k,1-6)\ \ S a ( » , m,k,T-0)/ =±/s^ (w,m,k,A) tS,(w,a,k,fr) The s i g n i s chosen i n egn (2. 14) to ensure c o n t i n u i t y o f S, and S v at &=TT/2. The s i g n i s given by (-1)*-"**1 where 1 i s the h a l f - i n t e g e r denumerating the k ei g e n v a l u e s f o r a given value of m (1=) m| ,| m+1| ,] m+2 !,••«) • 1 i s a l s o the v a l u e o f k 44 f o r the case aw=0. m i s the component of angular momentum along the r o t a t i o n a x i s (m=1/2 ,-1/2, 3/2,-3/2, 5/2, . • .) , while 1 i s , f o r a n o n r o t a t i n g black hole, the t o t a l angular momentum quantum number. The r a d i a l e q uations (2.10) have the asymptotic s o l u t i o n s (Unruh:1974) : (B* (w,m,k,r) , B*(w,k,m,r)) = 1//2* (A +(w,m,k)exp(iwr*) ,exp(-iw£)) = 1/J2K (0,B +(w,m,k)exp(-iwr)) r - * - «o (2.15) (B t (w,m,k,r) ,R~(w,k,m,r)) = 1/^37 (B_(w,m,k) exp(iw£) ,0) r-» = 1//2n (exp(iwr) ,A_(w,m,k) exp(-iW)) (2.16) The angular frequency w" i s d e f i n e d by w=w-mwH w ^ 3 / 0 . l A r * + (2.17) In the above a new r a d i a l c o o r d i n a t e £ has been i n t r o d u c e d : d£/dr= (r*+a*) / ^ or r=r+ln ( r - r v ) / 2 K ^ + l n (r - r _ ) /2K_ (2.18) r+ and r„ are the r a d i a l p o s i t i o n s of the event h o r i z o n s * «5 (given by egn (2 . 3 ) ) . K+. and K__ are the s u r f a c e g r a v i t i e s at these h o r i z o n s : K*= ( r ^ - r _ ) / 2 (r +z+az) K„=(r_-r^) /2(r_ 2+a 2) (2.19) As r i s decreased t o r + , i e approaches the out e r event h o r i z o n , r decreases towards negative i n f i n i t y . The s o l u t i o n ofegn (2. 15) r e p r e s e n t s a wave incoming from i n f i n i t y , p a r t l y r e f l e c t e d from (with amplitude A4(w,m,k) ) and p a r t l y t r a n s m i t t e d through (with amplitude B + (w,m,k) ) the g r a v i t a t i o n a l angular momentum p o t e n t i a l b a r r i e r surrounding t h e black h o l e . T h i s b a r r i e r can be found by w r i t i n g egn (2. 10) i n second order form and transforming away the f i r s t d e r i v a t i v e term. One f i n d s : d 2B (r)/dr* 2+? (r)E(r)=0 (2.20) with: i ) V(r) =A(r)+B(r) i i ) A (r) = - f 2 (r) -df (r) /dr i i i ) Bi (r) = (r 2+a 2) «^A _ , /*fi(r) i v ) f (r) = ( r 3 - 3 a r 2 + a 2 r * a 3 ) / ( r 2 + a 2 ) 2 v) B (r)= (K (K+i (r-H)) -A(k 2*2iMwr)) ^ (rz+a 2) 2 46 with • f o r R, , - f o r & x vi) K= w (r 2*a 2)+ma The p o t e n t i a l V (r) i s complicated and has an imaginary p a r t i n d i c a t i n g a t r a n s f e r of amplitude between R, (r) and R ( r ) . . The R- s o l u t i o n , which o r i g i n a t e s a t the outer h o r i z o n (£••-€*) , i s p a r t l y r e f l e c t e d back through the outer h o r i z o n and p a r t l y t r a n s m i t t e d t o i n f i n i t y . The f o l l o w i n g i d e n t i t i e s h o l d (Unruh, 197 4); (d/dr) ( B ; B 4 - B ^ B , ) = 0 (d/dr) ( R ' + R ^ + R ^ R , ) =0 (2.21) f o r any two s o l u t i o n p a i r s t o egns (2.10): R,(w,m,k,r), R z(w,m,k,r) and R'(w,m,k,r) R^(w,m,k,r). Applied t o the asymptotic forms R + and given by egns (2. 15) and (2.16), the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s , A and B , s a t i s f y : l A J z* I B . J 2=1 |JJ Z+|B.|2 = 1 (2.22) A 4 B , = A , B + B _ * (w,m,k)=-B +(w,m,k) ( 2 . 2 3 ) The f i r s t of these expresses c o n s e r v a t i o n of p r o b a b i l i t y f o r th e ingoing and outgoing waves. The symmetry of the r a d i a l e q u ations under complex c o n j u g a t i o n y i e l d s : 47 A + (-w,-m,k) =&+*(*,m#k) B„(-w,-m,k) = B..*(w,m,k) (2. 24) The i n g o i n g and outgoing r a d i a l s o l u t i o n s are o r t h o g o n a l : The mode s o l u t i o n s o f the separated n e u t r i n o e g u a t i o n s are now completely s p e c i f i e d by the s e p a r a t i o n c o n s t a n t s plus the l a b e l 7i f o r the r a d i a l f u n c t i o n s having the asymptotic forms eqn (2 .15 ) ( f o r A =•) and eqn ( 2 .16 ) ( f o r a =- ) : Vlw,m ,k xM) . These are or t h o g o n a l by egn (2 .9 ) and the o r t h o g o n a l i t y r e l a t i o n s eqns ( 2 . 1 3 ) and ( 2 . 2 5 ) : • B* (w,m,k,a,r) B (w, m,k,fl', r) ) =. (2 .25) \ , * , k.Til x*) }fo V ( k j A';X«) J^3d»x (2 . 26) 48 I I I . QDANTIZ ATION OF THE N EOTBINO FIELD; J** AND T ^ In the p r e c e d i n g s e c t i o n we d e a l t with c l a s s i c a l s o l u t i o n s ( r e l a t i v i s t i c but non-guantized) of the n e u t r i n o f i e l d e q u a t i o n s . Here the n e u t r i n o f i e l d i s quantized and e x p r e s s i o n s are d e r i v e d f o r the c u r r e n t operator J** and the energy momentum operator T^ v of the n e u t r i n o f i e l d . T h i s i s done i n a g e n e r a l manner independent of the s p e c i f i c form of the s o l u t i o n s *f*, which depends on the Kerr m e t r i c background of the n e u t r i n o f i e l d . The p h y s i c a l q u a n t i t i e s obtained from the o p e r a t o r s J"* and T M V depend, i n a d d i t i o n , upon the vacuum s t a t e chosen f o r the Kerr m e t r i c . T h i s t o p i c i s d i s c u s s e d i s s e c t i o n IV. In quantum f i e l d theory t h e n e u t r i n o f i e l d becomes an o p e r a t o r : ,5:(x M). I t i s assumed t h a t there e x i s t s a complete decomposition of the c l a s s i c a l n e u t r i n o f i e l d r^x*) i n terms of orthonormal modes r^(x*) . The index n i n d i c a t e s c o l l e c t i v e l y the mode i n d i c e s , eg f o r n e u t r i n o s i n the Kerr metric one has the modes *f*(w,m,k,^ ;x M) . O r t h o g o n a l i t y i s w r i t t e n as: (3.1) The f i e l d o p e r a t o r can then be expanded i n terms of products o f and a n n i h i l a t i o n or c r e a t i o n o p e r a t o r s a A,b^~: t 49 f ( x " ) = Z ( a ^ H M ^ x ^ + b ^ (3.2) The argument of %, w or d i s t i n g u i s h e s p o s i t i v e from n e g a t i v e freguency modes ( i e r p a r t i c l e from a n t i p a r t i c l e modes). a n ^ (w, x4*-) a n n i h i l a t e s a p a r t i c l e of mode n a t l o c a t i o n x** and ^(-w^x*) c r e a t e s an a n t i p a r t i c l e of mode n at x**. The vacuum s t a t e i s d e f i n e d v i a : a,jO> = 0 b,jO>=0 f o r a l l n (3.3) Thus the s t a t e J0> c o n t a i n s no p a r t i c l e s or a n t i p a r t i c l e s . What c o n s i s t s a p a r t i c l e or a n t i p a r t i c l e depends on the s e p a r a t i o n of p o s i t i v e from negative freguency modes 1 1. A l l o ther o b s e r v a b l e s are d e f i n e d i n terms of the f i e l d o p e r a t o r . The c u r r e n t operator J M a n d the energy momentum te n s o r T ^ are giv e n by (Schveber, 1961): J*(x<) = (1/2) [ (3.4) T^<x*)= (i/4) [ S ' ^ V y ) ^ ] +hermitian conjugate (3.5) The commutator a c t s on a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s t o 1 1 See s e c t i o n on p a r t i c l e c r e a t i o n i n chapter 1. 50 s u b t r a c t an i n f i n i t e c o n s t a n t from J** and T ^ . The round b r a c k e t s i n egn (3.5) i n d i c a t e s y a m e t r i z a t i o n ; For the black h o l e e v a p o r a t i o n process we are p r i m a r i l y i n t e r e s t e d i n how the c u r r e n t and energy momentum tensor behave when no p a r t i c l e s are present b e f o r e the b l a c k hole forms. Thus we wish t o c a l c u l a t e the vacuum s t a t e e x p e c t a t i o n values of J ^ a n d T ^ , where t h e vacuum s t a t e i s t h a t f o r an observer t o the past of the black h o l e . The e x p e c t a t i o n values are given by: (3.6) <0|J*(x*) |0>= (1/2) J > f t ( - ^ ( " ' X ^ O y ^ t W f X * ) (3.7) <0|T^ v(x*) !0>=(i/4) ^  (-Y?(w,x*)y<» }^V v J ^M"'**) • fc^-w***) jfo ^ Vv) VvJ-w^)) • h e r a i t i a n conjugate (3.8) The e x p e c t a t i o n v a l u e s f o r s t a t e s with a s i n g l e p a r t i c l e mode aA"+j0> or a n t i p a r t i c l e mode bNi"|0> i n v o l v e the vacuum e x p e c t a t i o n v a l u e s : 51 <Ola A J^a^J 0>=<0| J*J 0>* M£(w, jr«)£o f%l*.x«) (3.9a) < O j b ^ b ^ j 0>=<0| J*|0>- HiT(-w^^)^oyM v^j-w,**) (3.9b) <0|a„T M, a^jO>=<OlT^ vJO>*£ (i/2) V l«, • h e r m i t i a n conjugate] (3.10a) < 0 j b ^ v b ^ | 0 > = < 0 | T „ v j 0 > - [ (i/2) S^<-»,x*)yo^V K 3&{-w,x*) • h e r m i t i a n c o n j u g a t e ] (3.10b) From these one can c a l c u l a t e the c u r r e n t and energy momentum ten s o r f o r a one p a r t i c l e s t a t e , |A(1)>, d e f i n e d by: | A(1)>="2(A n(w)a^ •An(-w) b ^ ) J0> (3.12) T h i s i s normalized by i n t e g r a t i n g the d i f f e r e n c e between the e x p e c t a t i o n v a l u e s of the 0-component of the c u r r e n t , i n a one p a r t i c l e s t a t e and i n the vacuum s t a t e , over a s p a c e l i k e 3-surface: Jp^d^x (<A (1) | jo (x"*) J A (1) >-<0) j o (x«) J 0>) = F (A* (*) A^(w) •**(-»>**(-»)) =1 (3.13) V|>0 52 IV. THE CHOICE OF VACUUM STATE The c h o i c e of p o s i t i v e frequency i n the f i e l d o p e r a t o r expansion, eqn (3.2) determines the s e p a r a t i o n of modes i n t o p a r t i c l e and a n t i p a r t i c l e modes. T h i s c h o i c e a l s o d e f i n e s the c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s and thus determines the vacuum s t a t e . The c h o i c e i s not unique and, i n g e n e r a l , vacuum s t a t e s under d i f f e r e n t d e f i n i t i o n s of p o s i t i v e freguency are not e q u i v a l e n t * 2 . For the i n g o i n g modes (Y-(w, m,k,^) ,3=+) o r i g i n a t i n g a t i n f i n i t y , a n a t u r a l c h o i c e of p o s i t i v e frequency i s w>0. For the outqoing modes (^=- ) , a simple choice i s $>0, where "w i s given by egn (2.17). A h y p o t h e t i c a l observer, who i s j u s t o u t s i d e the h o r i z o n o r b i t i n g the black hole a t angular freguency wH, would see no outgoing p a r t i c l e s i n t h i s vacuum s t a t e . wH i s the a n g u l a r v e l o c i t y of dragging of i n e r t i a l frames at the h o r i z o n of a r o t a t i n g b l a c k hole. One can summarize t h i s p o s i t i v e frequency d e f i n i t i o n by: K(w,m,ft)>0 " i t h K=+1 :A=+,w>0 or ^=-,fi>0 K=-1 otherwise (4. 1) T h i s c hoice of p o s i t i v e frequency d e f i n e s the Boulware * 2 F u l l i n q (1977) d i s c u s s e s the d e f i n i t i o n of vacuum s t a t e s . 53 vacuum 1 3 . Another c h o i c e of p o s i t i v e frequency has been gi v e n by 0nruh(1976) f o r n o n r o t a t i n g black holes. T h i s i s motivated by the vacuum obtained by Hawking (1974, 1975) i n h i s d e r i v a t i o n of the black hole e v a p o r a t i o n process f o r black h o l e s formed by s t e l l a r c o l l a p s e . P o s i t i v e freguency i s d e f i n e d i n terms of an a f f i n e parameter 1* along the past h o r i z o n f o r the outgoing (A=- ) modes. An observer f o l l o w i n g a n u l l geodesic i n t o the black hole would see no p a r t i c l e s i n t h i s vacuum s t a t e . T h i s observer i s i n a more n a t u r a l s t a t e than the h i g h l y a c c e l e r a t e d observer who sees no p a r t i c l e s i n the Boulware vacuum s t a t e . A b r i e f s ketch o f the d e r i v a t i o n of the f i e l d o p e r a t o r e x p r e s s i o n f o r the Onruh vacuum i s given here. The o r i g i n a l arguments based on the S c h w a r z s c h i l d m e t r i c are extended t o the case of the Kerr m e t r i c . He w i l l make use of a Penrose diagram i n the d i s c u s s i o n . A Penrose diagram i s c o n s t r u c t e d so t h a t a l l n u l l rays ( l i g h t paths) are represented by l i n e s a t + or - 45 degrees. I n f i n i t y i s transformed t o a f i n i t e 1 3 Onruh(1974) uses t h i s vacuum s t a t e . ** An a f f i n e parameter, G , i s one f o r which the geodesic eguation has the form: dZx'Vao-** P ^ x ' / d o - X d x V d * ) =0 F o r t i m e l i k e (or spacelike) geodesies the a f f i n e parameter i s l i n e a r i n proper time (or d i s t a n c e ) . F o r a g e n e r a l parameter yo(<r) , the geodesic e q u a t i o n , which s p e c i f i e s p a r a l l e l t r a n s p o r t of a tangent v e c t o r along a curve parametrized by /> (eg f o r c e f r e e motion of an observers 4 - v e l o c i t y a l o n g a worldline) , is^more complicated i n form. 5H coordinate distance f r o i the o r i g i n . By preserving the l i g h t l i k e structure of a spacetime, Penrose diagrams allow an easy v i s u a l i z a t i o n of the behaviour of l i g h t rays and a l s o of the l i a i t s on t i i e l i k e carves (possible v o r l d l i n e s of observers), whose tangent vectors Bust l i e within the future directed n u l l cone. Ha wking, E l l i s (1973) discuss Penrose diagrams a ore f o l l y and B a k e extensive use of then. The derivation here i s applicable only along the syaaetry axis of the Kerr metric. The extension to o f f axis (&t0) i s more complicated since the n u l l geodesies on the past horizon do not depend on v only but involve <p as well. The derivation for o f f axis has been done bat i s not presented here. The r e s u l t below, egn (4.7), i s v a l i d f o r a l l 6 and 4>. , * Penrose diagram of a Kerr black hole along the symmetry axis(*=0) i s given i n f i g ore 1b i n Appendix a. J}+ and are future and past n u l l i n f i n i t y , a and v are n u l l coordinates which reduce to t+r and t - r at i n f i n i t y . ,I.e. a i s constant along r a d i a l l y ingoing l i g h t rays and v i s constant along r a d i a l l y outgoing l i g h t rays. u and v are given by: u=t«x v=t-r (4.2) •here r i s the transforaed r a d i a l coordinate given by eqn (2.18). One has r approaching negative i n f i n i t y f o r r approaching the oater horizon of 'the black hole at r+. ,The a and v coordinates diverge at r+ /Aong the past horizon (p.h.) and future horizon(f.h.), r e s p e c t i v e l y . E.g., along an S5 ingoing n a i l geodesic a i s constant vhile v increases froa at past n n l l infinity,$1 t o • oc at the future event horizoa,f.h. This holds both i n oar universe (region I) and i n the a i r r o r universe (region II) obtained by a n a l y t i c extension of the Kerr a e t r i c . Mull coordinates 0 and V can be constructed which cover both regions I and I I with one set of coordinates. In region I , one has 0<0«* #-«»<?<0 and i n region II : - o o < o<0, 0 < V < o « . 0 and a; V and v are related by: u=lnO/K+ i n I ; a=-ln(-D) /K+ i n II v=-ln(-V)/K+ i n I ; v=lnV/K+ i n I I (4.3) 0 i s the a f f i n e parameter along the future horizon and V i s the a f f i n e parameter along the past horizon. Surfaces of constant Schwarzschild t i n e , t , always pass through the crossover point of the past and future horizons i n the Penrose diagraa of the black hole, figure 1b. Kruskal time, defined b y 1 5 (U+7)/2, i s constant over surfaces which are horizontal s t r a i g h t l i n e s in f i g u r e lb. The discussion f o r the corresponding Penrose diagrams off the symmetry axis of the Kerr a e t r i c i s aore complicated. The important point i s that t increases i n the opposite d i r e c t i o n to Kruskal t i n e in region II of the Kerr a e t r i c . In f i g . 1b* 6, t approaches plus i n f i n i t y along the future horizon and along whereas t .approaches ainus i n f i n i t y »* Schvarzchild t i i e i s given by t={u*v)/2, see egn (4.2). 56 along the past h o r i z o n and along Near the past h o r i z o n , from egns (2.9), (2.16) and (4.2), one f i n d s : ytWr8>#K#3=-;x M)©< exp(-i'Sv) (4.4) Thus i n r e g i o n s I and I I one can w r i t e : ^ocexpUv-lnJ VJ/K +) V<0; «. exp ( i f r l n | V |/K+) ?>0 (4.5) where i s zero i n r e g i o n I I and ^ i s z e r o i n r e g i o n I. To c o n s t r u c t a pure p o s i t i v e freguency mode i n the V sense one can combine ^ and ^ by a n a l y t i c c o n t i n u a t i o n around the l o g a r i t h m i c s i n g u l a r i t y at V=0 (to do t h i s V i s taken t o be complex). One o b t a i n s the r e s u l t , p r o p e r l y n o r m alized: Vtw#m,k,- ;x M) = (exp M / 2 l U ) 4 i + e x p (-TTW/2K+) /(2cosh(irvVK +)) 1 / 2 (4.6) T h i s i s a n a l y t i c and bounded i n the lower h a l f complex V plane f o r a l l w and thus p o s i t i v e freguency f o r a l l w. E g u i v a l e n t l y , t o demonstrate t h a t the modes (4.6) are p o s i t i v e frequency i n the V sense f o r a l l w, one can F o u r i e r * 6 For a n o n r o t a t i n q black h o l e f i g u r e 1b i s v a l i d a t a l l angular c o o r d i n a t e s . 57 ana l y z e the nodes and f i n d only exp(-iwV) components with w>0. By i n t e r c h a n g i n g and i n eqn (4.6) , one ob t a i n s n e g a t i v e frequency modes, i n t h e V sense, f o r a l l v . The f i e l d operator e x p r e s s i o n , eqn (3.2), then becomes: <£>(**) =S.{ $ dw[ vftw»»»K,+ ;x*)a (w,m,k,*) •t(-w,-m,k,+;x*) bV.a.k,*) ] • i dw[exp(-ir1>/2K+)HlWrHi,k,-;x^)a(w,m,k,-) ottw •exp(-^/2K +) vf(w,a.k,-;x*) b^w,m,k,-) ] /(2cosh(irw/K + )) ) (4.7) V. THE VACUUM EXPECTATIOK VALUES OF J ~ AND F i r s t , conserved c u r r e n t s are d e r i v e d f o r the Kerr metric which correspond t o p h y s i c a l o b s e r v a b l e s of i n t e r e s t . The Ker r metric possesses the K i l l i n g v e c t o r s : C = ( 1 ' ° ' 0 ' 0 ) i^ = «>,0,1.0) <5-1> correspo n d i n g to the i n v a r i a n c e of the m e t r i c under t r a n s l a t i o n along t he t or ^  c o o r d i n a t e s . These symmetries g i v e r i s e t o the conserved c u r r e n t s : 58 The r a d i a l components (y^=3, see egn (2.1)) of these c u r r e n t s a r e the flow of energy and an g u l a r momentum, r e s p e c t i v e l y , i n y the outward r a d i a l d i r e c t i o n from the b l a c k hole: d 2 E (x*)/r 2dndt=<0|T 3 (x*) |0> d 2 L ( x * ) / ( r 2 d A d t ) = <OfT 3 C t r t(x") |0> (5.3) The net number l o s s of n e u t r i n o s i n the r a d i a l d i r e c t i o n i s d e r i v e d from the c u r r e n t o p e r a t o r : d 2N v.^ / ( r 2 d a d t ) = <0 J J 3 (x*) |0> (5.4) In t h i s s e c t i o n these p h y s i c a l g u a n t i t i e s w i l l be evaluated i n the l i m i t of l a r g e r using the r e s u l t s o f the past two s e c t i o n s . The asymptotic forms of B t and B x g i v e n by eqns (2.15) and (2.16) can then be used to e v a l u a t e the n e u t r i n o wavefunction V*. The vacuum e x p e c t a t i o n v a l u e s egns (5.3) and (5.4) w i l l be e v a l u a t e d f i r s t i n the Boulware vacuum f o r i l l u s t r a t i v e purposes. Later the e x p r e s s i o n s f o r the more p h y s i c a l Unruh vacuum w i l l be d e r i v e d . , The f i e l d o p erator i n the Boulware vacuum i s found d i r e c t l y from eqn (3.2), with egn (4.1) as the d e f i n i t i o n o f 59 p o s i t i v e f r equency: Jt**)^, S dw[*Hw,8>,K,a;**) a(w,m,k,a) • *(-w,-m,k,a ;x«) b ^ w ^ k , * ) ] (5.5) With the asymptotic forms of E , and Rx and the s e p a r a t i o n o f v a r i a b l e s , eqn (2 .9 ) , the e x p r e s s i o n f o r <0JJ3J0> becomes, f o r r-»«o: <01J 3(*,&) | 0> = 1 / ( 2 i r r 2 s i n 6 ) 2. ?«lw[ - (| A +(«,m,k) |2 HB_(w,m,k) | 2 ) S ,(w,m,k,&)2 + S 1 ( K , m, k,&)2 • (|A + (-»,-B,k) |2 • |B.(-w,-m,k) 12) S , (-w,-ra,k,&) 2 -S r(-w,-m,k,fr ) 2 ] (5.6) The s p i n oi=J/nz of t h e black hole has now been w r i t t e n e x p l i c i t l y as a parameter upon which the r a d i a l number c u r r e n t depends. Eqn (5.6) a l s o depends on the mass M of the black hole, but as w i l l be seen, t h i s dependence can be f a c t o r e d out. U t i l i z i n g the symmetries of St and Sx, eqn ( 2 . 1 4 ) , and of A + and (eqns (2.22) and ( 2 . 2 3 ) ) , the r a d i a l c u r r e n t s i m p l i f i e s t o : <0|J 3 («,&) lO> = 4/(2T , r 2 s i n&)Z k jL0a«M B_(w,m,k) J2 ( S l(w,m,k,fr)2-S^(w,m,k,&)2) (5.7) 60 E v a l u a t i o n of the vacuum e x p e c t a t i o n values o f the energy and angular momentum c u r r e n t s , egn (5.2), i n s i m i l a r manner leads t o : <0jT* («c,e)|0> =4/(2T,r2sin&)5 dw wj B _(w,m,k) J 2 (S f (w,m,k,s) 2+S Jw,m,k ,e) 2 ) (5.8) <0}T3 w(/,») |0> = V ( 2 n r 2 s i n s ) 5 t L dw |B_ (w,m,k) J 2 X[ (m-awsin 2£) (1/2) (S t (w,m,k, 0 2 + S Jw,m,k,&) 2) •ksinds^ (w,m,k,©.)S^(w,m,k,6) + (cos©/4) (S , (w,m,k,fl 2-S v(«,m,k,*) 2) ] =a/(2rr 2sinfr ) , 5 l j # dw J B_ (w,m,k) 1 2 X[m(S, (w,m,k,*) 2 +S v(w,m,k,*) 2) + (sinB/4) (d/da+cotfl.) (S , (w,m,k,&) 2 -S^(w,m,k,o) 2) J (5.9) Two forms are given i n egn (5.9) f o r the r a d i a l f l u x o f angular momentum. The f i r s t i s more con v e n i e n t f o r c a l c u l a t i o n . The second form d i s p l a y s the s i m i l a r i t y with the energy f l u x egn (5.8) (the second term of the second form vanishes i d e n t i c a l l y when i n t e g r a t e d over ft). Thus the r a d i a l energy and angu l a r momentum f l u x e s f o r g i v e n w,m,k,A mode are i n the r a t i o w/m as one would expect (see egn (5.10)). They have , however, a d i f f e r e n t angular d i s t r i b u t i o n . The l o s s r a t e s i n t e g r a t e d over a s p h e r i c a l 2 - s u r f a c e at 61 l a r g e r , i e ^ g d f r d ^ , a r e 1 7 d»v-v (•*)/<!t=0 dE(oc1/dt= (4/2TT)X [ dw w|B^(w,m,k) | 2 *»>© aL K)/dt=(U/2tr)^ E l r r dw m|B_(w,m,k) | 2 (5.10) *»/ W>0 I n the Boulware vacuum the frequency spectrum extends from w=0 t o a sharp c u t o f f at w=mwH with i n t e n s i t y |B„(w,m,k) J 2 . The range 0<w<mwH i s p r e c i s e l y t h a t range of f r e q u e n c i e s f o r which supe r - r a d i a n c e occurs f o r i n t e g e r s p i n p a r t i c l e s ( Z e l d o v i c h : 197 1; Press,Teukolsky: 1972) . Super-radiance i s the phenomenon where the r e f l e c t i o n c o e f f i c i e n t f o r an i n c i d e n t wave exceeds u n i t y . T h i s r e s u l t s i n more p a r t i c l e s being emitted than a r e i n the i n c i d e n t wave, i e s t i m u l a t e d emission. The more p h y s i c a l l y i n t e r e s t i n g case f o r the b l a c k hole e v a p o r a t i o n vacuum o f Onruh i s co n s i d e r e d next. The f i e l d o p e r a t o r was giv e n i n eqn (4.7) . The vacuum e x p e c t a t i o n value o f the c u r r e n t i s , by (3.7) : > 7 Onruh,1974 has a l s o obtained these r e s u l t s (eqn (5.10)). By h i s convention, h i s r e s u l t s c o n t a i n a minus s i g n with r e s p e c t t o eqn(5.10). 62 <0|J A(x<) |0>= <1/2) 2T ( Idw[-S» +(« rm rk,*;x' t)|o^S'(w #m,k,*;x* <) *S dw[ - (sinhnO/K* ) / (coshw&VK* ) X f ( W , m , k , - ; X * ) } O x * f t W , m , ] ) (5.11J An analogous e x p r e s s i o n , from egn (3.3), a p p l i e s f o r <0|T^ v(x* t) |0>. With the asymptotic forms f o r Y'for l a r g e r , one f i n d s , upon performing the algebra: <0| (ot,d-) |0>=2/(2Trr2sino)T 5 d w | B «.(w, m, k) 1 2/ (exp(2irw/K +) • 1) (S , (w,m,k,£) 2-S.Jti,m,k,e) 2) (5. 12a) <0JT» ( i C 9 ) | 0>=2/(2nr2sinfr) ,2. J dw} B_(«,m,k) J 2/ (exp (2nwVK.) +1) w(S ,(w,m,k,&)2+Sx(w,m,k,e)2) (5.12b) <0JT3 (oL,fr) |0>=2/(2-»r2sind) k^ J^dw |B_ (w, m, k) J2/ (exp(2n«v/K-v) +1) X[m(S, (w,m,k,ft.) 2<-s1(«,m,k#B) 2) • (sin»/4) (d/dft-+cot&) (S, (w,m, k,&) 2-S x(w, m,k,&) 2) ] (5.12c) I n t e g r a t i o n over a s p h e r i c a l 2-surface a t l a r g e r g i v e s the t o t a l l o s s r a t e s from the r o t a t i n g black h o l e : 63 dH v,-(*)/dt=0 dE(*g /dt=(2/2T,)Z f dw w|B^(w,n#k) } 2/(exp (2-BVK4)+1) dLCot)/dt=(2/2ir)X J dw m]B.(w,ra,k) I2 / (exp (2 • 1) | 5 . 13) The net number l o s s o f n e u t r i n o s i s zero. T h i s was expected due t o the egua l and o p p o s i t e l o s s r a t e from the two hemispheres. The energy spectrum of emitted p a r t i c l e s i s thermal with temperature K +/2T» The s t a t i s t i c a l f a c t o r (exp(w/T)+1)-* f o r fermions appears, along with a freguency dependent e m i s s i v i t y J B_ (w ,m ,k) J 2 . The e x t r a f a c t o r i n the exponent of 2vmw4|/K+ , i . e . t h e d i f f e r e n c e between w and wH, can be regarded as the 'chemical p o t e n t i a l ' of the d i f f e r e n t m modes with r e s p e c t t o the r o t a t i n g h o r i z o n . E m i s s i o n of p a r t i c l e s with p o s i t i v e angular momentum i s f a v o r e d . This guantum i n s t a b i l i t y i s a s s o c i a t e d with the r o t a t i o n of the bla c k hole. 64 VI. SUflHARY OF QUANTITIES FOB CALCULATION The l o s s r a t e s f o r the Unruh vacuum are reexpressed here t o g e t h e r with the e s s e n t i a l e q u a t i o n s f o r the r a d i a l and angular f u n c t i o n s . The equations are put i n d i m e n s i o n l e s s form using the f o l l o w i n g n o t a t i o n : x=Mw *=a/H z=aw=tfjc y-r/H & = A/M2 x^Mw^ H r = MK^ . X-MC" (6.1) Then one has: 2HW/1C,=2TT(2X[ U H"* 2) - 1 / 2 ]-m «<( 1-^.2, - i / 2 ) =2i rx/K 4 (6.2) The a n g u l a r e g u a t i o n s (2.11) are then w r i t t e n as: [ d/dft * (zsinfr-m/sinfii) ]S, (m, k # z , | ) =kS ^ (m,k,z,A) [ d/d* - (zsinfr-m/sinG) JS t (m ,k,z,* )=-kS ( (m,k,z,fr) (6.3) These c o n s t i t u t e an ei g e n v a l u e problem f o r Sx and S, with k the e i g e n v a l u e which depends on the parameters m and z. S, and S x a r e r e l a t e d to the s p i n weighted s p h e r o i d a l harmonics (Breuer ;1977) $ S * ( z , 0 ) by: S 4(m,k,z,e-)= (sinfr/2) 1 / 2_ vS^(z,&) 65 S v(m,k,z,0) = (sinS / 2 ) g ? l z # ) (6-4) 1 above denumerates the k ei g e n v a l u e s f o r given z and m (l=|m|, |a|+1, . . . ) . The r a d i a l eguations (2.10) i n dime n s i o n l e s s f o r a a r e : [d/dy-i/g(x(y 2+* 2)-m*) ]B , (*,x,m,k ,y) = (k/c ) R % (,<x, m,k, y) [d/dy+i / £(x(y 2+* 2)-m*) ]B f c(*,x, m,k,y) = ( k / c i ^ R , U,x,m,k,y) (6.5) Numerical s o l u t i o n of eqn (6.5) with the asymptotic forms o f egn (2. 16) as boundary c o n d i t i o n s y i e l d s the t r a n s m i s s i o n c o e f f i c i e n t f o r a wave o r i g i n a t i n g a t the past h o r i z o n t o escape t o i n f i n i t y : |B_ (<x, x, m,k) J 2 . The l o s s r a t e s eqn (5.12) are more c o n v e n i e n t l y w r i t t e n i n d i mensionless form as: « d 2 N v _ v (^,6) /dsidt=(1/2rOX f dxj B_(e4x, m,k) J 2 / (exp(2irS/K f ) +1) too X (2/sinW [S, (m,k,*x,6-) 2-S ^ (m, k,«<x^-) 2 ] (6.6a) M d 2 N v ^ (*,&•) /d*Ldt= ( 1 / 2 * ) ^ dxJB_(«<,x,m,k) | 2 / (exp (2nxVft.r) • 1) X (2/sinfr) [ S, (m,k,*ix£ ) 2+S^(m,k,**,£) 2 ] (6.6b) Hd 2E (*,&-) /dJLdt= (1/2*) ^ S^x! B_ (*,x ,m,k) ] 2 / (exp (2»x*/K*) • 1) X (2x/sin6-)[S t (m, k,«uc,d-) 2 * S ^ (m,k,<x^ ) 2 ] (6.6c) 66 MazM^fr) /aj£t={1/2rOr $ dx| B_( *x,m,k) \ V {exp(2-nx/K+) +1) X (2/sin©)[((a-,acsin2H/2)[Sl (m,k^x r&) 2+S Jm, k.ecxA) 2 ] + ksin£s, {mrk,pcxl ' ) S^(a#k,<xJ& ) + (cos 0-/4) [ S, (m,k,*x,&)2-S.^(m,k,eCX<fr)2 33 (6.6d) In the above the t o t a l neutrino plus antineutrino rate d 2N wj.v/dsult has been added. The loss rates integrated over angle are: MdNv_v (<*) /dt = 0 MdS v + 9 (oQ /dt=(1/2ir)T C dxj B.(<,x#m,k) ! 2 / ( e x p ( 2 T r x V ^ . ) + 1) * *>o MdE(<A)/dt= (1/2^)^5' dx xJB-(o<,x,m,k)l2/(exp(2nx/K.O+1) mo MdL(»0/dt= ( V 2 n ) T { dx m|B_(o<,x,m,k) | 2 / ( e x p ( 2 i r x V t L ) * 1 ) (6.7) The advantage of writing the rates i n dimensionless form i s that i t displays the minimum number of parameters needed to characterize the neutrino emission process. The mass does not appear anywhere on the right hand sides.. Thus the equations are v a l i d for any M, as long as one has M » 1 0 _ 5 gram (the Planck mass) so the e f f e c t s of back reaction of the emission process on the metric can be ignored. 67 VII. LOW FREQOENCY ANALYTIC APPROXIMATION When x i s s m a l l both the r a d i a l and angular e g u a t i o n s have a n a l y t i c s o l u t i o n s i n terms of known f u n c t i o n s . This a l l o w s one to c a l c u l a t e the low freguency behaviour of the number c u r r e n t , energy f l u x and angular momentum f l u x . The r a d i a l e g u a t i o n f o r x<<1 has s o l u t i o n s which can be w r i t t e n i n terms of hyper geometric f u n c t i o n s . The r e s u l t i n g t r a n s m i s s i o n c o e f f i c i e n t s 1 8 a r e , from P a g e 1 9 : |B„(m,k,oc,x) |2=[(1-1/2) 1(1*1/2) !/(21) !(21+1) ! ]* t f " L 1* ( x * / < n-V2)fr-,)* J ^ x ) * * - * 1 (7.1) with ft+ =MK+. 1 i s the h a l f i n t e g e r which denumerates the e i g e n v a l u e s k (see egn ( 6 . 3 ) ) . The e x p r e s s i o n above has f r a c t i o n a l e r r o r of order (K+x) 2** 1. The dominant c o n t r i b u t i o n i s from the modes with s m a l l e s t 1. The 1=1/2 and 1=3/2 modes have: JB_(m,k^,o<,x) J2=x* I B . l E , ^ , * , ! ) 1 2= ( W 2 / 9 ) (1-ot 2)x*/4 x « 1 (7.2) 1 8 S t a r o b i n s k y and C h u r i l o v ( 1974) c a l c u l a t e t r a n s m i s s i o n c o e f f i c i e n t s f o r s p i n s=0,1,2. 1 9 Page (1976a) g e n e r a l i z e s t o a r b i t r a r y s p i n s f i e l d i n the K e r r metric. 68 The angular e i g e n f u n c t i o n s can be approximated f o r x « 1 i n terms of s p i n weighted s p h e r i c a l harmonics by a p e r t u r b a t i o n method due t o Press and Teukolsky (1973) . The s p i n weighted s p h e r i c a l harmonics (Abramnowitz,Stegun:1964) $ Y t M ( ^ ) = e x p (-im<p) 5Y*(fr), s a t i s f y : { 1/sin©) (d/d©) (sin©d sY^(&)/d$) - (m2+sz+2mscos^)/sin*» ,Y^(&) =-1(1*1) /*(&) (7.3) They can be w r i t t e n i n terms o f elementary f u n c t i o n s a s : $Ye"v(W=[ ((l*m) ! (1-m) !/(l+s) ! (1-s) !) ( 21* 1) / 4 r ] ** Z (l-s)£1*^ X{-1) *-«•-* ( c o s ( 6 / 2 ) ) 2 * * » - ~ ( s i n ( < s / 2 ) ) 2 4 - 2 r - * + ~ (7.4) with max (0,m-s) <r<min (l-s,l+m). Here we w i l l work with the e i g e n f u n c t i o n s + s^"(z,&) r a t h e r than S t and S A. Egn (6.4) a l l o w s c o n v e r s i o n between the two forms f o r s=1/2. Press and Teukolsky (1973) g i v e the d i f f e r e n t i a l equation f o r t h e s p i n weighted s p h e r o i d a l harmonics j S ^ z , ^ ) : [ (1/sinfr) (d/d&) (sin&d/dfr) -(m 2*s 2*2mscos»)/sin 2 e • (a 2w 2cos 2»-2awscos©.) ] sS*(aw,6 ) = - ^ ( a w ) sS^(aw,©) (7.5) One can write t h i s i n the form: 69 (He+H,) iS^z^)=- sE*(z) $S«*(z,&) (7.6) with zero order e i g e n v a l u e s given by egn (7.3) H o » Y T { » ) = = " & W ( 0 ) sY<T«» (7.7) with 5E*(0) =1 (1 + 1) . The p e r t u r b a t i o n Hamiltonian i s then, from egn (7. 5) : H l=z 2cos 2fl--2zscosA- =-2zscos$ f o r z « 1 (7.8) F i r s t order p e r t u r b a t i o n theory then g i v e s : tS*(z ,e- ) = jl^(») • X^<sl«mlH, Islm>/{1 (1*1)-1« ( l ' * 1 ) ) ,!«?(*) Press and Teukolsky g i v e the matrix element of the p e r t u r b a t i o n Hamiltonian: <slmJH, jslm>=[ (21*1) / (21+1) ]»* (-z) x<l, 1,m,0|l,m><l, 1,-s,0|l,-s> (7. 10) The Clebsch-Gordon c o e f f i c i e n t s on the r i g h t hand s i d e of egn (7.10) can be eval u a t e d by standard but unwieldy formulae { B r i n k , S a t c h l e r : 1 9 7 3 ) . Since t h e s e are zer o f o r 1» not egual 70 t o 1-1,1 or 1 + 1, the sum of eqn (7.9) reduces to three terms. To lowest order i n x, by eqn (7.2), we only need c o n s i d e r the 1=1/2, m=+1/2 or -1/2 modes. For the case m-s=1/2 one f i n d s : *xk«i' = &>*'*zi ( 1 + 1 / 2 ) * 1 + 3 / 2> 3^/(21 + 2) z Si^, wC-^rf £ o r W 1' (7-11) The symmetries, eqn (2.14) ; are used to o b t a i n f o r s=-1/2 or m=-1/2: ^ £ < z , © ) = ^ ( z , * - f t ) (7.12a) ^(2,fr)=- yS v^(-z,iT-6-) (7.12b) ^ S ^ ( z , ^ ) = ^ S^(-z ,6) (7.12c) With eqns (7.9) and (7.11), one has to lowest order i n z: vS^(z,©)= ^ ( t f • < /2V9) ^YJJ(A.) (7. 13) The sY/"(t) can be c a l c u l a t e d from eqn (7.4) , y i e l d i n g : ^<&) = <1/(2i?)sin(6/2) t,X^> V j ^ J s i n (fr/2)cos& (7.14) 71 ^s'£(z,fr)=(1/j2?) sin {&- /2 )£ 1* (2z/9) c o s e ] (7.15) The d i f f e r e n t i a l form of the r a d i a l c u r r e n t , eqn (6.6a), i s given by: Md (d2N v_ 7 (*,&)/djidt) = (dx/2n) ~ JB_(*,x,m,k) J 2/(exp(2Tix/Kl) +1) XT v S~ (iU ,0-) 2 -_«S J W , ») 2 ] (7.16) S u b s t i t u t i o n of eqns (7 . 2 ) , (7.12) and (7.15) i n t o eqn (7.16) y i e l d s the low freguency l i m i t f o r t h e net l o s s of n e u t r i n o s from a r o t a t i n g b l a c k hole: -(dx/2n) (x2/2-n) {[exp (2TT(X-X h/2)/^)*1 ] - i X[cos2 ( » / 2 ) (1-(i»z/9) cosfi) -sin2 (fr/2) ( U (4z/9) cosO) J • [ e x p ( 2 T ( x + X r t / 2 ) / K ) • 11 * X[sin2{&/2) (1-(4Z/9) cosfr) - c o s 2 (*/2) ( U (4z/9) cos«) ]) (7.17) T h i s can be f u r t h e r s i m p l i f i e d i n the l i m i t s of sl o w l y (»t-f0) and r a p i d l y r o t a t i n g black h o l e s ( »c->1). (See eqn (6 .2) f o r the argument o f the e x p o n e n t i a l : 2HX/KV)« For P t « 1 the e x p o n e n t i a l f a c t o r reduces to exp (2Vx/K+ ) • 1= 2+2t(x/Kv-oeC/( W2) i/zj (7.18) 72 Then egn (7.17) becomes (m=*1/2 or m=-1/2): -di{x/2it) 2[uoC/2 (1-* 2) cos** (1+*x/K%) (4Z/9)COS0] or Hd(d 2N v_ v (*,&) /dndt) =-dx (x/2ir) 2 c o s f r ( W 2 ) ( 1 - < 2 ) ^ 2 * « 1 , x « 1 (7.19) For r a p i d l y r o t a t i n g b l a c k h o l e s , ( l - ^ 2 ) */2<<1, the e x p o n e n t i a l f a c t o r f o r m=-1/2 makes the second term i n egn (7.17) v a n i s h i n g l y s m a l l , l e a v i n g : -dx(x/2jr) 2cosP (1-4z/9) or Md (d 2^.,- (*s,^)/df^t)=-dx(x/2n) 2cos©" x « 1, (1-«?) »*<< 1 (7.20) The low freguency dependence on r o t a t i o n , oc, i s l i n e a r i n f o r s m a l l and reaches a c o n s t a n t (in oi ) f o r oC approaches 1. The angular dependence i n the x<<1 l i m i t i s independent of both x and oi and i s given by a simple c o s i n e ($) f a c t o r . T h i s v e r i f i e s t o lowest order i n x the p r e d i c t e d asymmetry i n the n e u t r i n o emission from r o t a t i n g b l ack h o l e s . The asymmetry vanishes as the r o t a t i o n approaches zero, as r e g u i r e d . The r e g i o n of net l o s s of n e u t r i n o s or a n t i n e u t r i n o s around the poles fr=0,5-=n, occupies a l a r g e s o l i d angle (about 1 s t e r a d i a n ) . Neutrinos dominate i n the hemisphere &>"n/2, a n t i n e u t r i n o s i n the hemisphere $.<T{/2. T h i s was expected s i n c e angular momentum i s l o s t from the black hole p a r t i a l l y i n the form of n e u t r i n o s p i n . 73 V I I I . NUMERICAL CALCULATIONS In t h i s s e c t i o n , we b r i e f l y d e s c r i b e the procedure i n v o l v e d i n c a l c u l a t i n g the f u n c t i o n s needed t o o b t a i n the e m i s s i o n r a t e s of a r o t a t i n g b l ack h o l e , as given by egns (6.6) and (6.7). The d e t a i l s of s o l u t i o n of the e q u a t i o n s are presented i n appendix B. l«_Angular E i g e n f u n c t i o n s The second order equations f o r S, and Sz have r e g u l a r s i n g u l a r i t i e s a t fr=0 and &=ir. Due to the symmetry egn (2.14c), the f i r s t order coupled eguations need only be i n t e g r a t e d f o r 0<6-<n/2. The s t a r t i n g values were o b t a i n e d by a power s e r i e s expansion about 6=0. A few e i g e n v a l u e s were found by t r i a l and e r r o r . The e i g e n v a l u e s used i n the f i n a l c a l c u l a t i o n were obtained by a polynomial f i t to s i x t h order i n z=aw (see eqn (6.3)). D.Page s u p p l i e d the t a b l e of c o e f f i c i e n t s f o r the f i t t o e i g e n v a l u e s he has c a l c u l a t e d . A check on these e i g e n v a l u e s was provided by r e q u i r i n g the s l o p e s and v a l u e s o f S, and t o match at fr=T/2, a c c o r d i n g t o egn (2.14c) . The c o n t i n u a t i o n method of Press and Teukolsky(197 3) to c a l c u l a t e S v and S„ and the k e i g e n v a l u e s was a l s o t r i e d . A 74 comparison of r e s u l t s of t h i s with d i r e c t i n t e g r a t i o n using Page's e i g e n v a l u e s (as d e s c r i b e d above) showed t h a t higher accuracy and l e s s computer time were r e q u i r e d when d i r e c t i n t e g r a t i o n was used. The d i r e c t i n t e g r a t i o n method a v o i d s the problem of f i n d i n g the e i g e n v a l u e s which the c o n t i n u a t i o n method s u p p l i e s as p a r t o f the c a l c u l a t i o n . The angular f u n c t i o n s were c a l c u l a t e d and s t o r e d f o r 41 value s of £ from .001 t o i r / 2 . A l l modes f o r which e i g e n v a l u e s were a v a i l a b l e were c a l c u l a t e d f o r aw=0,.2,.4,.6,.8,1. I n a d d i t i o n the modes (l,m) = (5/2,5/2) , (7/2,7/2), (7/2,5/2) and a l l of the 1=9/2 and 1=11/2 modes were c a l c u l a t e d f o r aw=1. 2, 1. 4, 1. 7,2.0, 2. 4, 3.0. T h i s r e s u l t e d i n a t o t a l of 180 (l,m,aw) modes f o r S, and S ^ a t the 41 & values. Cubic s p l i n e i n t e r p o l a t i o n was used t o f i n d S, and S x f o r i n t e r m e d i a t e values of aw. Sample angular e i g e n f u n c t i o n s are p l o t t e d i n appendix A ( f i g . 6). 2. The Transmission C o e f f i c i e n t s The e x p r e s s i o n s f o r the emission r a t e s i n v o l v e the t r a n s m i s s i o n c o e f f i c i e n t : |B„J Z. T h i s i s the p r o b a b i l i t y f o r a wave o r i g i n a t i n g a t the past h o r i z o n t o escape to i n f i n i t y . The wave i s expressed by the asymptotic form, eqn (2. 16). Due t o the r e l a t i o n (2.23b) one can ^ solve f o r iB+1 2 r a t h e r than 75 | B . l 2 . In t h i s case the asymptotic form f o r i n g o i n g Q=+) waves, eqn (2.15), together with t h e r a d i a l e q u a t i o n s (6.5) are to be s o l v e d . F o l l o w i n g a method due to Page (1977), both the dependent v a r i a b l e s (B , and Rx) and the independent v a r i a b l e y are transformed. The f i r s t t r a n s f o r m a t i o n makes the d i f f e r e n t i a l e q u a t i o n s r e a l , the second i s performed t o smooth the f u n c t i o n a l behaviour f o r e a s i e r numerical s o l u t i o n . A p u r e l y i n g o i n g wave at the h o r i z o n (r=r+) i s i n t e g r a t e d outward, then r e s o l v e d i n t o i n g o i n g and outgoing components a t l a r g e r a d i u s . |B + J 2 i s the r a t i o of i n t e n s i t y of the ingoing wave near r + to t h a t of the i n g o i n g wave at l a r g e r. The t r a n s m i s s i o n c o e f f i c i e n t s were c a l c u l a t e d f o r most o f 21 (l,m) modes and f o r each of the v a l u e s of oC s e l e c t e d . For each of these, the choice of values of the d i m e n s i o n l e s s freguency x, was decided by the f o l l o w i n g requirement: the t o t a l number r a t e ( n e u t r i n o s p l u s a n t i n e u t r i n o s ) per mode: « d N v ^ v ( ^ ^ , k ) / d t = (1/2ir) f dx|B-(.<,x,m, k) | 2/(exp(2tjx/K \0+1) (8.1) be a c c u r a t e to a t l e a s t one p a r t i n TO*. T h i s r e q u i r e d from 9 t o i n excess of 40 values of x per i n t e g r a l depending on m,l and «t (on average about 13 values were required) . The double p r e c i s i o n numerical i n t e g r a t i o n r o u t i n e OBC DDE2<> was used t o 76 s o l v e the r a d i a l equations f o r 1B>|2 with f r a c t i o n a l and a b s o l u t e e r r o r c r i t e r i a of 10-*. The v a l u e s o f J 2 / ( e x p (2nx"/K^.) *1) were saved and used i n the e v a l u a t i o n o f the other i n t e g r a l s i n eqns (6.6) and (6.7). 3, The Emission Bates These were c a l c u l a t e d u s i n g the s t o r e d v a l u e s of the f u n c t i o n \ B+ J z / ( e x p {2tiz/k+) * 1) . The s t o r e d values o f the angular e i g e n f u n c t i o n s were i n t e r p o l a t e d t o o b t a i n t h e i r v a l u e s at the r e q u i r e d values of z=o6c. A check on the r e s u l t s was provided by comparison with a l i s t of t o t a l number rate H dN-^y'dt and energy emission r a t e M 2 dE/dt ( i e egns (8.1) and (6.7c)) c a l c u l a t e d by D.Page. 2 « 0 B C Computing Centre Documentation: 0 B C- D D E (Subject Code 43.2) 77 IX. RESULTS AND DISCUSSION The r e s u l t s o f the c a l c u l a t i o n s are summarized by the f i g u r e s i n appendix & and r e f e r e n c e to these f i g u r e s w i l l be e s s e n t i a l t o the f o l l o w i n g d i s c u s s i o n . The v a l u e s f o r the r a d i a l number c u r r e n t , energy c u r r e n t , e t c , depend on a number of f a c t o r s . The prime f a c t o r i s the F e r m i - D i r a c s t a t i s t i c a l f a c t o r : [ exp (2tT(x-mx*) /£*.) +1 ]~» (9.1) T h i s f a c t o r g i v e s the emi s s i o n i t s c h a r a c t e r i s t i c thermal p r o p e r t i e s (the temperature, i n n a t u r a l u n i t s , i s K » / 2 T ) . I t i s p l o t t e d i n f i g u r e 2a together with I B J 2 f o r b l a c k hole r o t a t i o n •<=. 8 and angular mode (1, m)= (3/2,3/2). I t decreases e x p o n e n t i a l l y f o r x>mxH with an e - f o l d i n g r a t e of 2ir/K +. K+ i s very s e n s i t i v e to when * i s near u n i t y . Thus t h e t r a n s i t i o n toward z e r o i s much f a s t e r as o< approaches 1. The az i m u t h a l guantum number m i s of prime importance here. For m<0 the Fermi-Dirac f a c t o r i s w e l l on i t s way t o z e r o f o r x=0. Since the f a l l o f f to zero occurs around x=nx H ( f o r a>0) , the c u t o f f i n the spectrum i s l i n e a r i n m. F i g u r e 2a a l s o d i s p l a y s t y p i c a l behaviour of the t r a n s m i s s i o n c o e f f i c i e n t |B +| 2. F o r low freguency, the wave does not have enough energy t o surmount the g r a v i t a t i o n a l - angular momentum 78 b a r r i e r . At high f r e q u e n c i e s the t r a n s m i s s i o n c o e f f i c i e n t approaches u n i t y . The t r a n s i t i o n r e g i o n i s f o r x approximately l x M . The p o t e n t i a l i s complicated enough t h a t t h i s i s only a rough estimate. The product of the two f a c t o r s , i e : a*N (*,*)/dwdt = (V2n) |B 4 (<^x,m,l) |*/(exp (2T»X/KV) + 1) (9.2) has been c a l c u l a t e d f o r a l l the angular modes over a range o f f r e g u e n c i e s . In p a r t i c u l a r f o r ^=.8 , (1,m)=(3/2,3/2) the product i s g i v e n i n f i g 2b ( f o r comparison with f i g 2a). F u r t h e r r e s u l t s are d i s p l a y e d f o r black hole angular momentum parameters K=.3 ( i n f i g u r e 3) and<*=.999 ( i n f i g u r e 4 ) . A l l the important modes, i n t h e i r c o n t r i b u t i o n t o the t o t a l spectrum, are p l o t t e d . Note that f o r the l a r g e r value of many more a n g u l a r modes are important. As expected from the d i s c u s s i o n of f i g u r e s 2a and 2b above, the m= 1 modes a r e most important f o r O O . The t o t a l spectrum summed over angular modes i s p l o t t e d i n f i g u r e 5a: d 2 N ^ v (*,»)/dwdt)= (V2-n)5 |B^(»C,x,m,l) | */(ex p (2TTX/K*) • 1) (9.3) 79 T h i s i s the t o t a l number l o s s per u n i t time per u n i t frequency i n the r a d i a l d i r e c t i o n ( i e i n t e g r a t e d over & and <f>). The power spectrum, f i g u r e 5b, i s o b t a i n e d by m u l t i p l y i n g t h i s by x (see egn (6.7c)) and l o o k s s i m i l a r but with the high frequency part emphasized. The spectrum was c a l c u l a t e d f o r o£=0,.01,.3,.5,.7,.8,.9,.95,.99,.999 and shows a smooth p r o g r e s s i o n throughout the range o£=0 t o oc=„999. Forcx=.5 a s l i g h t h i n t of the h i g h e r modes (l=m>1/2) f i r s t appears. As <* reaches 0(=.99, modes up t o l=m=11/2 are c l e a r l y v i s i b l e i n the spectrum. They appear as d i s t i n c t s p i k e s near x = l x H as expected. T h i s i s due to the sharp c u t o f f at mxM o f the Fermi-Dirac s t a t i s t i c a l f a c t o r and t h e sharp r i s e of J B V J 2 f o r x > l x w as noted f o r f i g u r e s 2a and 2b. The o s c i l l a t i o n s w i t h i n the s p i k e s are due t o t h e freguency v a r i a t i o n o f |B+l 2. The f i r s t (1=1/2) peak i n the spectrum reaches a maximum s t r e n g t h f o r <*• approximately e q u a l to .8 and d e c l i n e s f o r l a r g e r oc . Also f o r *.>. 8 the higher modes c o l l e c t i v e l y dominate the lowest mode (1=1/2) i n terms of t o t a l power e m i t t e d . The angular dependence of the emission r a t e s i n v o l v e s t h e a d d i t i o n a l f a c t o r of the angular e i g e n f u n c t i o n s S, and S^. These a r e p l o t t e d i n f i g u r e 6 f o r the 1=5/2 and 1=3/2 modes. The parameter z=aw i n f i g u r e 6 takes on values of 0,.4,1,2,3 as i n d i c a t e d . The frequency dependence of the f a c t o r : 80 (1/2IT) |B +(cC,x,a,k) ! 2 / ( e x p (2nx7K\)*1) (9.4) ensures t h a t z=«Cx ranges o n l y over s m a l l values (eg see f i g u r e 3 f o r a t y p i c a l case and f i g u r e 4 f o r the extreme case oL=,999). The behaviour of S, and S t i s t y p i c a l of angular e i g e n f u n c t i o n s with t h e number of z e r o s (other than the pole) b e i n g l-|mj. S t and S a are not very s e n s i t i v e t o the parameter z and t h i s s e n s i t i v i t y decreases with i n c r e a s i n g 1 ( i e the l=1/2,m=1/2 modes are most s e n s i t i v e t o z ) . The angular modes with l a r g e r 1 are c o n c e n t r a t e d more towards the eguator. F i g u r e s 2 through 5 and the r e l a t e d d i s c u s s i o n emphasized the a s s o c i a t i o n of h i g h e r energy n e u t r i n o s with l a r g e r l,m modes, p r i m a r i l y due to the Fermi-D i r a c f a c t o r . Thus the power spectrum w i l l not appear l i k e f i g u r e 5b a t any g i v e n angle but be dominated by lower e n e r g i e s near the p o l e and higher e n e r g i e s near the eguator. The r a d i a l number c u r r e n t H d2H>_9 (<*.,&) /dJldt , i s p l o t t e d i n f i g u r e 7 f o r a r e p r e s e n t a t i v e c h o i c e of the v a r i o u s values of ot c o n s i d e r e d . The n e u t r i n o emission r a t e i s g r e a t e s t a t t h e poles f o r black hole a n g u l a r momentum up to c<=. 8, then i s s h i f t e d toward the eguator f o r higher * . T h i s can be e x p l a i n e d as f o l l o w s : the main l o s s mechanism f o r a n g u l a r momentum from the b l a c k hole i s spin angular momentum o f the emitted n e u t r i n o s f o r s m a l l o r b i t a l a ngular momentum of n e u t r i n o s f o r l a r g e oc. 81 The t o t a l number l o s s r a t e i n one hemisphere i n c r e a s e s monotonically with The d e t a i l e d behaviour of the l o s s r a t e can be understood with r e f e r e n c e to f i g u r e s 8a, 8b, and 8c which give the dominant modes c o n t r i b u t i n g t o the l o s s r a t e f o r the cases ofc=.1, .5 and ,999. Only the m=1/2 and -1/2 modes gi v e a nonzero c o n t r i b u t i o n at the po l e s . T h i s can be shown from a power s e r i e s expansion of S/ and around £=0 o r &=tr (see appendix B) , and t h e f a c t t h a t S , 2 / s i n f l and S ^ / s i n ^ appear i n the formulae (6.6) f o r the l o s s r a t e s . For ot=.1 there i s a s i g n i f i c a n t c a n c e l l a t i o n between m>0 and m<0 modes (the c a n c e l l a t i o n i s complete f o r <=0). , N e u t r i n o s are emitted i n the m<0 modes near the &=0 p o l e , and a n t i n e u t r i n o s i n t h e m>0 modes. For »C=.5 the m<0 modes are s t r o n g l y suppressed by the Fermi-Dirac f a c t o r . For the ec=.999 case the m<0 modes are completely n e g l i g i b l e , as are a l l modes f o r which m i s not equal t o 1. The predominance of l=m modes was noted i n s e c t i o n V I I i n c o n s i d e r i n g the low freguency behaviour of the t r a n s m i s s i o n c o e f f i c i e n t s . Hodes with 1 as l a r g e as 11/2 are now s i g n i f i c a n t i n t h e i r c o n t r i b u t i o n . The emis s i o n from the p o l a r r e g i o n s no l o n g e r dominates. The angle o f peak emission i n c r e a s e s with f o r ot>.8 as l a r g e r 1 modes become important (see f i g u r e 7 ) . The behaviour of the angu l a r e i g e n f unctions i s r e s p o n s i b l e f o r t h i s as t h e i r peaks s h i f t toward the equator with l a r g e r 1. The other p h y s i c a l q u a n t i t i e s of i n t e r e s t (emission 82 rat e s ) as given by egn (6.6) are p l o t t e d i n f i g u r e s 9, 10 and 11. The t o t a l number r a t e of n e u t r i n o s p l u s a n t i n e u t r i n o s ( f i g u r e 9) i s independent of angle for«C=0 (as i t must be due t o the s p h e r i c a l symmetry) . I t i n c r e a s e s s t e a d i l y with od, but r i s e s f a s t e r a t the equator than at the pole due t o the i n c r e a s i n g importance o f higher angular modes (l>1/2). Again f o r o<>.8 the lowest (1=1/2) mode i s suppressed so t h a t e m i s s i o n from the p o l e s decreases. The power d i s p l a y s s i m i l a r behaviour. One notes the evidence of the hardening of the spectrum with i n c r e a s i n g o^; the r a t i o of power ( f i g u r e 10) t o t o t a l number r a t e ( f i g u r e 9) i n c r e a s e s from about . 18 fo r e d = 0 t o about .8 f o r * . = .999 (see a l s o f i g u r e 5 ) . This r a t i o i s j u s t the average energy per emitted p a r t i c l e . The angular momentum l o s s r a t e i s d i s p l a y e d i n f i g u r e 11. The c h a r a c t e r i s t i c behaviour near the poles i s understood i n l i g h t of the p r e v i o u s d i s c u s s i o n s on the number l o s s r a t e , both net and t o t a l , and the power. Near the equator though, the l o s s r a t e of angular momentum i s neg a t i v e . From eqn (6.6) one sees t h a t when 1 i s l a r g e (these modes w i l l be more important f o r l a r g e r ot) and i s near the equator the c r o s s term ksin&S, S z i s dominant. S, i s of opposite s i g n as near the equator r e s u l t i n g i n the l o s s of n e q a t i v e angular momentum around the equator. O v e r a l l the i n t e g r a t e d l o s s remains p o s i t i v e , so that the r o t a t i n g b l a c k hole s p i n s down. 83 i e d e c r e a s e s 2 * . The l a s t f i g u r e , f i g 12, d i s p l a y s the asymmetry i n n e u t r i n o emission from the black hole f o r v a r i o u s v a l u e s of I t i s i d e n t i c a l l y z e r o , as r e g u i r e d , f o r a n o n r o t a t i n g black hole. However, i t i s s i g n i f i c a n t even f o r sm a l l e r v a l u e s . The asymmetry extends over a r e g i o n of or d e r one s t e r a d i a n i n angular extent around each pole and very c l o s e l y obeys the c o s i n e dependence found i n the low frequency a n a l y t i c approximation. X . SUMMARY The g e n e r a l l y c o v a r i a n t n e u t r i n o f i e l d e q u a t i o n s have been expressed e x p l i c i t l y f o r the Kerr m e t r i c , which d e s c r i b e s the g r a v i t a t i o n a l f i e l d of a r o t a t i n g black hole. The second q u a n t i z a t i o n procedure was c a r r i e d out to o b t a i n e x p r e s s i o n s f o r the number, energy, and angular momentum c u r r e n t s . The vacuum e x p e c t a t i o n v a l u e s were taken i n the Unruh vacuum s t a t e . T h i s s t a t e corresponds to the b l a c k hole e v a p o r a t i o n vacuum s t a t e o f Hawking f o r black holes formed by s t e l l a r c o l l a p s e . 2 * To show t h i s , one must demonstrate t h a t the a n g u l a r momentum l o s s i s f a s t e r than the mass l o s s so t h a t a/M decreases. Page(1976b) has done t h i s . 84 Asymptotic s o l u t i o n s to the r a d i a l e q uations f o r the n e u t r i n o f i e l d were used t o w r i t e the r a d i a l components of these c u r r e n t s f o r l a r q e r i n terms of t r a n s m i s s i o n c o e f f i c i e n t s . A low frequency a n a l y t i c approximation was c a r r i e d out t o o b t a i n a p r e l i m i n a r y estimate o f the number l o s s from the black hole as a f u n c t i o n of p o l a r angle & . Next numerical c a l c u l a t i o n s were performed t o o b t a i n the f u l l r e s u l t s on number, energy, and angular momentum l o s s f o r r o t a t i n g b l a c k holes of any mass and of any of s e v e r a l values of angular momentum. The c a l c u l a t i o n s r e v e a l a marked d e v i a t i o n from s p h e r i c a l l y symmetric emission { about 50 percent f o r J/H 2 of . 3 ) . N e u t r i n o s are emitted p r i m a r i l y from one p o l e and a n t i n e u t r i n o s from t h e other p o l e of the a x i s of r o t a t i o n . The emission r a t e s i n c r e a s e s t e a d i l y as J / H 2 i s i n c r e a s e d andj f o r J/H 2>.8, the peak of n e u t r i n o emission s h i f t s away from the p o l e s due to predominance o f n e u t r i n o s with high o r b i t a l a n gular momentum. Fig u r e 13 s c h e m a t i c a l l y summarizes the emission process f o r n e u t r i n o s by r o t a t i n g b lack holes. At the top i s given the thermal Fermi-Dirac f a c t o r where Si and T=K^/2-K are the angular v e l o c i t y and temperature of the black h o l e . J i s the angular momentum of the black h o l e . E i s the n e u t r i n o energy and j'Jl/TL i s the component of n e u t r i n o a n g u l a r momentum along t 85 t h e black hole r o t a t i o n a x i s . C and p are the s p i n and ( l i n e a r ) momentum of the p a r t i c l e . The diagram emphasizes the predominant l o s s of a n t i n e u t r i n o s (which have p o s i t i v e h e l i c i t y , fi*p>0) from the upper hemisphere, and n e u t r i n o s (which have n e g a t i v e h e l i c i t y ) from the lower hemisphere of the r o t a t i n g black h o l e . The asymmetry i n n e u t r i n o emission i s important f o r the e n v i r o n s of a r o t a t i n g black hole o n l y when the e m i s s i o n i s i n t e n s e enough. For J/M 2=.5 the i n t e g r a t e d power e m i s s i o n i s a b o u t 2 2 4 x 1 0 _ s / f l 2 i n Planck u n i t s . The mean energy per p a r t i c l e i s a b o u t 2 3 . 2 / f i . For a s o l a r mass (fl of 1 0 3 8 ) black hole t h i s y i e l d s a t o t a l r a t e o f 1 0 _ 1 * erg/sec and a p a r t i c l e energy of 10-* 7 Rev. T i n y black holes (H<102 o=2x10 1 5 grams) r e s u l t i n a l a r g e e m i s s i o n power (>10* 7 erg/sec) and p a r t i c l e energy (>10Hev). These could have been formed o n l y i n the e a r l y s t a g e s o f the u n i v e r s e when the energy d e n s i t y was extremely h i g h , and would have completely evaporated by now 2*. However the r e s u l t a n t e f f e c t s of the n e u t r i n o emission on the hot high d e n s i t y medium c o u l d have s u r v i v e d u n t i l the p r e s e n t time. 2 2 T h i s can be estimated from f i g . 10: f o r oc=0 and .8 the power i s about 4TTX10- 5/M 2 times .7 and 1.7, r e s p e c t i v e l y . 2 3 From f i g . 5 b the mean energy f o r #=0 and .8 i s . 18/H and .3/M, r e s p e c t i v e l y . See a l s o f i g . 5 a - the tf=.5 curve i s much c l o s e r to t h a t f o r ei=Q. 2 * The l i f e t i m e of an e v a p o r a t i n g black hole i s roughly mass/power or fl3. See egn (7) of chapter 1. 86 Consider the early universe when the temperature was high enough that nucleons and antinucleons coexist. The cross-sections for neutrinos on nucleons and f o r neutrinos on antinucleons are d i f f e r e n t . Thus the neutrino beams from primordial rotating black holes would have pushed matter and antimatter apart. This might explain why the universe l o c a l l y seems to have matter but no antimatter. The next chapter i s devoted to examining t h i s mechanism. 87 3. HATTER ANTIMATTER SEPARATION IH yHE_EABLY UNIVERSE BY ROTATING BLACK HOLES I. INTRODUCTION In t h i s chapter a mechanism i s s t u d i e d whereby e v a p o r a t i n g black h o l e s i n the e a r l y universe push matter and a n t i m a t t e r a p a r t . Others have s t u d i e d the p o s s i b i l i t y of a c h i e v i n g separate r e g i o n s of matter and a n t i m a t t e r , i n an universe with i n i t i a l l y e g u a l amounts of matter and a n t i m a t t e r , u n i f o r m l y d i s t r i b u t e d . The a s s o c i a t e d models o f t h e e a r l y u n i v e r s e are r e f e r r e d to as baryon symmetric cosmologies. Baryon Symmetric Cosmologies A baryon symmetric cosmology has two b a s i c f a c t s t o e x p l a i n . The f i r s t i s the presence of matter, and v i r t u a l l y no a n t i m a t t e r , to a d i s t a n c e a t l e a s t as f a r away as the next galaxy and probably as f a r as the next c l u s t e r of g a l a x i e s . T h i s f a c t i s a s c e r t a i n e d from the observed l a c k o f gamma r a d i a t i o n which would be c h a r a c t e r i s t i c of matter a n t i m a t t e r a n n i h i l a t i o n . T h i s s e t s lower l i m i t s to the d i s t a n c e of the 88 n e a r e s t a n t i m a t t e r (Steigman, 1976) . , The second f a c t to e x p l a i n i s the value of the baryon to photon r a t i o , which i s known to be 10~ 9 ( w i t h i n an order of magnitude)., T h i s r a t i o i s j u s t the number of baryons d i v i d e d by the number of photons f o r the u n i v e r s e ( i . e . , f o r a l a r g e enough volume t h a t the inhomogeneities- s t a r s and g a l a x i e s - have been averaged o v e r ) . The i n v e r s e of the baryon t o photon r a t i o i s a measure of the entropy o f the universe or the amount of d i s s i p a t i o n which has taken pl a c e . A S an i l l u s t r a t i o n c o n s i d e r an i n i t i a l s t a t e with a number d e n s i t y n(1 + f ) / 2 of baryons and n ( 1 - f ) / 2 of a n t i b a r y o n s ( 0 < f « 1 ) . A f t e r a n n i h i l a t i o n , there w i l l be nf baryons and n(l'-f) photons ( f o r sake of argument take 2 photons per a n n i h i l a t i o n ) . Then f i s the f i n a l baryon to photon r a t i o as w e l l as the i n i t i a l f r a c t i o n a l baryon asymmetry, flore w i l l be s a i d below when we d i s c u s s F r i e d a a n n models, as to why the baryon t o photon r a t i o i s an important g u a n t i t y f o r the u n i v e r s e and why i t remains very n e a r l y constant d u r i n g the expansion. A l s o r e l a t e d t o the g u e s t i o n of the o r i g i n of the baryon- a n t i b a r y o n asymmetry of the u n i v e r s e are some recent developments i n grand u n i f i e d t h e o r i e s f o r elementary p a r t i c l e p h y s i c s . In these t h e o r i e s , a massive (of o r d e r 10* 5 GeV) gauge p a r t i c l e , c a l l e d the X-boson, mediates t r a n s i t i o n s between guarks and l e p t o n s , and thus r e s u l t s i n baryon number v i o l a t i o n . The p r e d i c t e d proton decay l i f e t i m e i s of order 89 1 0 3 1 years. However, e a r l y i n the u n i v e r s e , a t high enough temperature, the lepto-guark t r a n s i t i o n s would be r a p i d . For s u c c e s s f u l baryon number ge n e r a t i o n , baryon nonconservation, CP v i o l a t i o n , and a p e r i o d o f d i s e g u i l i b r i u m are a l l r e g u i r e d ( f o r d e t a i l s see e.g. Weinberg, 1979). The l a t t e r i s p r o v i d e d by the r a p i d expansion of the u n i v e r s e , which a l l o w s the X-bosons to decay i n t o unegual numbers of baryons and a n t i b a r y o n s , while i n h i b i t i n g the i n v e r s e r e a c t i o n s . A baryon t o photon r a t i o of 1 0 - 8 t o 10-* 3 (Hanopoulos and Weinberg, 1979) , c o n s i s t e n t with the observed v a l u e , i s obtained with reasonable e s t i m a t e s f o r the p r o p e r t i e s of the X-boson. Yoshimura ( 1978) was the f i r s t t o c o n s i d e r t h i s mechanism f o r baryon number ge n e r a t i o n . Barrow (1979) r e c e n t l y has c o n s i d e r e d the e f f e c t s an a n i s o t r o p i c expansion o f the u n i v e r s e would have, to put l i m i t s on the mass o f the X-boson and on the time of i s o t r o p i z a t i o n of the expansion. L a r g e l y as a r e s u l t of the above developments, i n t e r e s t has waned i n baryon symmetric cosmologies i n which the matter and a n t i m a t t e r are s p a t i a l l y s e p a r a t e d by some mechanism 1. One of the most ambitious of the baryon symmetric cosmologies i s that due to Omnes (1972). Omaes co n s i d e r e d matter a n t i m a t t e r s e p a r a t i o n due to a phase t r a n s i t i o n i n the high d e n s i t y , high temperature n u c l e a r matter and r a d i a t i o n 1 Thus these cosmologies remain baryon symmetric f o r a l l time. 90 present b e f o r e the end of the hadron e r a 2 of t h e e a r l y universe. He i s one of few t o c l a i m success i n a c h i e v i n g s e p a r a t i o n on a s u f f i c i e n t s c a l e (the requirements w i l l be d i s c u s s e d below). By n e c e s s i t y , h i s model depends on numerous p h y s i c a l processes, which may not proceed as he d e s c r i b e s 3 , and thus i s q u i t e s p e c u l a t i v e . The p h y s i c a l processes o c c u r i n g d u r i n g t h e e a r l y stages of the u n i v e r s e are understood only very roughly, yet the d e t a i l s a r e necessary t o determine the e v o l u t i o n of the matter and a n t i m a t t e r r e g i o n s t i l l the present time. The model presented i n t h i s chapter i s a l s o of t h i s s p e c u l a t i v e nature. However a negative r e s u l t i s o b t a i n e d here- the matter i n the present u n i v e r s e cannot be e x p l a i n e d as r e s u l t i n g from a baryon symmetric i n i t i a l s t a t e due t o black hole e v a p o r a t i o n s . The s p e c u l a t i v e nature of the p h y s i c s i n v o l v e d does not i n v a l i d a t e t h i s c o n c l u s i o n , s i n c e the most f a v o r a b l e assumptions f o r the s u c c e s s of the model were chosen. However i t d i d mean t h a t the achievement of a d e f i n i t e n e gative r e s u l t was more d i f f i c u l t . Matter and a n t i m a t t e r are separated by n e u t r i n o c u r r e n t s from e v a p o r a t i n g b l a c k h o l e s . The s c e n a r i o i s as f o l l o w s . 2 The hadron e r a i s t h e p e r i o d during which the temperature i s g r e a t e r than the hadron (e.g. proton) r e s t mass. E.g. Harrison,1973c and Weinberg,1972 d i s c u s s the thermal h i s t o r y of the e a r l y u n i v e r s e . 3 Due to a s e r i e s of c r i t i c a l papers by Steigman (reviewed i n Steigman,1976) the Omnes model has been discounted. 91 D e n s i t y f l u c t u a t i o n s i n a r o t a t i n g medium i n the extremely e a r l y stages of the b i g bang (of order 1 0 - 3 1 seconds a f t e r the i n i t i a l s i n g u l a r i t y ) w i l l r e s u l t i n the f o r m a t i o n of r o t a t i n g b l a c k h o l e s . These t i n y black h o l e s (of order 4x10 7 grams i n mass) evaporate completely by the end of the hadron e r a (which occurs at about* 10-* sec) , e m i t t i n g i n t e n s e beams of high energy n e u t r i n o s . The n e u t r i n o s s c a t t e r baryons more s t r o n g l y than antibaryons and thus can push matter and a n t i m a t t e r a p a r t . Opposite i n d i r e c t i o n t o the n e u t r i n o beam i s an a n t i n e u t r i n o beam o f e q u a l i n t e n s i t y , which s c a t t e r s a n t i b a r y o n s more s t r o n g l y . The s i t u a t i o n i s i l l u s t r a t e d by f i g u r e 14 (appendix A). The long arrows r e p r e s e n t the n e u t r i n o , a n t i n e u t r i n o , baryon, and ant i b a r y o n c u r r e n t s , whereas the short arrows r e p r e s e n t the r o t a t i o n axes o f the ev a p o r a t i n g black holes. The i n d i v i d u a l r o t a t i n g black h o l e s each emit n e u t r i n o s and a n t i n e u t r i n o s as i l l u s t r a t e d by f i g u r e 13. When the n e u t r i n o c u r r e n t s cease and the hadron e r a ends ( i . e . , the temperature i n the e a r l y u n i v e r s e drops below about 1Gev, the proton mass) , any baryons and a n t i b a r y o n s remaining mixed together w i l l a n n i h i l a t e . Subsequently there w i l l be separate r e g i o n s of pure baryons or pure a n t i b a r y o n s whose c h a r a c t e r i s t i c s i z e and d e n s i t y depends on the i n i t i a l * A l l times are measured from the i n i t i a l s i n g u l a r i t y at t=0. black hole d i s t r i b u t i o n . These r e g i o n s w i l l expand and c o o l , as i n the standard b i g bang model, and p o s s i b l y form g a l a x i e s , s t a r s , e t c . The above mechanism p r o v i d e s a technique of c o n v e r t i n g r o t a t i o n a l inhomogeneities i n t o matter a n t i m a t t e r inhomogeneities. Hodel For The E a r l y O n iverse We nos d e s c r i b e the model of the universe t h a t i s used here, ft Friedmann models f o r the big bang expansion i s assumed to be a v a l i d d e s c r i p t i o n f o r the average p r o p e r t i e s o f the u n i v e r s e . These p r o p e r t i e s i n c l u d e the average energy d e n s i t y , the s c a l e f a c t o r , the temperature, and the number d e n s i t y of p a r t i c l e s . A Friedmann model i s a homogeneous, i s o t r o p i c s o l u t i o n to the E i n s t e i n f i e l d eguations t o g e t h e r v i t h an equation o f s t a t e f o r the matter. The spacetime m e t r i c f o r the Friedmann universe has the form o f the fiobertson-Walker m e t r i c : d s 2 = a t 2 - R ( t ) 2 ( d r 2 / ( 1 - k r 2 ) + r 2 d 5 2 + r 2 s i n 2 f r d ^ 2 ) (1.1) k= + 1, 0, and -1 are the c l o s e d , f l a t , and open models, r e s p e c t i v e l y . The dynamical equation f o r fi(t), r e s u l t i n g from 5 Byan and Shepley(1975) d i s c u s s the g e n e r a l p r o p e r t i e s of Friedmann models. 93 E i n s t e i n ' s e quations (see eqn (1), chapter 1), i s : (dB/dt) z«-k=8Y>fi2/3 (1.2) The e f f e c t of k i s n e g l i g i b l e when the uni v e r s e was younger than about 1/200th of i t s present age (Weinberg, 1972). For our u n i v e r s e , the o b s e r v a t i o n a l data i s s t i l l i n s u f f i c i e n t to determine k. The equation o f c o n s e r v a t i o n of energy (from c o n s e r v a t i o n of the energy momentum tensor) i s : (d/dB) lpX*)=-3?R2 (1.3) With an equation o f s t a t e , t h i s determines f as a f u n c t i o n o f B. The eguation o f s t a t e i s here taken as p-jf^ $ with ^=1/3 f o r r e l a t i v i s t i c p a r t i c l e s (or photons) and # = 0 f o r p r e s s u r e l e s s dust (e.g., c o l d matter or b l a c k h o l e s ) . Here p i s the energy d e n s i t y of matter and p i t s pressure. F o r times before the end of the hadron e r a , the strong i n t e r a c t i o n can d r a s t i c a l l y modify the eguation of s t a t e . Carr(1975) d i s c u s s e s t h i s problem. Since no d e f i n i t e knowledge d i c t a t e s the eguation o f s t a t e , we assume p=/d/3 f o r the p a r t i c l e component ( i . e . , a l l but t h e black h o l e s ) . The baryon t o photon r a t i o i s cons t a n t d u r i n q the 94 expansion*. , The s t a n d a r d model, and our model f o r the u n i v e r s e t h a t c o n t a i n s black h o l e s , has an i d e a l gas (the baryons) i n e q u i l i b r i u m v i t h r a d i a t i o n . The pressure and energy d e n s i t y are g i v e n by: p=nT+ (1/3) gaT* (1.4) >^=nm+ (r-1) - 1nT*gaT*4yO k l x n i s the p a r t i c l e number d e n s i t y , a i s the mean p a r t i c l e mass, a i s the r a d i a t i o n c o n s t a n t , r i s the s p e c i f i c heat r a t i o 7 , and g i s the number 8 of r e l a t i v i s t i c p a r t i c l e s p e c i e s . The b l a c k hole energy d e n s i t y yo^ , i s g i v e n by eguation (2.7b) belov and s c a l e s as 1/B3 so t h a t i t does not e f f e c t eqn (1.3), compared to the case when th e r e are no black h o l e s . The p a r t i c l e number d e n s i t y a l s o s c a l e s as 1/R3. S u b s t i t u t i n g egn (1.4) i n t o the c o n s e r v a t i o n e g u a t i o n (1.3) y i e l d s : dT/T=-(dE/E) (u+1) / (u+1/3 (r-1)) (1.5) where u i s the s p e c i f i c entropy per p a r t i c l e : u=4gaT 3/3n. For 6 T h i s argument p a r a l l e l s t h a t o f Weinberg, 1972. 7 5/3 f o r a monatomic gas l i k e hydrogen. 8 Times s t a t i s t i c a l f a c t o r s of 7/16 f o r each fermion h e l i c i t y s t a t e and 1/2 f o r each boson h e l i c i t y s t a t e , so t h a t f o r example photons c o n t r i b u t e 1 t o g. 95 u « 1 , eqn (1.5) g i v e s T p r o p o r t i o n a l to E - 3 ^ - 1 ^ which i s j u s t the a d i a b a t i c law f o r an i d e a l gas: T V r - * = constant. F o r u » 1 , one has T p r o p o r t i o n a l to 1/B, so that u remains c o n s t a n t i n time. Using the r e l a t i o n f o r number d e n s i t y of photons i n thermal r a d i a t i o n (g=1) : ny=3.7aT 3, and u s i n g a baryon number d e n s i t y of ng=1.8n ( i . e . about 27 percent Helium by mass), u i s r e l a t e d to the baryon to photon r a t i o ^ , by: u=Q.67g/V£. The constant of p r o p o r t i o n a l i t y w i l l vary with time. Before p o s i t r o n - e l e c t r o n a n n i h i l a t i o n when r e l a t i v i s t i c e* and e~ are abundant, g w i l l have the value 1*4 (7/16) =2.75, and e a r l i e r yet g w i l l be l a r g e r due t o the presence of more s p e c i e s of r e l a t i v i s t i c p a r t i c l e s . For our u n i v e r s e u i s l a r g e , of order 10 8 t o 10», so t h a t the baryon t o photon r a t i o i s constant i n time. Sharp changes by f a c t o r s of order u n i t y w i l l occur when the v a r i o u s r e l a t i v i s t i c p a r t i c l e s p e c i e s a n n i h i l a t e and produce more photons'. The baryon to photon r a t i o i s constant only i f the r e l a t i o n n gB 3=n 8 f t B d 3 holds f o r the p a r t i c l e s ( t h i s was used i n o b t a i n i n g (1.5)). l e i n t e r p r e t n & a s the l o c a l value of baryon number d e n s i t y minus a n t i b a r y o n number d e n s i t y , so t h a t n^B 3 i s constant a f t e r s e p a r a t i o n . Thus the baryon t o photon r a t i o i s c o n s t a n t i n the model here only a f t e r the s e p a r a t i o n of matter and matter. • The l a s t which do so are the e + - e - a t a cosmic time o f about 1 second. 96 Black Holes I n The E a r l y Universe He now c o n s i d e r the b l a c k hole formation p r o c e s s . Nothing d e f i n i t e i s known about the u n i v e r s e a t the e a r l y times (of order 1 0 ~ 3 1 sec) r e l e v a n t f o r f o r m a t i o n of the t i n y b l a c k holes r e q u i r e d (mass of order 4x10 7 g) . The elementary p a r t i c l e p h y s i c s determines the e g u a t i o n of s t a t e which determines the dynamical e v o l u t i o n of the Friedman expansion. Here, f o r s i m p l i c i t y and l a c k of a b e t t e r c h o i c e , a hard eguation of s t a t e ( a p p r o p r i a t e t o r a d i a t i o n ) i s assumed. Carr(1975) d i s c u s s e s the case where the Hagedorn(1970) model i s chosen. A hard equation of s t a t e i s one of the form p=JT/5 with a nonzero c o n s t a n t . F o r hard eguations of s t a t e only black holes of the same order of s i z e as the c a u s a l horizon*o can form from d e n s i t y f l u c t u a t i o n s (Carr, 1975) . B l a c k holes then form with s i z e t a t time t . The b l a c k hole forms from matter w i t h i n i t s S c h w a r z s c h i l d r a d i u s of about 2M. Thus black holes of mass H form i n the e a r l y u n i v e r s e at cosmic time t=H. To put upper l i m i t s on the e f f e c t i v e n e s s of the matter a n t i m a t t e r s e p a r a t i o n i t i s assumed t h a t a n g u l a r momentum and d e n s i t y f l u c t u a t i o n s of the r e q u i r e d t y p e are present i n order t h a t r o t a t i n g black holes form. 1 0 The c a u s a l h o r i z o n i s the d i s t a n c e t h a t a l i g h t s i g n a l c o u l d have t r a v e l l e d s i n c e t = 0 , i . e . c t where c i s the speed o f l i g h t (1 i n n a t u r a l u n i t s ) . T h i s d e f i n e s the l i m i t o f the r e g i o n c a u s a l l y a f f e c t e d by events, at a p o i n t . 97 To produce d i r e c t i o n a l n e u t r i n o c u r r e n t s the black holes oust have c o n s i d e r a b l e angular momentum when formed (see cha p t e r 2 o f t h i s t h e s i s ; V i l e n k i n , 1 9 7 8 ; Leahy Unruh,1979). F O E example, the angular momentum and mass of our galaxy, which seems a t y p i c a l s p i r a l galaxy, are estimated to be 2x10 7* gm cm 2/sec (Innanen, 1966) and 1.4x10** s o l a r masses (Allen , 1 9 7 3 ) . Angular momentum i s conserved i n the expansion o f the u n i v e r s e , so t h a t the angular momentum i n g a l a x i e s must have been present a t such e a r l y t i m e s * 1 . About I Q 3 0 times as much angular momentum i s present as the amount co n t a i n e d i n the black h o l e s * 2 (with 1 pa r t i n 10* 3 o f the mass of the galaxy i n black h o l e s , when they form). Furthermore, the q u a n t i t y (angular momentum)/(mass) 2 i n di m e n s i o n l e s s u n i t s , which i s e q u a l to the asymmetry parameter H/V(0<Rn<1) f o r a black h o l e * 3 , has the value 1000 f o r the galaxy, i n d i c a t i n g a huge amount of angular momentum. I t i s assumed here t h a t the angular momentum n e i t h e r i n v a l i d a t e s the use of a Friedmann model nor s e r i o u s l y a f f e c t s the black hole formation process. The f o r m a t i o n ** E.g. See Jones 1976. He a l s o reviews the problem of o r i g i n o f g a l a c t i c a n g u l a r momentum. * 2 T h i s a l s o means that the angular momentum of the galaxy cannot be e x p l a i n e d by p r e s e r v i n g p r i m o r d i a l angular momentum i n b lack hole r o t a t i o n . However, o r b i t a l motions of the black holes may be s u f f i c i e n t . 1 3 See f i g . 12 i n Appendix A and s e c t i o n X of chapter 2. The asymmetry parameter Bn. i s oL of chapter 2 and i s ro u g h l y the same as the value of Hy_^/NV4.p averaged over a hemisphere. 98 process c o u l d exclude v i r t u a l l y a l l the angular momentum from th e b l a c k h o l e s , but only (as yet unperformed) dynamical c a l c u l a t i o n s w i l l determine t h i s . &n advantage of b l a c k h o l e s forming with s i z e s i m i l a r t o t h a t of the c a u s a l h o r i z o n (see p r e v i o u s paragraph), i s t h a t t h e r e i s no s i g n i f i c a n t c o l l a p s e phase, so t h a t most of the an g u l a r momentum w i t h i n the c a u s a l h o r i z o n w i l l be i n c o r p o r a t e d i n the black h o l e . The angular momentum i s taken t o be uniform over a le n g t h s c a l e 1 a t the formation time, so t h a t the bl a c k h o l e s r o t a t e (more or l e s s ) i n p a r a l l e l throughout a r e g i o n of c h a r a c t e r i s t i c dimension 1. 1 i s d e f i n e d as comoving so t h a t i t expands as the universe does, i n p r o p o r t i o n to the s c a l e f a c t o r fi. 1 i s taken as the s c a l e on which p r o t o g a l a x i e s form a t a much l a t e r time, which i s taken as 10 times a t y p i c a l g a l a c t i c diameter. 1 i s 200 k i l o p a r s e c s ( 6 x 1 0 2 3 cm) at the present time (t=17x10 9 y e a r s ) , which r e s u l t s i n a s i z e o f 6 k i l o p a r s e c s (3x10** cm) a t the time of galaxy f o r m a t i o n 1 * . 1 w i l l be only .1 cm at t=10~ 3° sec when the black holes form. Other than at the time of fo r m a t i o n 1 may net correspond to the s c a l e of angular momentum s i n c e e f f e c t s o t h e r than the expansion of the universe may be important. For example, the angular momentum s c a l e i s a f f e c t e d by *• Galaxy formation i s here taken t o occur at t=10* y r , although i t c o u l d have taken p l a c e anywhere i n the range 10* t o 10* y r . 0 99 t u r b u l e n c e , which cascades energy t o s m a l l e r and s m a l l e r s c a l e s where i t i s d i s s i p a t e d (Jones, 1976) • The black h o l e s evaporate p a r t i c l e s t h e r m a l l y . Soon a f t e r black hole f o r m a t i o n , the temperature of the r e s t of the u n i v e r s e (the medium of r e l a t i v i s t i c p a r t i c l e s ) drops below that o f the black h o l e s , so t h a t any a c c r e t i o n of matter by the b l a c k h o l e s can be i g n o r e d * 5 . The t y p e s and numbers of p a r t i c l e s emitted by the low mass bl a c k h o l e s of i n t e r e s t here (of o r d e r 10 8 gm) are not known with any c e r t a i n t y , due to the u n c e r t a i n t y i n our knowledge of the p a r t i c l e spectrum at high energy. The number of d i f f e r e n t t y p e s of p a r t i c l e s which'can be emitted by the b l a c k h o l e s i s assumed to be s m a l l so t h a t n e u t r i n o s form a reasonable f r a c t i o n of the t o t a l e m ission**. Due to the alignment of the r o t a t i o n axes of the black h o l e s i n a r e g i o n , the n e u t r i n o f l u x e s w i l l be u n i d i r e c t i o n a l throughout the r e g i o n . The r e s u l t i n g l a r g e s c a l e n e u t r i n o f l u x and o p p o s i t e l y d i r e c t e d a n t i n e u t r i n o f l u x w i l l r e s u l t i n opposing baryon and antibaryon c u r r e n t s due t o the asymmetry i n weak s c a t t e r i n g of n e u t r i n o s on baryons. Baryons and * s C a r r and Hawking,1974, have shown t h a t between the time of formation and the time t h a t the black hole temperature i s h i g h e r , the a c c r e t i o n w i l l r e s u l t a t most i n an o r d e r of magnitude i n c r e a s e i n the black hole s i z e . »* Carr,1976 i n T a b l e 1 l i s t s the types and numbers of p a r t i c l e s emitted by b l a c k h o l e s f o r masses g r e a t e r than 3 x 1 0 1 3 gm. The l e a s t massive of these w i l l f i n i s h e v a p o r a t i n g a f t e r the end of the r a d i a t i o n e r a . 100 a n t i b a r y o n s w i l l flow i n t o d i f f e r e n t s p a t i a l r e g i o n s u n t i l e i t h e r the b l a c k h o l e s have f i n i s h e d e v a p o r a t i n g or u n t i l the cosmic temperature drops low enough f o r the baryons and a n t i b a r y o n s not yet separated t o a n n i h i l a t e ( i e the end of the hadron e r a ) . The baryon to photon r a t i o can be c a l c u l a t e d f o r t h i s time. The subsequent e v o l u t i o n of matter and a n t i m a t t e r r e g i o n s w i l l r e s u l t i n mixing and a n n i h i l a t i o n , i e a decrease i n the baryon to photon r a t i o . However the bulk of the a n n i h i l a t i o n has a l r e a d y taken place unless the matter r e g i o n s are so s m a l l so t h a t mixing a t the b o u n d a r i e s i s important. Chapter O u t l i n e The o u t l i n e of the c h a p t e r i s as f o l l o w s . S e c t i o n I I expresses the baryon to photon r a t i o i n terms of the v a r i o u s p h y s i c a l g u a n t i t i e s i n v o l v e d (eg black hole mass H, cosmic s c a l e f a c t o r B). I n S e c t i o n I I I , the baryon to photon r a t i o i s e v a l u a t e d f o r the case that the presence of black holes does not a l t e r the course of the r a d i a t i o n dominated Friedman expansion ( r e f e r r e d t o as 'small black hole d e n s i t y ' ) . Upper l i m i t s are obtained f o r two p l a u s i b l e e s t i m a t e s of the baryon-antibaryon c r o s s s e c t i o n a t high energy. In S e c t i o n I ? , m o d i f i c a t i o n s of the cosmic expansion due to the presence o f b l a c k h o l e s are c o n s i d e r e d and, i n S e c t i o n V, these are 0 10 1 used t o put upper bounds on the baryon t o photon r a t i o f o r the case of l a r g e b l a c k hole d e n s i t y . S e c t i o n VI reviews the r e s u l t s obtained. Unless otherwise s t a t e d , n a t u r a l (or dimensionless) u n i t s are u s e d 1 7 . In n a t u r a l u n i t s , the fundaaental c o n s t a n t s c, G and k are taken to have value u n i t y . The u n i t s of mass, l e n g t h , time, temperature, energy, and d e n s i t y are 2.1x10-5 gram, 1.6x10-" ce n t i m e t e r , 5.4x10-** second, 1.4x1032 K e l v i n , 1.2x1022 HeV, and 5. 1x10« 3 gram/cm 3, r e s p e c t i v e l y . The s i z e o f t h e s e u n i t s makes them a b i t unwieldy, except f o r use i n t h e o r e t i c a l c a l c u l a t i o n s and perhaps cosmology, e x p l a i n i n g why t h e i r use i s not widespread. I I . CALCULATION OF THE BARYON TO PHOTON RATIO F i r s t we c o n s i d e r t h e a b s o l u t e upper l i m i t t o any mechanism of matter a n t i m a t t e r s e p a r a t i o n o p e r a t i n g before the end of the hadron era at t K = 10 3* (1 second i s 10* 3 n a t u r a l u n i t s ) . By c a u s a l i t y , the best p o s s i b l e s e p a r a t i o n at t ^ i s 100 percent over a d i s t a n c e of the speed of l i g h t times t w , i . e . 1 0 3 9 (c=1, 1cm=10 3 3 n a t u r a l u n i t s ) . The s i z e of the 1 7 See eg Appendix C or the t a b l e on r e a r l e a f o f Hisner et a l ( 1 9 7 3 ) . 102 r e g i o n p e r p e n d i c u l a r t o the d i r e c t i o n of the matter f l o w can be a r b i t r a r i l y l a r g e . Allowing f o r the cosmic expansion, 100 percent s e p a r a t i o n of matter from a n t i m a t t e r could occur over a d i s t a n c e of 1 0 5 0 ( 1 0 1 7 cm) a t the present time. However the baryon t o photon r a t i o i s now about 10~ 9 so that c o n c e i v a b l y a matter r e g i o n as l a r g e as 10s» (10 megaparsec) a c r o s s could r e s u l t . Thus the a b s o l u t e upper l i m i t t o the s i z e of a matter r e g i o n i s of the order of s i z e of a s u p e r c l u s t e r of g a l a x i e s . Since only matter e x i s t s i n a r e g i o n a t l e a s t as l a r g e as a galaxy, the s e p a r a t i o n mechanism must be extremely e f f i c i e n t f o r any baryon symmetric cosmology to stand up to o b s e r v a t i o n . The baryon to photon r a t i o i s c a l c u l a t e d here as a f u n c t i o n of as few as p o s s i b l e parameters needed t o c h a r a c t e r i z e matter a n t i m a t t e r s e p a r a t i o n by e v a p o r a t i n g black h o l e s . The important phase of the black hole e v a p o r a t i o n takes p l a c e d u r i n g the hadron e r a . ...-First the n e u t r i n o f l u x from b l a c k holes i s c a l c u l a t e d , a baryon f l u x a r i s e s from n e u t r i n o s c a t t e r i n g on baryons due t o the asymmetry i n the weak i n t e r a c t i o n . The value of the baryon to photon r a t i o i s then c a l c u l a t e d from the baryon f l u x by use o f the c o n t i n u i t y equation. 103 The Heutrino Cureent From Rotating Black Holes The neutrino current produced i s continuous over length 1 i f the separation between the evaporating black h o l e s i s l e s s than the neutrino Bean free path n^-rtKly (2.1) where i s the black hole number density . For a current that has sources and s i n k s , the c o n t i n u i t y eguation i n an expanding universe i s : V.(f*) *ty/bt+ (3/R) (dR/dt) =source-sink p i s the energy d e n s i t y . We can assume homogeneity i a a region of black holes because of egn (1.1) above, so that V*(p v) i s z e r o . A l s o , f o r a neutrino beam (v=1) the number c u r r e n t j v i s r e l a t e d to the energy density by j , -f/\ . The energy of the neutrinos of i s r e d s h i f t e d during the expansion: E v i s p r o p o r t i o n a l tol/R. This r e s u l t s i n : fy>/dt=E,fcjv/dt-jv(Ev/R) (dR/dt) Here the s c a l e f a c t o r R i s taken to have a power law dependence: R= At* , to allow for r a d i a t i o n (z=1/2) and matter 104 (z=2/3) dominated expansions. The net r e s u l t i s t h a t j v i s governed by: d3v/dt+(2z/t) j v = ( d j v / d t ) k h - j v / r v (2.2) ( d j y / d t ) ^ i s the black h o l e source term f o r the n e u t r i n o c u r r e n t . 3)v/ty expresses the l o s s e s due t o s c a t t e r i n g of n e u t r i n o s out o f the c u r r e n t beam (or absorption) . .The mean c o l l i s i o n time f o r n e u t r i n o s , £ , i s much l e s s than the cosmic time t(2y«t) durin g the hadron era. Thus n e u t r i n o s c a t t e r i n g l o s s e s dominate over the d i l u t i o n due to the expansion of the universe i n determining j v : Jv= V d ^ / a t ) b k (2.3) T h i s can be a l s o v e r i f i e d by an exact s o l u t i o n t o egn (2.2) using egn (2.4) below. The black hole source term ( d j v / d t ) f c ^ i s the product o f the black hole number d e n s i t y n ^ ; the emission r a t e per bla c k hole xH / ( E v T y,)J and the asymmetry p a r a m e t e r 1 8 Mjb.(0<nit<1) : ( d j v / d t ) b l = n b K H / ( E v ^ ) xan- (2.4) 1 8 See f i g . 12 i n Appendix A and s e c t i o n X of chapter 2. The asymmetry parameter Hjx i s «J of chapter 2 and i s ro u g h l y the same as the value of Nv_y /N v*v averaged over a hemisphere. . 105 x i s the f r a c t i o n of emitted p a r t i c l e s which c o n s i s t s of n e u t r i n o s * * ; E v i s the mean n e u t r i n o e n e r g y 2 0 ; and ,7j^i=MVy i s the black hole l i f e t i m e . SI i s the angular v e l o c i t y of r o t a t i o n of the b l a c k h o l e s . Here we take a b l a c k hole l i f e t i m e of 100H 3 (y=.01), energy of emitted n e u t r i n o s of 1/H, and f r a c t i o n of e m i s s i o n i n the form of n e u t r i n o s of • 06. Each of these i s accurate to an order of magnitude. , The energy and r a t e of n e u t r i n o s emitted from the e v a p o r a t i n g black h o l e s , as well as x and MA-, are taken to be c o n s t a n t f o r the e n t i r e black h o l e l i f e t i m e . S ince most of the emission occurs f o r H and SL almost a t t h e i r o r i g i n a l v alues t h i s i s a good approximation (see a l s o Carr,1976). Here we estimate the black hole l i f e t i m e , which i s necessary t o e v a l u a t e the n e u t r i n o c u r r e n t , by egns (2.3) and (2.4). The black hole l i f e t i m e i s only a weak f u n c t i o n o f r o t a t i o n . I t s c a l e s as H 3 as found by d i v i d i n g the mass H by t h e emission r a t e 2 * . The e m i s s i o n r a t e i s p r o p o r t i o n a l t o the number of p a r t i c l e s p e c i e s emitted ( i n c l u d i n g d i f f e r e n t »• See e.g. Table 1 o f Carr,1976 f o r a l i s t of x vs black h o l e mass. zo The mean n e u t r i n o energy depends on the temperature of the b l a c k hole: T=K +/2-n= (1+ (1-« 2)-**)-*/(4TTH ) (see egn (9.1) of chapter 2), as w e l l as weakly on the r o t a t i o n parameter hn o f the black hole. For a n o n r o t a t i n g hole i t i s .18/H (Page, 1976a) and f o r a near maximally r o t a t i n g b l a c k hole i t i s about 1/M (Leahy,Unruh,1979) 2 * The emission r a t e i s given by the area o f approximately 16itM2 times the f o u r t h power of the temperature of approximately (8TTH)~ 4 times the number of p a r t i c l e s p e c i e s e m i t t e d . See egn (6) of chapter 1. 106 h e l i c i t y s t a t e s o f each p a r t i c l e ) , s i n c e each s p e c i e s i s e m i t t e d independently a t a thermal r a t e . Only e l e c t r o n s and p o s i t r o n s , e l e c t r o n and muon n e u t r i n o s and a n t i n e u t r i n o s , photons, and g r a v i t o n s are emitted (a t o t a l of 12 h e l i c i t y s t a t e s ) by b l a c k holes of mass 10»»<H<1022. i h e i r l i f e t i m e i s : 9x10 2H 3 f o r a n o n r o t a t i n g black h o l e ; 4x10 2H 3 f o r a maximally r o t a t i n g b lack hole (see f i g . 3 o f Page,1976b). For t h e much l e s s massive black h o l e s c o n s i d e r e d here, the number of p a r t i c l e s p e c i e s i s c o n s i d e r a b l y l a r g e r . T h i s number depends on the number of elementary p a r t i c l e s p e c i e s with r e s t mass l e s s than the black hole temperature 1/8TTM. For H=2x10 1 2 (using a l i f e t i m e of 100H 3), which corresponds to a black hole which j u s t f i n i s h e s e v a p o r a t i n g a t the end of the hadron e r a , the b l a c k hole temperature i s 2x10-** o r 2x10»Gev. T h i s l e a v e s c o n s i d e r a b l e room f o r s p e c u l a t i o n on the number of emitted p a r t i c l e s p e c i e s . In the guark model f o r example, the elementary p a r t i c l e s are ( c u r r e n t l y ) 3 l e p t o n s (each with i t s neutrino) and 6 quarks (each i n 3 c o l o r s ) f o r a t o t a l of (3x2*3x1*18x2)x2=90 d i f f e r e n t p a r t i c l e s and a n t i p a r t i c l e s , c o u n t i n g d i f f e r e n t h e l i c i t i e s ( n e u t r i n o s have onl y one h e l i c i t y ) . T h i s does not make allowance f o r any of the gauge p a r t i c l e s . The photon, H*,Z°,H - bosons, and the g r a v i t o n add 2*9+2 more h e l i c i t y s t a t e s . In quantum chroraodynamics (QCD) , an o c t e t o f c o l o r e d v e c t o r ( s p i n 1) 107 gluons i s r e s p o n s i b l e f o r the s t r o n g f o r c e . These are massless. The p r o p o s i t i o n i n QCD t h a t a l l p h y s i c a l s t a t e s are c o l o r l e s s , prevents a c o l o r e d gluon from reach i n g an observer ( s i m i l a r l y the c o l o r e d quarks are unobservable)., Thus f o r quarks and gluons, f r e e p a r t i c l e e m i s s i o n from a b l a c k hole doesn't make sense. Here we simply take the quarks as having no c o l o r and i g n o r e the gluons (without f u r t h e r j u s t i f i c a t i o n ) , so t h a t the number of emitted p a r t i c l e s t a t e s i s (3x2+3x1+6x2)x2=42 fermions p l u s 13 bosons. In a d d i t i o n there may be numerous other bosons; i n grand u n i f i e d t h e o r i e s a d d i t i o n a l s c a l a r (spin 0) Biggs bosons are present (24+5+45 i n the SO(5) u n i f i c a t i o n of S e o r g i and Glashow, 1974) , as w e l l as the superheavy X bosons (of mass of order 10* 5 Gev, mentioned i n the i n t r o d u c t i o n t o t h i s c h a p t e r ) , lie i g n o r e these here, i n a d d i t i o n to the gluons, so t h a t we may be c o n s i d e r a b l y o v e r e s t i m a t i n g the b l a c k hole l i f e t i m e . On the other hand, the black hole l i f e t i m e may be i n c r e a s e d due to strong i n t e r a c t i o n e f f e c t s between emitted p a r t i c l e s . The number of p a r t i c l e s i s c o n s i d e r a b l y more i n o t h e r models ; e.g. the e x p o n e n t i a l i n c r e a s e of p a r t i c l e number with energy i n the model of Uagedorn (1970). Here we assume the number of emitted p a r t i c l e s p e c i e s i s not l a r g e and take 55 (as argued above) as an estimate. S c a l i n g the l i f e t i m e of about 6x10 2H 3 f o r 12 emitted s p e c i e s t o 55 s p e c i e s y i e l d s 10 2M 3 f o r the black h o l e l i f e t i m e . Thus * 108 we take y t o have a value of 1 0 - 2 . The number of emitted p a r t i c l e s p e c i e s a l s o p r o v i d e s an estimate f o r x as .06, from 3 n e u t r i n o s out of a t o t a l of 55 emitted p a r t i c l e s . To e v a l u a t e the n e u t r i n o c u r r e n t , we need the n e u t r i n o mean c o l l i s i o n time. For n e u t r i n o baryon s c a t t e r i n g the mean c o l l i s i o n time i s : 1r*/l*t<?vt) (2.5) From egn (2.3), the c u r r e n t of n e u t r i n o s from the b l a c k h o l e s i s then: j v = n k K xyMn/(n g<5^H) (2.6) n e i s the number d e n s i t y of baryons and <Tfg i s the n e u t r i n o baryon t o t a l c r o s s s e c t i o n . i e now e v a l u a t e the number d e n s i t y of black h o l e s f o r eqn (2.6). The energy d e n s i t y i n black holes can be expressed i n terms of the bl a c k hole number d e n s i t y a t the time of formation t . At times o f formation r e l e v a n t here a hard eguation of s t a t e i s assumed, i . e . p=yfl/3 where p i s the pressure and p i s the energy d e n s i t y o f the cosmic medium. T h i s i m p l i e s t h a t o n l y black h o l e s of h o r i z o n s i z e K=t 6 can form from d e n s i t y f l u c t u a t i o n s at , as d i s c u s s e d i n the i n t r o d u c t i o n . The energy d e n s i t y 2 2 i n the Friedman model with 109 p=p/3 i s ft-2 f with f=3/32* = .03, and the s c a l e f a c t o r i s B=At l /* where A i s a constant. Depending on the amplitude of the d e n s i t y f l u c t u a t i o n s , a f r a c t i o n of the t o t a l energy d e n s i t y w i l l go i n t o forming b l a c k h o l e s 2 3 at t 0=H. T h i s r e s u l t s i n black holes s c a t t e r e d throughout the cosmic medium with an energy d e n s i t y and number d e n s i t y of a t the i n s t a n t o f f o r m a t i o n . Subsequently, s i n c e R=AM*fe a t tt , d i l u t i o n due to the expansion g i v e s : n ^ ^ a f f l - 3 * (B/A)- 3 yOfcK=MnbK f o r H<t< ? h K (2.7b) The net n e u t r i n o f l u x eqn(2.6) i s then given by: J v = p H " S A (B/A)- 3(n 0 < 5- v t f)-»fxyH^ (2.8) The c o n d i t i o n t h a t the black hole s e p a r a t i o n i s l e s s than the n e u t r i n o mean f r e e path, eqn (2.1), must be s a t i s f i e d f o r t h i s t o be v a l i d . I n v e c t o r boson exchange models, the n e u t r i n o 2 2 T h i s can be determined from eqn (1.2) f o r dR/dt, by using the r e l a t i o n />oeR~* (which r e s u l t s from egn (1.3)). 2 3 Here we are not concerned with the r e l a t i o n between the amplitude of the d e n s i t y f l u c t u a t i o n s and the value of A (for t h i s see C a r r , 1975). « 110 baryon c r o s s s e c t i o n i s 2 * : C&g = e * s / ( s * H w 2 ) 2 with s=2E«E v=2T/M (2.9) Hyj i s the mass of the W boson, e i s the e l e c t r o n charge (e=. 1) , and s i s the square of the 4-momentum t r a n s f e r . The energy of ambient baryons Ev i s the temperature T of the cosmic medium and the mean n e u t r i n o energy E i s given by 1/H as d i s c u s s e d e a r l i e r . , T h e r e s u l t i n g c r o s s s e c t i o n i s : & v S =4x10-sa/T f o r s>HK,2=(70 G e v ) 2 (2.10) The r e s t r i c t i o n , eqn (2.1), t h a t the n e u t r i n o c u r r e n t be continuous then r e s u l t s i n a lower l i m i t t o the i n t e n s i t y o f black hole f o r m a t i o n ^ : >2x10-» 2M»An3 (B/A) 3/T3 (2.11) «• See Appendix D; a l s o S c i u l l i (1974) with GM v 2=g 2/4f7 and g=e/sinOw# sin2&y=. 23 (g i s the dimensionless weak c o u p l i n g c o n s t a n t , eg see G a i l l a r d (1979) , Abers and Lee (1973)). I l l The Baryon C u r r e n t Produced B Y Neutrino S c a t t e r i n g The net baryon c u r r e n t , j g , r e s u l t i n g from the n e u t r i n o c u r r e n t , depends on the d i f f e r e n c e between n e u t r i n o baryon and n e u t r i n o a n t i b a r y o n s c a t t e r i n g . The problem of f i n d i n g the baryon c u r r e n t i s s i m p l i f i e d by c o n s i d e r i n g only events with l a r g e momentum t r a n s f e r , i e of the same order as t h a t of the i n c i d e n t p a r t i c l e . The m a j o r i t y of the n e u t r i n o baryon s c a t t e r i n g events f a l l i n t h i s category as do the hadron hadron s c a t t e r i n g events. What may be important but i s ign o r e d i s s c a t t e r i n g o f secondary n e u t r i n o s . However p a r t o f t h i s averages to zero due to the p a r t i a l i s o t r o p y o f the secondary n e u t r i n o s , and p a r t i s taken i n t o account by assuming t h a t a l l of the primary n e u t r i n o momentum i s t r a n s f e r r e d to the baryon component of the cosmic medium. In t h i s approximation, baryon baryon s c a t t e r i n g events do not change the magnitude of the baryon c u r r e n t . C o l l i s i o n s o f the baryon component of the baryon c u r r e n t with ambient a n t i b a r y o n s and of the a n t i b a r y o n component of the baryon c u r r e n t with ambient baryons are most imp o r t a n t i n d i s s i p a t i n g t h e net baryon c u r r e n t . The baryon antibaryon c r o s s s e c t i o n <5QQ depends on the s t a t e of the baryons. at temperatures g r e a t e r than iGev and d e n s i t i e s g r e a t e r than 1 0 1 6 gm/cm3, i t i s not c l e a r i f the baryons would be i n the form of f a m i l i a r p a r t i c l e s (protons, 112 neutrons etc) or i n some other form such as quarks. E s t i m a t e s f o r the c r o s s s e c t i o n are given below ( i n egn (2.22)). Since the n e u t r i n o mean c o l l i s i o n time i s much l e s s than the t i m e s c a l e f o r change i n the e a r l y u n i v e r s e (see d i s c u s s i o n f o l l o w i n g egn (2.2)), a steady n e u t r i n o f l u x i s present.. Balance of the source and l o s s terms f o r the baryon c u r r e n t then g i v e s : 0 = ( n 6 < r v f t ~ n f ^ 5 > ^ " n 6 % v«3g (2.12) Before the end of the hadron era one has roughly n=n^=n-where n i s the number d e n s i t y of any p a r t i c l e s p e c i e s with r e s t mass l e s s than the ambient temperature. Due t o the high n e u t r i n o energy(>10 6 Gev) and r e s u l t i n g high energy o f the s c a t t e r e d baryons, the baryon c u r r e n t has a v e l o c i t y o f very n e a r l y u n i t y . The baryon c u r r e n t i s then g i v e n by; ^ =<^»fe- < yvl)/C> e > (SV<Tj$ ) l v (2.13a) =otfxyHA£ H-V2(B/A)- V(n<r,j ) (2. 13b) The baryon c u r r e n t i s independent of the ne u t r i n o baryon c r o s s s e c t i o n but depends on the c r o s s s e c t i o n asymmetry 2 S c(-2 S See Appendix D. 113 = 2/3 f o r s<H ( V« (2 .14a) = 2 / (31n (s/H w«)) = 2 / (31n ( 2 T / 1 0 - " M ) ) f o r s > f l w 2 (2 .14b) For t y p i c a l values of ambient temperature; T=IO-*9=1 Gev, and black hole mass: H= (ytj,) » ^ = 2 x 1 0 » 2, one has t/=. 1. , Since*/" depends only l o g a r i t h m i c a l l y on energy one can takee< = . 1 f o r a l l e n e r g i e s o f i n t e r e s t . For black h o l e temperatures of 1/(8TTH)= 2 x 1 0 - i * = 2 x 1 0 S G e v ( f o r M = 2 x 1 0 » 2 ) , o n e can take the f r a c t i o n of energy emitted as n e u t r i n o s , x , a s 2 * . 0 6 . We d e f i n e B by: B=«cf xyHrt (2. 15) The r a p i d l y r o t a t i n g b l ack h o l e s c o n s i d e r e d here have fl-rt o f about .5, so t h a t B t y p i c a l l y has a value o f about 1 0 - * . The Baryon Number Density and paryon To Photon R a t i o The baryon number d e n s i t y a f t e r s e p a r a t i o n can be estimated from the c o n t i n u i t y eguation f o r the expanding u n i v e r s e ; 2 * See e a r l i e r d i s c u s s i o n on x f o l l o w i n g egn (2.4). 114 X). j =d/dt(n R(B/B 9) 3)=§n,/dt*n €(3/fi) dfi/dt](R/fO* (2.16) n ^ ( B / B p ) 3 i s the baryon number per u n i t comoving volume. The s c a l e f a c t o r o b e y s 2 7 : For r o t a t i n g b l a c k h o l e s a l i g n e d over a r e g i o n of c h a r a c t e r i s t i c s i z e 1, the n e u t r i n o c u r r e n t and t h u s the baryon c u r r e n t w i l l f a l l from t h e i r steady s t a t e v a l u e s , egns(2.8) and (2.13) r e s p . , i n the c e n t r a l p a r t of the r e g i o n to zero w i t h i n a few n e u t r i n o mean f r e e paths beyond the edge o f the r e g i o n . Thus the divergence of the baryon c u r r e n t i s approximated by j^/1 over the s c a l e 1: j & / l = d n 6 / d t * n 6 / t (2.18) Si n c e the baryon c u r r e n t d u r i n g the black hole e v a p o r a t i o n p e r i o d i s f a i r l y c o n s t a n t 2 8 the baryon number d e n s i t y i n egn (2. 18) w i l l grow u n t i l i t reaches the v a l u e : 2 7 See comment a f t e r egn(2.2). 2 8 In egn (2. 13) B ~ 3 n _ l i s constant i n time.. S i n c e the p h y s i c a l c o n d i t i o n s have power law dependences,, C j j and et^xMoj are almost co n s t a n t f o r most of the p e r i o d of b l a c k hole' e v a p o r a t i o n . See eg Carr,1976. (3/B)dB/dt=3z/t (2.17) n f t = j f t t / l = j t B f / ( R l ^ ) (2.19) 11.5 Here 1^ and B^ are the values of 1 and fl a t any f i x e d t i n e t j . T h i s t i a e can be chosen a t w i l l s i n c e the r a t i o 1/B r e n a i n s c o n s t a n t i n t i a e . The time t h a t matter a n t i m a t t e r s e p a r a t i o n ends i s tj=min ( t w , 100H 3) , i . e . , e i t h e r the end of the hadron e r a or the end of black hole e v a p o r a t i o n , whichever i s f i r s t . The baryon t o photon r a t i o , ij , a t time t<ts i s : ^n^/n^B^M-Vz <B/A)-V(n2*B S) t B ^ / ( E l f ) (2.20) The number d e n s i t y of photons n^, i s approximately e q u a l to n f o r times l a t e enough t h a t the number of p a r t i c l e s p e c i e s i s not very l a r g e . The end o f the hadron e r a s a t i s f i e s t h i s c c n d i t i o n . He now est i m a t e the baryon a n t i b a r y o n c r o s s s e c t i o n , r e q u i r e d f o r egn (2.20). The value of the baryon a n t i b a r y o n c r o s s s e c t i o n i s unknown a t e n e r g i e s of 1/H>10* Gev (10 3 Gev c e n t r e of mass). Cosmic ray experiments i n d i c a t e ( G a i s s e r e t al,1978) t h a t the proton a n t i p r o t o n t o t a l c r o s s s e c t i o n i s approximately constant (or l o g a r i t h m i c a l l y i n c r e a s i n g ) a t a value of about 10*o (40 m i l l i b a r n s ) . C o n s i d e r a t i o n of quark a n t i g u a r k s c a t t e r i n g due to massless v e c t o r gluon exchange, eg a p p r o p r i a t e to a quark medium, i n d i c a t e s much lower c r o s s s e c t i o n s : = «^ 2/s where the strong c o u p l i n g c o n s t a n t *^ is*«i 116 e^^(s) = 12 T T/(25ln(s/A 2) )=1.5/ln(s/.03Gev2)=.09 f o r s=10*Gev« (2.21) Thus the baryon a n t i b a r y o n c r o s s s e c t i o n can be w r i t t e n as a power law: ffjj =as-^ with s=2T/H (2.22a) with: a=10*° b=0 f o r proton a n t i p r o t o n (2.22b) and: a=10~2 b=1 f o r quark antiquark (2.22c) s i s the f o u r momentum t r a n s f e r squared. T, the temperature, i s the energy of ambient baryons, and 1/H i s the energy of a n e u t r i n o d r i v e n baryon (see comment f o l l o w i n g eqn ( 2 . 9 ) ) . The r e s u l t i n g baryon to photon r a t i o eqn (2.20) i s : f ^ B t ( B ^ / A ) / ( l ^ a ) ) p H-s*-*>Tfc ( B / A ) - * n - * t (2.23) 2 9 See Bures e t a l (1978). Here A i s a constant taken by thei t o be .17 Gev. 117 I I I . CASE A. SHALL BLACK HOLE DENSITY In t h i s section, we consider the case that the presence of black holes does not a l t e r the standard radiation dominated Friedmann expansion. T,B,and n have the time dependences of the standard Friedmann model i f the energy density i n black holes remains l e s s than the energy density i n radiation f o r t<H3/y. For t>H3/y the black holes have evaporated completely and no longer are present. I f the black holes dominate, the Friedmann model becomes matter dominated (F=0). The energy density i n black holes decreases from i t s i n i t i a l value, given by egn (2.7a), according to: /3tK=(J f a- 2(B 0/B)3 (3.1) The energy density i n radiation decreases more r a p i d l y than t h i s i n the standard model: / ^ r = f f 2 . For a r a d i a t i o n dominated expansion B i s proportional to tVz , giving ^=f (B/A)-*. Hith egn (2.7b) f o r the black hole density, the time that drops to the value of i s found to be t=H/^2. Beguiring t h i s to be greater than the black hole l i f e t i m e , so that always remains less than prt gives the requirement: fk <y1/2/M (3.2) 118 To e v a l u a t e the baryon to photon r a t i o , eqn (2.23), the temperature and number d e n s i t y f o r the medium need t o be known. For r e l a t i v i s t i c p a r t i c l e s i n thermal e q u i l i b r i u m , the energy d e n s i t y and temperature are r e l a t e d by (e.g. Weinberg, 1972) : ^=((7/8)n y,*(7/U)n f ) aT* (3.3a) where a i s the r a d i a t i o n c o n s t a n t : a = T f 2 k * / (15h 3c 3) or a.=T 2/15 i n n a t u r a l u n i t s . n y i s the number of n e u t r i n o s p e c i e s , njs i s the number of (other) fermion s p e c i e s , and n f c i s the number of boson s p e c i e s . At high enough t e m p e r a t u r e 3 0 a l l p a r t i c l e s are present- n v=3, n^=3 l e p t o n s +18 quarks, 1^ = 4 (omi t t i n g the graviton) . T h i s r e s u l t s i n the r e l a t i o n yO=28.7T*. A f t e r e l e c t r o n - p o s i t r o n a n n i h i l a t i o n (but p r i o r t o the end of the r a d i a t i o n e r a a t t = t r ) , only photons and n e u t r i n o s c o n t r i b u t e t o the energy d e n s i t y . With n y=3, n £ = 1 # egn (3.3a) y i e l d s ^)=2.39T*. As a s i m p l i f i c a t i o n , we take fr=T* at a l l times t < t r 3 4 . Since T depends on the f o u r t h r o o t of the r e s u l t i n g e r r o r i n T i s s m a l l . The number d e n s i t y of p a r t i c l e s i s approximated f a i r l y well by f/T, With the r e l a t i o n 3 2 f o r the energy d e n s i t y : ^ f t ~ 2 . 3<> See d i s c u s s i o n p r i o r to eqn (2.5). 3 * For s e c t i o n s I I I and IV (see t a b l e 1), we use y^28T* f o r very e a r l y times ( t < t , ) . 119 f=.03, the r a d i a t i o n dominated Friedman model has: T=f Y+t-tfc H= At V2 (3.3) The r e s u l t i n g baryon t o photon r a t i o , eqn (2.23), i s : ^ = 2 ' f * » - 3 2 B ( ( R £ / A ) / ( l . £ a ) ) ^ (3.4) The upper l i m i t on i s found by maximizing egn (3.4) with r e s p e c t to p, H, and t . I.e. one f i n d s the optimum i n t e n s i t y of b l a c k hole f o r m a t i o n , b l a c k hole mass, and cosmic time f o r s e p a r a t i o n of matter from a n t i m a t t e r . For b<2 i n the baryon a n t i b a r y o n c r o s s s e c t i o n , eqn(2.22), the baryon t o photon r a t i o i s l a r g e s t f o r black holes which f i n i s h e v a p o r a t i n g a t the end of the hadron e r a - t=H3/y=t|,, and f o r an i n t e n s i t y of f o r m a t i o n - ^=y 1 / 2/H: I j < 2 H y ~ ( H - 2 V 3 B ((B^/A) / (Ifa)) t Cs*)a -v.) (3.5a) (3.5b) (3.5c) 3 2 See d i s c u s s i o n p r i o r to eqn (2.7a). 120 The d i f f e r e n t c r o s s s e c t i o n s cause the maximum e s t i m a t e s f o r ^ t o vary d r a s t i c a l l y , i e by a f a c t o r of 10»°. The c o n s t a n t f a c t o r s y(=10 - 2) and f(=.03) are a l s o contained i n B, so the net dependences of ^ on y and f are only y^-H1^ and f</»-vfc, where b has the value 0 or 1 (see eqn (2.22)). The case f o r A>y*£/M does not i n c r e a s e the upper l i m i t s of eqn (3.5), as w i l l be demonstrated i n the f o l l o w i n g s e c t i o n s . F i r s t we e v a l u a t e the upper l i m i t s to the baryon t o photon r a t i o i n eqn (3.5). Since the value of the length s c a l e of angular momentum a t the time of black hole f o r m a t i o n , 1, i s not predetermined here, we allow 1^ t o take the value r e q u i r e d at time t f t o gi v e matter r e g i o n s a t the present time l a r g e r than a galaxy. The l e n g t h s c a l e r e q u i r e d f o r r e g i o n s of e v a p o r a t i o n of black h o l e s r o t a t i n g i n p a r a l l e l , 1 , depends on the e v o l u t i o n of the matter and an t i m a t t e r r e g i o n s produced by the n e u t r i n o c u r r e n t s . I t i s e a s i e r t o sepa r a t e matter from a n t i m a t t e r on s m a l l s c a l e s , as seen from eqn (3.5). F i r s t we c o n s i d e r the p o s s i b i l i t y t h a t the c o a l e s c e n c e mechanism of Omnes(1972) 3 3 operates. In Omnes' model. 3 3 The coalescence mechanism depends on a n n i h i l a t i o n pressure a t the boundary between matter and a n t i m a t t e r r e g i o n s . T h i s prevents f u r t h e r mixing a t the boundaries and r e s u l t s i n growth of matter or an t i m a t t e r r e g i o n s . Coalescence can, i n the Omnes model, r e s u l t i n growth o f matter r e g i o n s to a s i z e l a r g e r than c l u s t e r s of g a l a x i e s . I t has not been demonstrated t h a t c o a l e s c e n c e can a c t u a l l y occur. 121 c o a l e s c e n c e i s e f f e c t i v e a f t e r the end of the l e p t o n e r a , a t t i n e t j . I t i s then s u f f i c i e n t to separate matter from a n t i m a t t e r on a s c a l e g r e a t e r than the d i f f u s i o n d i s t a n c e d a t t=tjt. Any matter r e g i o n s s m a l l e r than d w i l l have mixed by d i f f u s i o n with adjacent a n t i m a t t e r r e g i o n s during the time between the end of the s e p a r a t i o n p e r i o d and the end of the l e p t o n e r a , and w i l l have been a n n i h i l a t e d before c o a l e s c e n c e begins. In the l e p t o n era, protons s c a t t e r predominantly o f f e l e c t r o n s . However p r i o r to n e u t r i n o d e c o u p l i n g 3 * a t t=10**, protons and neutrons f r e e l y i n t e r c o n v e r t . Neutrons thus determine d with t h e i r much l o n g e r path l e n g t h 3 5 3 4 . For a baryon to photon r a t i o of 1 0 - 9 , neutron e l e c t r o n and neutron nucleon c r o s s s e c t i o n s are approximately equal. The r e s u l t i n g d i f f u s i o n length and thus the minimum r e q u i r e d value o f 1 a t t=10** i s 3 * : l^=d=(VX,t/3) i/2=103» with E f/A=tva= 102* (3.6) I f no mechanism e x i s t s which causes s m a l l matter o r a n t i m a t t e r r e g i o n s t o merge with other l i k e r e g i o n s , 1 must 3 * E.g. see de Graaf,1970. 3 5 Eg, see Steigman, 1976. 3 6 The l e p t o n era ends at a time of 10* 3, but neutron d i f f u s i o n i s important u n t i l n e u t r i n o d e c o u p l i n g . Due to the approximate nature of the c a l c u l a t i o n here t h i s d i f f e r e n c e i s not g r e a t l y s i g n i f i c a n t . Omnes chose the end of the l e p t o n e r a to e v a l u a t e d i f f u s i o n r a t h e r a r b i t r a r i l y and omitted the e f f e c t of neutrons. 122 be a t l e a s t as l a r g e as a t y p i c a l comoving g a l a c t i c s i z e s c a l e . He e v a l u a t e 1^ and a t the present time: 1{=200 kpc=4x10s* with R^=102«cm=10* i (3.7) He have allowed f o r c o l l a p s e of the r e g i o n during galaxy f o r m a t i o n by a f a c t o r of 10 i n l i n e a r e x t e n t , to reach a present diameter of 20kpc f o r a galaxy.. The value of a depends on H and p, the present Hubble constant and matter d e n s i t y , and i s a p p r o x i m a t e l y 3 7 5 x 1 0 2 9 . Thus reasonable values to r e g u i r e of B^/(&lf) are i n the range: 5x10-«*<B£/(Alr) <10-»* (3.8) The baryon to photon r a t i o i s s e n s i t i v e to how the e v o l u t i o n of matter a n t i m a t t e r r e g i o n s a f t e r s e p a r a t i o n by the black h o l e s occurs. T h i s i s not s u r p r i s i n g s i n c e the c o a l e s c e n c e mechanism of Omnes(1972) would be extremely e f f i c i e n t . H i t h egns (3.5c) and (3.8), the baryon to photon r a t i o has an upper l i m i t l e s s than 10-**. T h i s i s l e s s than the observed 37 at the p r e s e n t time (t=17x10*years=10*» i n n a t u r a l units) the u n i v e r s e i s matter dominated with s c a l e f a c t o r H = c t ^ = 1 0 * » (approximately). T h i s g i v e s C=2x 10 2 ° . P r i o r to the time of r e d s h i f t z (t) =B (now)/B (t) -1 of about 2x10 3 ( t h i s v a lue depends on H and/> ) , the u n i v e r s e was r a d i a t i o n dominated with B=At»*. One then f i n d s & t o be about 5 x 1 0 2 « . 123 value by about 7 o r d e r s of magnitude. T h i s assumes t h a t the co a l e s c e n c e mechanism comes i n t o play at the end of the l e p t o n e r a . Arguments {Steigman, 1976) i n d i c a t e t h a t i t does not, i n which case the upper l i m i t on the baryon t o photon r a t i o a t t a i n a b l e through black hole e v a p o r a t i o n s i s reduced t o 5 x 1 0 - 2 * , i . e . , completely n e g l i g i b l e . The next two s e c t i o n s are devoted t o d i s c u s s i n g t h e case ^>yi/2/H. i n t h i s case the form of the Friedmann expansion i s a l t e r e d by the presence of the l a r g e number o f b l a c k h o l e s , and must be r e c a l c u l a t e d . I t i s shown t h a t i n c r e a s i n g p above y t £ / a does not i n c r e a s e the upper l i m i t to the baryon t o photon r a t i o . i s the f r a c t i o n of matter t h a t goes i n t o forming black holes a t time H. Thus the upper l i m i t s o f eqns (3.5b) and (3.5c) are s t i l l v a l i d . IV. FRIEDMANN EXPANSION WITH BLACK HOLE FORMATION EVOLUTION OF THE ENERGY DENSITIES OF RADIATION(^ 1^ BLACK HOLES f ^ ) . AND BLACK HOLE EMISSION ( feJ_i THE SCALE FACTOR (R) ; THE NUMBER DENSITY (*7) I AND THE TEMPERATURE (T) 124 A Friedmann model of the uni v e r s e i s assumed h e r e 3 8 . The r a d i a t i o n (which i n c l u d e s r e l a t i v i s t i c p a r t i c l e s ) has a hard e g u a t i o n of s t a t e (pressure p=^/3) whereas the black holes have a s o f t eguation of s t a t e (p=0). Wr i t i n g p= xy> , t h e time dependence i s as f o l l o w s (see e.g. C a r r , 1975): pFft-2 R=At2/t3<»*5^ T=gt-« (4.1) f has the value .03 and g i s a f a c t o r of order u n i t y which depends on the d e t a i l s of the eguation of s t a t e . He take g=1 here (see d i s c u s s i o n around egn (3.3a) ). Equation (4.1) can be d e r i v e d from egns (1.2) and (1.3). The r e l a t i o n f o r T holds f o r an a d i a b a t i c expansion (T p r o p o r t i o n a l to R-3*) and ceases to hold during the period t h a t h e a t i n g by black hole e m i s s i o n i s s i g n i f i c a n t . However t h e r m a l i z a t i o n of the black hole e m i s s i o n i s r a p i d (see Appendix E) so t h a t the energy d e n s i t i e s of the e m i s s i o n and of the background r a d i a t i o n , ^J^and r , are i n d i s t i g u i s h i b l e and, a f t e r the end of black hole e v a p o r a t i o n , the r e l a t i o n i n egn (4.1) f o r T i s v a l i d . For r a d i a t i o n of mean energy T, the number d e n s i t y i s simply ^d/T. The expansion of the u n i v e r s e with formation o f black h o l e s at time fl and with energy d e n s i t y =/J f+ can be 3 8 See i n t r o d u c t i o n . Ryan and Shepley(1975) a l s o d i s c u s s the p r o p e r t i e s of Friedmann models. 125 d i v i d e d i n t o f i v e stages ( f o r ^ >y1^2/M) p r i o r to the end of the r a d i a t i o n e r a . T h i s i s d i s p l a y e d i n f i g u r e 15b (in Appendix A) and t a b l e 1 (on page »3*f ) • F i g u r e 15a i l l u s t r a t e s the case of low black hole d e n s i t y (^<y»^/M) foe c o o p a r i s o n . The f i r s t stage i s the time before b l a c k hole f o r m a t i o n . During the second stage the energy d e n s i t y i n black holes i n c r e a s e s r e l a t i v e to t h a t i n r a d i a t i o n (which i n c l u d e s a l l r e l a t i v i s t i c matter o u t s i d e the black h o l e s ) . At time t ( , the energy d e n s i t y i n r a d i a t i o n f a l l s below t h a t i n b l a c k h o l e s , r e s u l t i n g i n the eguation of s t a t e changing from hard (f = 1/3) t o s o f t (^=0). T h i s a l t e r s the expansion through egn (4.1). During the time H<t<H3/y the black h o l e s emit r a d i a t i o n roughly at a c o n s t a n t r a t e 3 9 , so t h a t the energy d e n s i t y i n e m i s s i o n i s a p p r o x i m a t e l y * 0 ( t y / M 3 ) ^ ^ . At time t ^ , the energy d e n s i t y due to b l a c k hole emission i s as l a r g e as that i n the background r a d i a t i o n , so f o r t > t ^ the e m i s s i o n dominates the cosmic medium. The medium c o n s i s t s of a t h e r m a l i z e d plasma of r e l a t i v i s t i c p a r t i c l e s s i n c e the t h e r m a l i z a t i o n time i s r a p i d (see Appendix E) . The temperature and number d e n s i t y of p a r t i c l e s are then found from: 3 9 See d i s c u s s i o n f o l l o w i n g egn (2.4). 4 0 T h i s r e s u l t i n c l u d e s the r e d s h i f t e f f e c t f o r r a d i a t i o n emitted a t times e a r l i e r than t , but the r e d s h i f t does not a f f e c t the r e s u l t except f o r a f a c t o r of 2/3 which i s dropped here. 126 n = ^ /T (4.2) S t a t i s t i c a l weighting f a c t o r s i n the r e l a t i o n between T ana pe^, which depend on the number of p a r t i c l e s p e c i e s present at the ambient temperature, a r e of order u n i t y . He use ^=T* f o r t>t, , y?=28T* f o r t<t,, f o r s i m p l i c i t y * * . a f t e r time t=H 3/y, black h o l e s are no l o n g e r p r e s e n t and the expansion i s r a d i a t i o n dominated a g a i n * 2 . The major change compared with the case of s m a l l b l a c k hole d e n s i t y (^yV^/H) i s an i n c r e a s e i n the s c a l e f a c t o r B by a f a c t o r (^ H) t*/yi*. V. CaSE B, UPPER LIMITS ON THE BARYON TO PHOTOH BaTIQ FOB In s e c t i o n I I , an e x p r e s s i o n , egn (2.23), was d e r i v e d f o r the baryon to photon r a t i o . Here upper l i m i t s are found f o r the baryon to photon r a t i o f o r the case t h a t the black hole d e n s i t y i s l a r g e enough to a l t e r the Friedmann expansion, so t h a t the arguments of s e c t i o n I I I need to be •* See d i s c u s s i o n a f t e r eqn (3.3a). • 2 The t r a n s i t i o n from black h o l e dominated ( p r e s s u r e l e s s ) to r a d i a t i o n dominated expansion occurs g r a d u a l l y and i s centred a t time t=M 3/2y, but t h i s only i n t r o d u c e s f a c t o r s of order u n i t y , which are not of concern i n the e n t i r e c a l c u l a t i o n here. 127 r e c o n s i d e r e d . The f a c t o r T 1 , ( H / a ) - • n - 2 i n eqn (2.23) i s expressed here i n t e r n s of t , ^ and H with the a i d of t a b l e Is T t(B/A)-*n-z=50x.2 b t»-*/« f o r t<t, (5.1a) = 200x. 4 1 | 5< 2 - ^ 3 H 0— «t'*t fc-*0CaT) f o r t f < t < t a (5. Jb) 2x105 X. T^-van-si/b-aiAt-**-''* f o r t x<t<100M> (5.1c) =9x.<* fa&-**t*-v* f o r 100fl3<t<t r (5. Id) The l i m i t on the s e p a r a t i o n time i s t<100B 3. We d e f i n e the constant C by: C=2 bB(R^/A)/(l^a) (5.2) The baryon t o photon r a t i o becoaes: ^=50x.2^H - * * - H 2-W2 f o r t<t, (5.3a) 17=2001.^C^C'-«»V3H- ti**5fc>/*t f o r t, <t<t ; (5.3b) 7J=2x10*x. I^C^-vaa^-^Mt-V*- • f o r t a<t<1G0H3 (5.3c) 128 From eqn (2.23), t h i s i s found d i r e c t l y by ^  tC times egns (5.1) . To maximize the baryon to photon r a t i o , t i s taken as t , , t z , and t x r e s p e c t i v e l y f o r the three time p e r i o d s i n egns{5.3). Hext i t i s assumed t h a t one has b<1 f o r t,<t<10QM 3 and b<3 f o r t<t,. The baryon t o photon r a t i o then has the upper l i m i t s : ^<50x.2 iC^4- 3B-»A-3*/2 f o r t=t, i n (5 . 3 a ) (5.4a) <5xlO*x.Q2 bCH5Ci-*»y2 ^<10Sx.06 bC^ &-i*/5 H2 c i - w i o ) / 3 f o r t = t x i n (5.3b) (5.4b) <2x10«x.04 1 , CBS ^lOSx.Oe 4 , C ^ 0»-tVsfl2 c t - k / i o>/3 f o r t=t^ i n (5.5c) (5.4c) <2x10 sx. 04^ CH S Ci—w)/2 The values i n the second column r e s u l t from s e t t i n g ^=1/10H t o maximize the upper l i a i t t o ^ » S e t t i n g yS=1/10M a l s o r e s u l t s i n t=t ( =t v=100fl 3. The d i f f e r e n c e i n numerical f a c t o r s between eqn (5 . 4 a ) and eqns (5.4b), (5.4c), r e s u l t s from using p=28T* i n t a b l e 1 f o r t<t, and ^=T* f o r t>t, (see eqn (3.3a)). The net e f f e c t of i n c l u d i n g more p a r t i c l e s p e c i e s i s a r e d u c t i o n i n <[ by a f a c t o r o f 2 to 4. For b=1 one has ^<C, whereas f o r b=0, l a r g e s t H i s d e s i r a b l e c o n s i s t e n t with t=100» 3, i . e . H=t h V 3/5=2x10 » 2 . The time t h a t the hadron e r a ends, t . , i s a f f e c t e d by 129 the presence o f black holes. He used the value f o r t^ which i s a p p l i c a b l e f o r the case ^<1/10H, when the bl a c k h o l e s do not a l t e r the expansion. T h i s i s p e r n i s s a b l e f o r e v a l u a t i n g t h e maximum of the baryon t o photon r a t i o , s i n c e we are t a k i n g the l i m i t ^ = 1/1 OH f o r which the r e l a t i o n between the temperature T (which determines when the hadron era ends) and the time t , i s not a f f e c t e d . In g e n e r a l , the end of the hadron e r a occurs when the temperature drops to T s=300HeT =3x10- 2o. Hhen t h i s occurs i n stage 4 of the expansion with b l a c k holes (see t a b l e 1), the time t h a t i t occurs i s : t^=3x10 7VM3 f o r T (M3/y) <3x10-2o<T (t^) T h i s would be a p p l i c a b l e as the time t h a t s e p a r a t i o n ends t o c a l c u l a t e the baryon to photon r a t i o f o r the case o f egn (5.3c). For more massive b l a c k h o l e s , the hadron e r a ends bef o r e t z : tk=6x102»P!i/»|j-*2 T ( t 4 ) <3X10 - 2 0 <T (t, ) which would be a p p l i c a b l e f o r egn (5.3b). Eqn (5.3a) i s not a p p l i c a b l e f o r t^<t,, s i n c e we used the r e l a t i o n p=28T* (see egn (3.3a) and t a b l e 1) which i s a p p l i c a b l e f o r high temperature. I f we had used p=T*, egn (5.3a) would a p p l y f o r t i <t, , but the numerical f a c t o r would be 200x.4 l >. 130 S e t t i n g 4 3 B=10-*, the upper l i m i t t o the baryon to photon r a t i o , from egn (5.4b) or (5.4c), becomes: ^<2x10-io B ^ / U l f ) f o r b=0,a=10*o (5.5a) ^<2B^/(A1^) f o r b=1,a=10-z (5.5b) These upper l i m i t s on the baryon t o photon r a t i o are t h e same as those of eqn (3.5). The e v a l u a t i o n of R^/(hlf) i s s i m i l a r t o t h a t of s e c t i o n I I I , except f o r the m o d i f i c a t i o n (see t a b l e 1) o f the s c a l e f a c t o r . However, s i n c e the change i n B i s 2(pM)V3 and one has/J=/10H f o r maximum baryon to photon r a t i o , the net r e s u l t i s unchanged. I t seems s u r p r i s i n g t h a t i n c r e a s i n g the i n t e n s i t y of b l a c k hole f o r m a t i o n ^ above 1/10M a c t u a l l y decreases the baryon to photon r a t i o . However black hole d e n s i t i e s l a r g e r than f o r |2 = 1/10H r e d u c e ^ by the i n c r e a s e i n r a d i a t i o n due to t h e i r emission. • 3 See egn (2. 15) . 131 f l . DISCUSSION In s e c t i o n I I , the baryon to photon r a t i o was c a l c u l a t e d f o r matter a n t i m a t t e r s e p a r a t i o n by n e u t r i n o s from r o t a t i n g b l a c k holes. The case f o r which the presence of black holes does not a l t e r the expansion from the standard model was c o n s i d e r e d i n s e c t i o n I I I and m o d i f i c a t i o n s due to the presence of black h o l e s were taken i n t o account i n s e c t i o n s I¥ and V. The maximum baryon to photon r a t i o o ccurs f o r the case i n which the black holes j u s t f i n i s h e v a p o r a t i n g a t the end of the hadron era (100B 3=t^) and f o r a f r a c t i o n ^=1/101! (see egn (3.2)) of the mass d e n s i t y going i n t o black h o l e s when they form at time M. T h i s maximum value i s given by eqn (5.5). I t depends on the form of the s t r o n g c r o s s s e c t i o n (egn (2.22)) and on the s c a l e 1 over which matter a n t i m a t t e r s e p a r a t i o n i s r e q u i r e d . For the most f a v o r a b l e case ( i e quark ant i q u a r k s c a t t e r i n g , egn(2.22c), and e f f i c i e n t coalescence) the upper l i m i t to the baryon t o photon r a t i o a t t a i n a b l e by the black holes mechanism i s 10-**. A more r e a l i s t i c e s t imate would not assume coalescence but would s t i l l allow the more f a v o r a b l e s t r o n g c r o s s s e c t i o n , egn(2.22c). The upper l i m i t t o the baryon t o photon r a t i o i s then 5 x 1 0 - 2 * . T h i s r u l e s out the proposed mechanism as the b a s i s f o r a v i a b l e baryon symmetric cosmology. The main reason f o r f a i l u r e of the 132 mechanism of t h i s chapter, i s t h a t the time a v a i l a b l e f o r s e p a r a t i o n i s only 10 -* sec ( i . e . , the d u r a t i o n of the hadrom e r a ) . I f an upper l i m i t on the baryon to photon r a t i o o f g r e a t e r than 10~» had r e s u l t e d , i t would be necessary to argue that the s c e n a r i o i s probable. I.e., one r e q u i r e s the formation of r a p i d l y r o t a t i n g mini b l a c k h o l e s , each of mass o f order 4x10 7 gm, with s p i n axes a l i g n e d over a p r o t o g a l a c t i c 4 * s i z e s c a l e ( s c a l e d t o the time o f the hadron e r a , i . e . 10* 2 cm a t t^=10~* s e c ) . However s i n c e the i n t e n s i t y of black hole formation r e q u i r e d f o r the a l i g n e d s e t of black h o l e s i s of order 1/10M o r only about 5 x 1 0 - 1 * , a higher d e n s i t y of black h o l e s with only a s m a l l f r a c t i o n a l a n i s o t r o p y could y i e l d the d e s i r e d net e f f e c t . To e v a l u a t e t h e r e s u l t f o r t h i s case would r e q u i r e a more d e t a i l e d model. However, the e x t r a non a l i g n e d black h o l e s would tend to reduce the barycn t o photon r a t i o , i n s i m i l a r manner to the e x t r a a l i g n e d black holes of s e c t i o n s IV and V. The c a l c u l a t i o n s here are only order of magnitude. Hany u n c e r t a i n t i e s a r i s e , e s p e c i a l l y as t o how t o t r e a t the p a r t i c l e p h y s i c s d u r i n g the hadron e r a when the temperature i s g r e a t e r than 1 Gev and the d e n s i t y g r e a t e r than n u c l e a r d e n s i t y . Complex processes a f f e c t the subseguent e v o l u t i o n of ** I.e., the s i z e s c a l e corresponding to a galaxy b e f o r e i t c o l l a p s e s , d u r i n g galaxy f o r m a t i o n , t o i t s present s i z e . 133 the matter, such as Omnes' (1972) c o a l e s c e n c e mechanism, which are d i f f i c u l t t o ana l y z e . Other unforeseen e f f e c t s probably a f f e c t the outcome. The bl a c k h o l e s e n v i s i o n e d here form at a time of order 1 0 _ 3 » second a f t e r the i n i t i a l s i n g u l a r i t y (assuming a Friedmann expansion) and have evaporated t h e i r 4x10 7 gram mass by the end of the hadron era a t 10-* second. E f f e c t s of the hot dense medium on the e v a p o r a t i n g b l a c k h o l e s have been ignored s i n c e the temperature o f the bl a c k h o l e s i s much higher s t i l l (of order 10* Gev). S i g n i f i c a n t refinement of models f o r bl a c k holes and f o r the e a r l y u n iverse i n these c o n d i t i o n s i s u n l i k e l y t o occur u n t i l the p a r t i c l e p h y s i c s i s d e f i n i t e l y known. The e f f o r t r e g u i r e d i n t r y i n g to r e a l i s t i c a l l y i n c l u d e as many e f f e c t s as p o s s i b l e i s not j u s t i f i e d , e s p e c i a l l y because o f the s t r o n g n e g a t i v e r e s u l t obtained here. Another consequence o f n e u t r i n o c u r r e n t s from .black holes i n a c o s m o l o g i c a l context i s magnetic f i e l d g e n e r a t i o n . The next chapter i n v e s t i g a t e s the p o s s i b i l i t y t h a t the l a r g e s c a l e ( g a l a c t i c ) magnetic f i e l d was generated by charged c u r r e n t s r e s u l t i n g from black hole e v a p o r a t i o n . . 134 T a b l e 1. Expansion With Black Hole Formation- ^o>y VVM STAGE * I A w B 1. P r i o r t o black hole formation*; t<fl, y-1/3 f t - z At 2. Black h o l e s form at t i m e 3 t=H: H<t<t t as above (ty/H 3)/)* ^ y f M-7*t"*J as above /V/T = . 1 5 t - ^ as above .2 t - * * as above 3. At t, £ r a d i a t i o n energy dens i t y drops below b l a c k h o l e energy d e n s i t y * : t7=M/p*. t,<t<t^. y=5" A A V V * t V l f (B/A) ~pr^*i i^t f t - * (ty/M 3)AK = yfH -V t A / T = .07^ M *t 4. At t ^ = p-v5H-n/5y-a*5 m the enerqy d e n s i t y i n 'background' r a d i a t i o n drops below that of the_black hole emission: t r < t < f l 3 / y . *=0 as above as above as above as above = .002tf*'<t*V 5. At t=H 3/y. black h o l e e v a p o r a t i o n ends: H3/y<t<t«-,2f = 1 /3 f t - 2 A ^ H ^ t V 4 = . Q 7 t - 3 / 2 = .4t-»« At t r , r a d i a t i o n era ends. 1 The eguation o f s t a t e i s p=*p. z For very e a r l y times (t<t,) we use yO=28T*, see eqn (3.3a); also f has the value .03, see d i s c u s s i o n before eqn (2.7). 3 See eqn (2.7b) . y=.01 f o r black holes of i n t e r e s t t o chapter* 3 and 4, see d i s c u s s i o n f o l l o w i n g egn (2.4). * At t h i s time we r e v e r t t o using f>=T*t see eqn (3.3). 135 tt. COSHOLOGICAL HAGNETIC FIELD GEBEBATIOM BY BOXATIHG_BLACK HOLES I. INTBODUCTION F i r s t we b r i e f l y review present i d e a s on the o r i g i n of magnetic f i e l d s i n the e a r l y u n i v e r s e . These are based on models of the expansion which have random p e c u l i a r motions, r a t h e r than d e n s i t y f l u c t u a t i o n s , as the source of the inhomogeneities which are r e s p o n s i b l e f o r galaxy formation. Jones(1976) reviews the two major t h e o r i e s of galaxy f o r m a t i o n - G r a v i t a t i o n a l I n s t a b i l i t y Theory, which depends on d e n s i t y f l u c t u a t i o n s , and Cosmic Turbulence Theory, which depends on random p e c u l i a r motions i n the u n i v e r s e . Turbulence w i l l occur i n the l a t t e r case and, i n the expanding r a d i a t i o n dominated plasma, w i l l generate weak magnetic f i e l d s on the same s c a l e s as the motions. F u r t h e r a m p l i c a t i o n o f the magnetic f i e l d by the t u r b u l e n t motions o f the plasma occurs. Harrison(1973a,b) e s t i m a t e s t h a t a f i e l d of o r d e r 1 gauss i s produced at the time of egua l matter and r a d i a t i o n energy d e n s i t i e s . T h i s f i e l d would be l a r g e enough t o prevent fragmentation of the inhomogeneities a t the time t h a t protons 136 and e l e c t r o n s recombine to form hydrogen 1., The r e s u l t i n g i n t e r g a l a c t i c f i e l d a t the present time would be 1 0 - 8 gauss over a l e n g t h s c a l e of 1 megaparsec., Recently, Baierlein{1978) has done a more d e t a i l e d a n a l y s i s and f i n d s t h a t the magnetic f i e l d i n t e n s i t y a t the time of equal matter and r a d i a t i o n energy d e n s i t i e s i s only 10 -* gauss. T h i s i s d y n a m i c a l l y i n s i g n i f i c a n t f o r galaxy f o r m a t i o n and e v o l u t i o n . Thus t u r b u l e n t g e n e r a t i o n seems to be i n s u f f i c i e n t t o account e i t h e r f o r present g a l a c t i c magnetic f i e l d s or f o r galaxy f o r m a t i o n . Astronomical o b s e r v a t i o n s (Bees, B e i n h a r d t , 1972) determine only an upper l i m i t of 1 0 - 8 gauss t o the l a r g e s c a l e ( i e megaparsec) magnetic f i e l d e x i s t i n g i n i n t e r g a l a c t i c space at the present epoch. The present g a l a c t i c magnetic f i e l d has been measured as a few a i c r o g a u s s (Heiles,1976). In t h i s chapter a mechanism i s proposed which r e s u l t s i n a magnetic f i e l d being generated i n the r e g i o n s of high angular momentum (or t u r b u l e n c e ) . T h i s mechanism can provide a s i g n i f i c a n t magnetic f i e l d where g a l a x i e s are formed but very l i t t l e f i e l d i n i n t e r g a l a c t i c space. T h i s mechanism i s the g e n e r a t i o n of the g a l a c t i c and i n t e r g a l a c t i c magnetic 1 Fragmentation has been a problem f o r the cosmic t u r b u l e n c e theory (Peebles,1971). Ihen the sound speed drops d r a s t i c a l l y a t recombination, the t u r b u l e n c e becomes s u p e r s o n i c , r e s u l t i n g i n s m a l l dense knots of matter. 137 f i e l d s by the e v a p o r a t i o n of r o t a t i n g black h o l e s . The proposed s c e n a r i o i s as f o l l o w s . Black h o l e s form i n the e a r l y u n i v e r s e from d e n s i t y f l u c t u a t i o n s . T h i s process i s c o n s i d e r e d by Carr,1975. The black holes are r o t a t i n g i n p a r a l l e l throughout a region of c h a r a c t e r i s t i c s i z e 2 1 a t the time of t h e i r f o r m a t i o n . We c o n s i d e r times of f o r m a t i o n and masses f o r these b l a c k h o l e s such t h a t they evaporate v i a the Hawking process d u r i n g the r a d i a t i o n era of the expansion of the u n i v e r s e . T h i s i m p l i e s black h o l e s of mass of o r d e r 1 0 1 2 grams forming a t times of order 1 0 - 2 * seconds'. . The e v a p o r a t i o n of these black h o l e s r e s u l t s i n a net n e u t r i n o c u r r e n t throughout the r e g i o n of s i z e 1, a n t i p a r a l l e l to the b l a c k hole r o t a t i o n axes*. During the r a d i a t i o n e r a the n e u t r i n o c u r r e n t i n t e r a c t s with e l e c t r o n s and protons i n the cosmic plasma. The d i f f e r e n c e i n n e u t r i n o - proton and a n t i n e u t r i n o - proton c r o s s s e c t i o n s r e s u l t s i n a net proton c u r r e n t p a r a l l e l to the neutrino c u r r e n t . A net e l e c t r o n c u r r e n t p a r a l l e l t o the n e u t r i n o c u r r e n t i s a l s o produced, f o r the same reason. F i g . 14 ( i n Appendix A) i l l u s t r a t e s the 2 1 i s taken to be comoving, i . e . , 1 i n c r e a s e s i n p r o p o r t i o n t o the s c a l e f a c t o r B so i t c o n t i n u e s to d e s c r i b e the s i z e of the black hole r e g i o n s . 3 A l l times are measured from the i n i t i a l s i n g u l a r i t y of the b i g bang. * See chapter 2 of t h i s t h e s i s and Leahy, Dnruh(1979), where the d e t a i l s of n e u t r i n o emission from r o t a t i n g b l a c k h o l e s a r e c a l c u l a t e d . Vilenkin(1978) has a l s o demonstrated the e x i s t e n c e of a n e u t r i n o c u r r e n t from a r o t a t i n g black h o l e . 138 s i t u a t i o n , except that i n s t e a d of a baryon(B) c u r r e n t there a r e proton and e l e c t r o n c u r r e n t s , and t h e r e are no a n t i b a r y o n (B) c u r r e n t s . The long arrows i n f i g . 14 r e p r e s e n t c u r r e n t s and the s h o r t arrows represent the s p i n axes of the e v a p o r a t i n g black h o l e s s . The e l e c t r o n and proton c u r r e n t s are d r i v e n by the n e u t r i n o s from the b l a c k hole and d i s s i p a t e d by c o l l i s i o n s with the background r a d i a t i o n . Due t o the g r e a t e r mass of the proton, i t r e c e i v e s more of the momentum i n a c o l l i s i o n with a n e u t r i n o and a l s o i s slowed down l e s s by the photon background. The outcome i s a s u b s t a n t i a l l y l a r g e r proton c u r r e n t than the e l e c t r o n c u r r e n t , r e s u l t i n g i n a net flow of charge d u r i n g the r a d i a t i o n e r a . T h i s charge c u r r e n t generates magnetic f i e l d s , a f t e r the black hole e v a p o r a t i o n ends (at about 10* y e a r s ) , the f i e l d s remain f r o z e n i n t o the plasma 6. The f i e l d s t r e n g t h decreases as the plasma expands with the u n i v e r s e . The above mechanism i s s u f f i c i e n t t o account f o r the observed g a l a c t i c magnetic f i e l d , which i s of order 10-s t o 10~ 6 gauss (He i l e s , 1 9 7 6 ) . The model o f the u n i v e r s e t h a t i s used here i s * The alignment of the black hole s p i n axes i s d r a s t i c a l l y o v erestimated. As estimated below, the product of alignment with d i m e n s i o n l e s s r o t a t i o n parameter i s c l o s e r to 1 0 ~ 1 6 t o 1 0 ~ 2 0 than u n i t y . 6 The plasma remains s u f f i c i e n t l y conducting, even a f t e r combination of protons and e l e c t r o n s to form hydrogen, f o r t h e magnetic f i e l d t o be f r o z e n i n (e.g. See S p i t z e r , 1 9 6 8 ) . 139 d e s c r i b e d 7 i n the i n t r o d u c t i o n t o chapter 3. The assumptions made about the fo r m a t i o n and e v a p o r a t i o n of r o t a t i n g black h o l e s are a l s o d e s c r i b e d t h e r e . That d i s c u s s i o n i s a p p l i c a b l e here, with the f o l l o w i n g changes. The black h o l e s considered i n t h i s c h a p t e r a re more massive ( 1 0 1 2 gm vs. 10 8 gm). The e f f e c t i v e r o t a t i o n parameter (product of an. and alignment) i s not assumed t o be of order u n i t y . I t s value i s estimated a t the end of s e c t i o n I I I . The o u t l i n e of the chapter i s as f o l l o w s . In s e c t i o n I I the charge c u r r e n t produced by the ev a p o r a t i o n of r o t a t i n g black holes d u r i n g the r a d i a t i o n e r a i s c a l c u l a t e d . The magnetic f i e l d generated over l e n g t h s c a l e 1 i s estimated i n s e c t i o n I I I , using Haxwell*s eguations, i n terms of the i n t e n s i t y of black hole f o r m a t i o n and the mass of the black h o l e s . The r e s u l t s are di s c u s s e d i n s e c t i o n 17. Due to the approximations made numbers a r e accurate t o an or d e r of magnitude only and equ a t i o n s should be i n t e r p r e t e d a c c o r d i n g l y . N a t u r a l u n i t s (see Appendix C o r the t a b l e i n s i d e the r e a r l e a f o f H i s n e r e t a l ) are used unless otherwise i n d i c a t e d . 7 The baryon t o photon r a t i o i s not of concern here., 140 II..CALCULATION OF THE CHARGE CUBBENT In t h i s s e c t i o n , we c a l c u l a t e the charge c u r r e n t t h a t r e s u l t s from r o t a t i n g b l a c k h o l e s which evaporate d u r i n g the r a d i a t i o n e r a . The times of i n t e r e s t f o r magnetic f i e l d g e n e r a t i o n are between the end of the l e p t o n e r a a t time t^=10* 3 (about 1 second) and the recombination of p r o t o n s and e l e c t r o n s a t a r e d s h i f t of about 2000 or at time t r=10s* (about 10* y e a r s ) . Thus black h o l e s of i n t e r e s t are those which evaporate d u r i n g the r a d i a t i o n e r a . P r i o r to the end of the l e p t o n e r a the charged c u r r e n t i s much reduced: the n e u t r i n o s from the black h o l e s s c a t t e r mainly on e l e c t r o n s and p o s i t r o n s , which then outnumber protons by about ^09 to 1. The r e s u l t i n g p o s i t r o n and e l e c t r o n c u r r e n t s , having i d e n t i c a l s c a t t e r i n g from n e u t r i n o s and d i s s i p a t i o n on r a d i a t i o n , c a n c e l i n t h e i r c o n t r i b u t i o n to the charge c u r r e n t . A f t e r recombination the matter i s n e u t r a l . However generat i o n of charged c u r r e n t s which can produce magnetic f i e l d s i s not s i g n i f i c a n t l y more d i f f i c u l t due t o the high n e u t r i n o e n e r g i e s i n v o l v e d . The mean f r e e path of the protons which make up the c u r r e n t t h a t generates the magnetic f i e l d i s l a r g e r than a p r o t o g a l a c t i c s i z e f o r t > 2x10 S 5. Thus t r i s chosen as an upper time l i m i t f o r black hole e v a p o r a t i o n . A l s o , g a l a x i e s are thought to have formed not l o n g a f t e r 141 recombination. Host of the e v a p o r a t i o n o c c u r s f o r times of o r d e r the black bole l i f e t i m e 7 of 100H 3. Thus the masses of i n t e r e s t a re given by t^<100H 3<t r o r : 5x10» 3<H<10»« (2.1) The charged c u r r e n t i s the d i f f e r e n c e between the proton and e l e c t r o n f l u x e s , f and f e : j = e ( f f - f e ) (2.2) I n the absence of source and s i n k terms, the each f l u x per u n i t comoving c r o s s - s e c t i o n a l area i s constant: 0=d(fB2)/dt =B* d f / d t +2B(dB/dt)f B i s the s c a l e f a c t o r of the u n i v e r s e . H i t h source and s i n k terms, the f l u x e s obey: d f / d t * f / t =n(<r v-c~)f - f / t (2.3) T h i s i s s i m i l a r to eqn (2.2) of chapter 3, with z=1/2 f o r a 7 See d i s c u s s i o n i n chapter 3 a f t e r eqn (2.4). 142 r a d i a t i o n dominated expansion. The source f o r the proton and e l e c t r o n f l u x e s i s the n e u t r i n o f l u x from the b l a c k holes, and the s i n k i s s c a t t e r i n g on ambient p a r t i c l e s (protons, e l e c t r o n s , neutrons ( i n He*) , and photons). The proton o r e l e c t r o n mean l i f e t i m e i s l e s s than the expansion t i m e s c a l e - H/(dB/dt)=2t ( t h i s i s demonstrated below when we c o n s i d e r s p e c i f i c s c a t t e r i n g p r o c e s s e s ) . Thus the proton and e l e c t r o n f l u x e s are estimated i n the steady s t a t e approximation, fp and f f e are then g i v e n by dropping the l e f t hand s i d e of egn (2. 3) : 0= (<^yf-<3yf) n f f , - f f / t f 0= (<SVtt -o~i^) n e f v - f e / T _ (2.4a) (2.4b) In the above and i n what f o l l o w s , c and n are the c r o s s s e c t i o n and number d e n s i t y of the v a r i o u s s p e c i e s a t time t : n e u t r i n o s (v) , a n t i n e u t r i n o s (^), protons (p), neutrons (n) , e l e c t r o n s ( e ) and p h o t o n s ^ ) . The Neutrino Flux The proton and e l e c t r o n c u r r e n t s a re d i r e c t l y p r o p o r t i o n a l to the n e u t r i n o f l u x . The n e u t r i n o c o l l i s i o n t i m e s c a l e fc, i s l o n g e r than the expansion t i m e s c a l e t f o r the 143 e n t i r e r a d i a t i o n e r a . T h i s means t h a t most of the n e u t r i n o s emitted by the black h o l e s never i n t e r a c t . From chapter 3, eqn (2.2), the n e u t r i n o f l u x from black h o l e s i s then g i v e n by t ( d j / d t ) b ^ r a t h e r then t ^ d j / f t ) ^ . T h i s can be v e r i f i e d by an exact s o l u t i o n to eqn (2.2) of chapter 3. T h i s r e s u l t s i n the e x p r e s s i o n 8 : f v = 3 x 1 0 - » t ^ H-s * ( E/A)-3 Xftxv n^-^K-Xy (2.5) In (2.5) A v i s the n e u t r i n o mean f r e e path, ^ i s the f r a c t i o n o f energy d e n s i t y going i n t o black h o l e s a t time of formation II, x i s the f r a c t i o n of n e u t r i n o s i n the emitted p a r t i c l e s from the black h o l e 9 , and n b h i s the b l a c k hole number d e n s i t y . i s the ^ r o t a t i o n parameter of the b l a c k holes (0<HfL<1)» where J i i s the angular v e l o c i t y of dragging of i n e r t i a l frames a t the bl a c k h o l e event h o r i z o n . The c o n d i t i o n i n egn(2.5), nnf* / 3<3 v# r e q u i r e s the b l a c k hole s p a c i n g , Rh>t~*/3* t o b e l e s s than the n e u t r i n o mean f r e e path. When t h i s holds, the n e u t r i n o f l u x i s conti n u o u s on s c a l e s l a r g e r than the n e u t r i n o mean f r e e path, i . e . , on the s c a l e 1 o f the r e g i o n of a l i g n e d b l a c k h o l e s . However, here, as opposed to chapter 3, the n e u t r i n o mean f r e e path i s s o long 8 The c o n s t a n t s f and y of chapter 3 have the values .03 and .01, r e s p e c t i v e l y . 9 x=.3 f o r black hole masses of i n t e r e s t here, eg see Carr,1976. 144 t h a t t h i s c o n d i t i o n i s u n i a p o r t a n t . To c a l c u l a t e the n e u t r i n o mean f r e e path we must know the n e u t r i n o c r o s s s e c t i o n . The n e u t r i n o c r o s s s e c t i o n on ambient protons o r e l e c t r o n s , cs v , can be approximated b y 1 0 : S^e+Sy/lSy + RvJ*) 2 s v=2E ym (2.6) The nean energy of n e u t r i n o s , E v , emitted from the r o t a t i n g b l a c k h o l e s i s a p p r o x i m a t e l y 1 1 1/M. F o r black h o l e s of i n t e r e s t here, t h i s g i v e s a 4-momentum t r a n s f e r sguared f o r n e u t r i n o s on p r o t o n s 1 2 , s ^ , o f : 2x10-3 7 (=30 (Gev) 2 ) < s v ^ < 4 x 1 0 - « (=6x10» (Gev) 2 ) The f o u r momentum t r a n s f e r f o r n e u t r i n o s on e l e c t r o n s , s ^ , i s l e s s by a f a c t o r o f the r a t i o o f e l e c t r o n to proton masses: me/m^ =5x 10~*« The H boson mass, H^, i s taken as 70 Gev= 6 x 1 0 - 1 8 . By egn(2.6) the n e u t r i n o s are stopped p r i m a r i l y by protons (tf^>o\«.) with a mean f r e e path #j={n^<fy ) " * • The s c a t t e r e d e l e c t r o n s and protons w i l l be produced with e n e r g i e s of about /s, and thus w i l l be h i g h l y >« See Appendix D, a l s o S c i u l l i (1974) with GH(l>2=g2/4/2' and g=e/sin&„, sin 2£*=.23; g i s the dime n s i o n l e s s weak c o u p l i n g c o n s t a n t , eg see G a i l l a r d , 1 9 7 9 1 1 See the summary o f chapter 2. 1 2 The mass of the proton i s m ^ S x I O - 2 " or .9 Gev., * 145 r e l a t i v i s t i c . He e v a l u a t e the n e u t r i n o f l u x between the end o f the le p t o n e r a , at time t ^ , and recombination, a t t r . We assume the energy d e n s i t y of b l a c k h o l e s i n the universe i s s m a l l . Then the standard r a d i a t i o n dominated Friedman expansion i s a p p l i c a b l e * 3 : s c a l e f a c t o r B=Atv* # energy d e n s i t y ^J=.03t-*, number d e n s i t y n^=.07t - 3 / S, and r a d i a t i o n temperature T=.4t~V2. The r a d i a t i o n e r a ends a t the time when the energy d e n s i t i e s of matter and r a d i a t i o n a re egual and i s approximately the same time t h a t recombination o c c u r s . Here we take both times t o be the same: t r = 1 0 s * (about 10* y e a r s ) . Upper l i m i t s on the black h o l e energy d e n s i t y a r e now found so t h a t the b l a c k h o l e s dc not dominate the expansion. The black hole energy d e n s i t y a t the time of f o r m a t i o n (at t=H) i s a f r a c t i o n of the t o t a l energy d e n s i t y .03t-« (=.03H - Z a t t=H) . I t t h e r e a f t e r decreases as B - 3 . Then the black hole energy d e n s i t y i s given by: =.03£H-" 2t- 3* b e f o r e the f i r s t of t = t r or t=100H 3 (the black hole l i f e t i m e ) . The upper l i m i t t o the i n t e n s i t y of b l a c k hole f o r m a t i o n of ^ <1/10H i s found»• by r e q u i r i n g the energy d e n s i t y i n black h o l e s to be l e s s than t h a t i n r a d i a t i o n f o r a l l times l e s s than 100H 3. T h i s c o n c l u s i o n i s v a l i d f o r black 13 see eqn (3.3) of chapter 3 and the d i s c u s s i o n j u s t p r i o r t o i t (f=.03). ** T h i s i s a l s o d e r i v e d i n ch a p t e r 3, eqn (3.2). 146 h o l e s which evaporate completely b e f o r e the end of the r a d i a t i o n e r a . A f t e r the r a d i a t i o n e r a , matter dominates the energy d e n s i t y and the energy d e n s i t y i s no l o n g e r r e d s h i f t e d by a f a c t o r of E with r e s p e c t to the b l a c k hole energy d e n s i t y . The a p p r o p r i a t e l i m i t on the i n t e n s i t y of black hole f o r m a t i o n f o r bl a c k hole masses g r e a t e r than (t^/100) i/3=io»8 i s t h e n ^ < 1 0 - 1 9 , independent of H, so t h a t black h o l e s do not dominate a t the time tr or l a t e r . The c o n d i t i o n f o r n e u t r i n o f l u x c o n t i n u i t y over a r e g i o n o f a l i g n e d black h o l e s , (the i n e g u a l i t y i n egn ( 2 . 5 ) ) , i s e v a l u a t e d here. For black h o l e masses M>5x1Q 1 5, one has s v < ><H w 2. By eqn (2.6), one then has <r„e =e*s/H H* =7x10**/M. The proton number d e n s i t y nf i s g i v e n by ^ ^=7x10"* »t -3te , where /^=10-9 i s the baryon t o photon r a t i o . The b l a c k hole number d e n s i t y nfcv^ i s given by pkK/&. The c o n d i t i o n f o r n e u t r i n o f l u x c o n t i n u i t y , n t K - » / 3 < A i n eqn (2.5), then becomes a lower l i m i t on the i n t e n s i t y of black hole f o r m a t i o n , ^ : yj>5x10-»* (H/5x10iS)-** ( t / t t ) -3 T h i s lower l i m i t i s completely n e g l i g i b l e and does not c o n f l i c t at a l l with the upper l i m i t on ^ of ^<1/1 OH. T h i s c o n c l u s i o n can be v e r i f i e d a l s o f o r H<5xl0> 5, u s i n g the f u l l formula f o r <rv<>. Thus r e g u i r i n g n e u t r i n o f l u x c o n t i n u i t y i n 147 the r a d i a t i o n dominated period does not f u r t h e r r e s t r i c t fl. Me are i n t e r e s t e d i n magnetic f i e l d s generated on a p r o t o g a l a c t i c s c a l e . T h i s i s taken to be 200 k i l o p a r s e c s a t t h e present time and i s d i r e c t l y p r o p o r t i o n a l to B ( t ) . The n e u t r i n o mean f r e e path i s A, = i^fft-f ) ~*• T h i s r e s u l t s i n 3„ = 2x10-"Ht*2 H>5x10»s J|y=4x10-*t*2/H fl<5x10*s Since %>t f o r the e n t i r e r a d i a t i o n e r a , the lower l i m i t t o the s e p a r a t i o n s c a l e i s t , by c a u s a l i t y . The p r o t o g a l a c t i c s c a l e i s l=2x10 2 5tvfc (see eqn (3.7) below). In the case t < l , the n e u t r i n o s have not had enough time t o t r a v e l more than 1. T h i s l i m i t s t f o r the n e u t r i n o c u r r e n t t o : t<4x10«° Inhomogeneities i n the d e n s i t y of matter don't h e l p t o l i m i t the extent of the magnetic f i e l d r e g i o n . T h i s i s because both the source and s i n k terms f o r the c u r r e n t j are p r o p o r t i o n a l t o the matter d e n s i t y . Thus the n e u t r i n o f l u x a t time t i s then given by egn (2.5) when r a d i a t i o n dominates the expansion: f v = 3 x 1 0 - » t - ^ H - s ^ x f l « . H>5x10i3 t < V (2.7) 148 For the p r o t o g a l a c t i c f i e l d , one has t < 4 x 1 0 5 0 . The Proton and E l e c t r o n L i f e t i m e s The k i n e t i c equations (2.4a), (2.4b) g i v e the proton and e l e c t r o n f l a x e s f ^  and f e . Here we compare the d i f f e r e n t s c a t t e r i n g mechanisms t o determine the dominant one. The important q u a n t i t y t o compare i s the energy l o s s r a t e r a t h e r the c o l l i s i o n t i r a e s c a l e s f o r d i f f e r e n t processes. T h i s i s because some c o l l i s i o n s (e.g. proton-photon) occur f r e q u e n t l y but have l i t t l e e f f e c t , whereas other c o l l i s i o n s (e.g.,proton-neutron) d r a s t i c a l l y a l t e r the energy and d i r e c t i o n of the i n c i d e n t p a r t i c l e . Here, proton-proton and e l e c t r o n - e l e c t r o n s c a t t e r i n g s are i g n o r e d . & c o l l i s i o n of a r e l a t i v i s t i c p a r t i c l e with an ambient l i k e p a r t i c l e i n c r e a s e s the charge c u r r e n t . . A f t e r the c o l l i s i o n , one has 2 f a s t moving charged p a r t i c l e s r a t h e r than 1. .. By momentum c o n s e r v a t i o n one has: 2 f m T = x . a v i *&avA. where 1 and 2 r e f e r t o the 2 p a r t i c l e s a f t e r the c o l l i s i o n and $ = ( 1 - v 2 ) - * ' a i s the r e l a t i v i s t i c f a c t o r . The charge c u r r e n t i s given by ev before and e(v, *v 2) a f t e r the c o l l i s i o n , but v,+v^>v f o r $>1. The maximum i n c r e a s e i n the c u r r e n t i s by a f a c t o r of y, a f t e r any number of c o l l i s i o n s with l i k e p a r t i c l e s . In p r a c t i c e , the i n c r e a s e w i l l be much l e s s due t o c o l l i s i o n s with u n l i k e particles.„,, Here y f o r 4 149 protons ranges from 5 to 800, depending on when the black holes f i n i s h e v a p o r a t i n g ( t r to t ^ ) . Thus we ignore the above e f f e c t . F i r s t we c o n s i d e r s c a t t e r i n g on photons. The r e l a t i v i s t i c Compton s c a t t e r i n g c r o s s s e c t i o n (eg Heitler,1954) f o r the p a r t i c l e s of the f l u x e s s c a t t e r i n g on ambient r a d i a t i o n i s : C5y = « C T = c*V S « 1 =<SV3(ln2S + 1/2)/(8fr) S » 1 (2.8) with 6=Ej/m. E y i s the photon energy i n the p a r t i c l e r e s t frame: Ej=T(E y/m), where T i s the r a d i a t i o n temperature. Er i s the n e u t r i n o energy (roughly the same as the p a r t i c l e energy) and m i s the p a r t i c l e mass (proton or e l e c t r o n ) so t h a t photon energy i s u p s h i f t e d from T by roughly the f a c t o r Er/m. S=(m*Ht*^)-» ranges from 2x10* to 3x10~» f o r the proton f l u x , 6x10« to 10-2 f o r the e l e c t r o n f l u x , f o r n>5x1Qi3, t < t r (t=100H 3). . <Cp i s the Thompson c r o s s s e c t i o n : Cv=8Trr e2/3 with r , = e V " =4x10**(electrons) = 103s (protons) (2.9) 150 The energy loss i n a p or e c o l l i s i o n Afy i s roughly gzl» The temperature T=. I t - * ' 2 i s the energy of an ambient photon, and y=E/m (E=1/H) i s the r e l a t i v i s t i c factor of the p or e: AZn =103«t-*>*H-2 (proton) £E.j =2x10**t-*«M-2 (electron) The d e f l e c t i o n of the p or e i n a c o l l i s i o n with a photon i s small (of order 4Ey / E ) . The proton electron cross section for large momentum transfer i s 1 5 : c^s.=e*/s with (2. 10a) s=2js^J f o r r e l a t i v i s t i c p on ambient e (2.10b) s=2/s^ ay fo r r e l a t i v i s t i c e on ambient p (2. 10c) The proton neutron cross section i s almost independent of energy, with a value of 40 millibarns or 10+° i n natural units. For electrons c o l l i d i n g with ambient protons, the 1 5 This can be viewed as being egn (2.6) with Hw=0, i n the s p i r i t of unified weak and electromagnetic theories.. 151 d e f l e c t i o n angle i s l a r g e so the e f f e c t i v e energy l o s s f o r f^ i s the o r i g i n a l e l e c t r o n energy, 1/H. For protons c o l l i d i n g with e l e c t r o n s the proton energy l o s s i s roughly 2^ f f lc : ^ ^ = 7 x 1 0 * 5 8 - 2 F i n a l l y , f o r protons c o l l i d i n g with neutrons {in He*), the d e f l e c t i o n angle i s again l a r g e so t h a t the e f f e c t i v e energy l o s s i s of order Ep = 1/H per c o l l i s i o n . The energy l o s s r a t e due to a giv e n process i s g i v e n by r=dE/dt= 4E/t t with the c o l l i s i o n time g i v e n by t t=1/n«v. Si n c e the protons and e l e c t r o n s are r e l a t i v i s t i c , we have v=1. The baryon to photon r a t i o i[ i s d e f i n e d by n , , = ^ n y . Since the cosmic gas i s about 1/4 He* and 3/4 H by mass, the neutron to proton r a t i o n i s 1/7. Thus one has n r = n c = Y u y - ^ 1 1 * -The r a t i o s of energy l o s s r a t e s r e l e v a n t t o egns(2.4a) and (2.4b) a r e : r n /Vft =&En a f y /AEffSk = 10 « H- V21- & (2. 11 a) r p z r / r r n =lA*n<?iv/&fqG+ =7x 10** ^M~it-*« (2. l i b ) C . y / C e f = A E € / < r « , / A E € ^ ^ = 3 x 1 0 * » ^ e H - 3 * t - ^ (2.11c) 1 0 - « < ^ < 1 10-8<o^.<1 with << f o r protons or e l e c t r o n s d e f i n e d by the photon s c a t t e r i n g c r o s s s e c t i o n egn (2.8). 152 Host of the black hole e v a p o r a t i o n occurs f o r t o f order 100H 3. The r a t i o s of energy l o s s r a t e s from eqns (2.11) become: From t h i s i t i s seen that p-y d i s s i p a t i o n i s always more important than pe d i s s i p a t i o n ( f o r t<10*°=2x10* y e a r s , by egn (2.12a)). For t > 1 0 5 2 pn d i s s i p a t i o n i s dominant over For charge c u r r e n t s on the s c a l e of a protogalaxy we are l i m i t e d to t <4x10 5 0, so we w i l l be always i n t e r e s t e d i n pjr d i s s i p a t i o n . Thus the dominant d i s s i p a t i o n mechanism f o r the proton and e l e c t r o n c u r r e n t s i s c o l l i s i o n with the photons i n the background r a d i a t i o n r a t h e r than with ambient protons, neutrons, or e l e c t r o n s . Dominance Of The Proton Current Here we c a l c u l a t e the magnitude of the proton and e l e c t r o n c u r r e n t s . F o r e n e r g i e s of n e u t r i n o s emitted by b l a c k h o l e s such t h a t s v<M w 2, the c r o s s s e c t i o n asymmetry f o r n e u t r i n o s on protons or e l e c t r o n s : rf> ' rr« = 3 x i o » 3 t f f t - % * r ^ / r ^ a x l O ' o ^ t - i (2.12c) (2. 12b) (2.12a) d i s s i p a t i o n , whereas f o r t < 1 0 5 2 p^ dominates. 153 (<Tv*-<Tv* >/<*;* <*=P or e) has a value of roughly 2/3 (appendix D with A<<1, or S c i u l l i , 1974). For higher e n e r g i e s (s y>H ( ( f 2) the asymmetry only decreases l o g a r i t h m i c a l l y with energy. Thus we can ignor e a n t i n e u t r i n o s c a t t e r i n g by dropping ff^x i n egns (2.4). The proton and e l e c t r o n f l u x e s due to bl a c k hole n e u t r i n o s are then given by eqns (2.4a) and (2 . 4 b ) a s 1 * : f f =7 fyCTyp /<Sfn =150^H-W2t-»*xfl,r7. -t> (2.13a) f f = ^ f v<y> P/(<T7, 4E„/E r) t<10« 2 (2.13b) = 2x10-*» /SM-**xHJl M>5x10»s = 9 X 1 0 - 7 2 ^ H - » / 2 X H J T M<5x10»s £e=1jzv<rve. /(<JiydE^/F i,) = 10-5^H-»2xaA/Ai. (2. 13c) For H>5x10»s i n egn (2.13b) and f o r (2.13a), s v<M w 2 has been used i n the n e u t r i n o proton c r o s s s e c t i o n , eqn (2.6). For s m a l l e r H, i n eqn (2. 13) b, the approximation S y > H „ 2 i s used. For eqn (2.13c), s y<M w 2 h o l d s f o r a l l black hole masses of i n t e r e s t (eqn (2.1)) because of the sm a l l mass of the e l e c t r o n . »* For s i m p l i c i t y I s e t ^,= 1 f o r H>5x10»s and c£=.1 f o r H<5x10is i n eqn (2.13b) (see eqn (2.8) ) . 154 The r a t i o o f e l e c t r o n f l u x t o proton f l u x i s found to be very s n a i l : f e / f , = i&re.G'rx)/(o-^O^y) = ( B e / H p ) 3=10-*o t<10sz (2.14a) f« /% = ( o V * * " , , ) / ^ * ^ - /Sy)=(B*./ni f ) 2x10 - n / * e t>1Qsz (2.14b) The proton f l u x d o a i n a t e s the charged c u r r e n t j : j = e f r H>5x10i3 t < t r (2.15) The f a c t o r xHA i s roughly constant f o r Bost o f the b l a c k hole l i f e t i m e changing o n l y r a p i d l y i n the f i n a l e x p l o s i v e phase of the black hole evaporation. R o t a t i n g black h o l e s of masses o f i n t e r e s t here have roughly x=.3. e has the value . 1 i n n a t u r a l u n i t s (see Appendix C). S u b s t i t u t i n g t=1008 3 then g i v e s the charge c u r r e n t d u r i n g the a a j o r p a r t o f the black hole e v a p o r a t i o n f o r any given i n i t i a l b l a c k hole mass H: j = 3x10-**yffH- »*Mn 5x10i3<H<5x10is (2.16a) j=6x10-*3^n-s«H/t 5x10i«<H<5x10i* (2.16b) j=.4^a-*a>t 5x10i*<H<10ie (2.16c) Si n c e we are i n t e r e s t e d i n the f i e l d generated on the s c a l e 1 of a protogalaxy, only egns (2. 16a) and (2. 16b) are 155 applicable (due to the requirement t<l) and eqn (2.16b) only f o r H<1.6x10i*. Pith the simplifying assumptions described i n t h i s section, the charge current j i s characterized by just 3 parameters- the mass of the black holes, t h e i r i n t e n s i t y of formation, , and their e f f e c t i v e rotation parameter. M». In the next section the magnetic f i e l d generated by t h i s charge current i s estimated. I I I . GENERATION OF THE MAGNETIC FIELD The e l e c t r i c and magnetic f i e l d s generated by a charged current j can be calculated from Maxwell's equations: c u r l E • dB/dt=0 c u r l B - dE/dt=<*-irj (3.1) with E and B the e l e c t r i c and magnetic f i e l d s , respectively, i e assume the black holes have aligned spin axes over scale 1, so that the neutrino and charge currents are homogeneous on scale 1. The currents drop from t h e i r equilibrium values (egns (2. 7) , (2. 13), and (2.16)), around the centre of the region, to zero within a few neutrino mean free paths beyond the boundary. Thus one can approximate c u r l by 1/1. Because of the power law dependences of the various quantities (eg 156 n = . 0 7 t - 3 « , B p r o p o r t i o n a l to t ^ j , d/dt can be approximated by 1/t. For a charge current j s t a r t i n g at time much l e s s than t and of d u r a t i o n of at l e a s t t , the magnetic f i e l d at time t over the scale 1 i s found from eqn (3. 1) to be: B = 4 T r J l t * / ( l 2 * t * ) (3.2) The magnetic f i e l d generated by the proton c u r r e n t of egn (2.16) i s then of order: B=4irjt 2/1 t^<t<l(t) (3.3a) B=4irjl l(t)<t<t,. (3.3b) T h i s r e s u l t s i n s e v e r a l cases, depending on M and 1, s ince the charge c u r r e n t has 3 cases (eqns (2.16)) and 1 has 2 cases. The s c a l e 1 s a t i s f i e s the c o n d i t i o n : 1 (**) >tl=2x10tOcm for (3. 3a) 1 ( t r ) <t r=2x10 2 3cm f o r (3.3b) (3.4) Eqn (3.3a) i s the case where the length scale of the c u r r e n t s and magnetic f i e l d i s l a r g e r than the c a u s a l h o r i z o n a t time t and eqn (3.3b) i s the case where the magnetic f i e l d scale i s smaller than the causal h o r i z o n . Egn (3.4) expresses the 157 l i m i t s on 1 imposed by r e q u i r i n g the magnetic f i e l d g e n e r a t i o n to occur d u r i n g the r a d i a t i o n e r a ( t f c < t < t r ) . For g e n e r a t i o n of the g a l a c t i c magnetic f i e l d , only egn (3. 3a) i s a p p l i c a b l e (t<l) . The magnetic f i e l d w i l l grow u n t i l the b l a c k h o l e s f i n i s h e v a p o r a t i n g at time t (=100M 3. when the f i e l d c eases to grow a t time t , , i t w i l l have the value B,: B,=1. 2x105 j H V l ( t , ) 5x10» 3<W<l» 3 (t,)/4.6 (3.5a) B, = 12 j l ( t , ) 1*« (t,)/4.6<M<10»« (3.5b) Eqn (3.5a) corresponds t o the case (3.3a) where the magnetic f i e l d s c a l e 1, at the time when the c u r r e n t s cease, i s l a r g e r than the c a u s a l h o r i z o n . Eqn (3.5b) corresponds t o the case (3.3b) where 1 (t,) i s smaller than the c a u s a l h o r i z o n . A f t e r the g e n e r a t i n g c u r r e n t s s t o p , the magnetic f i e l d w i l l evolve with f l u x c o n s e r v a t i o n : BR 2=constant, as the cosmic medium expands with the s c a l e f a c t o r B. T h i s holds s i n c e the magnetic f l u x i s f r o z e n i n t o the cosmic medium. Even a f t e r the time of recombination, the number d e n s i t y o f f r e e e l e c t r o n s and protons i s s u f f i c i e n t t o l o c k the f i e l d l i n e s i n t o the medium (e.g. S p i t z e r , 1968). The e v o l u t i o n of the s c a l e f a c t o r with time i s g i v e n by: 158 B = A t V 2 t<t r ; R=Bt«^ t>t,. (3.6) T h i s f o l l o w s from egn (1. 3) o f chapter 3, which g i v e s yO(R) , with p=/)/3 f o r r a d i a t i o n and p=0 f o r c o l d matter, together with eqn (1.2) of chapter 3 to y i e l d the time dependence. Since the energy d e n s i t y of matter and r a d i a t i o n i s g r e a t e r than t h a t of the magnetic f i e l d , the magnetic f i e l d s c a l e 1 expands d i r e c t l y i n p r o p o r t i o n to JB. With t, l e s s than t r , the present value of the magnetic f i e l d l e n g t h s c a l e 1^ i s : l ^ = ( t P / t r ) * * ( t r / t x ) »*l(t,) (3.7) = 2x10* (1056/t, ) ^ l ( t j Here I have taken the present time as t f=10* 1 =5.4x10* 7 sec =17x10* y r . The l i m i t s on l p corresponding to cases (3.3a) 1, >t, and (3.3b) l,<t, are r e s p e c t i v e l y : l„>10zo cm and l . < 4 x 1 0 « cm The present magnetic f i e l d B f i s then found by using Bf =3x10~*3tj B, =3x 10-*»M3B (3.8) 159 Magnetic F i e l d Generated over A _ P r o t o q a l a c t i c Scale He are i n t e r e s t e d i n the case t h a t lf corresponds t o g a l a c t i c s i z e s c a l e . The c o l l a p s e phase of the f o r m a t i o n of a galaxy i s taken i n t o account. During c o l l a p s e the l o c a l medium does not expand as the s c a l e f a c t o r S. The g a l a x i e s a r e thought to have r e s u l t e d from c o l l a p s e of a r e g i o n t h a t i f comoving with the expansion would be of order 10 times the g a l a c t i c s i z e . Thus we take 1^, = 200 k i l o p a r s e c s =4x10 S 6. Since B r 2 i s constant, where r i s the s i z e of the magnetic f i e l d r e g i o n , the f i e l d i n a galaxy has been a m p l i f i e d by a f a c t o r of order 100 over the estimate of egn(3.8). l i t h the r e s t r i c t i o n l^=4x10 s*, the magnetic f i e l d a t time t, =100M 3 i s now found. From egn (3.7), one has l ( t , ) = 2 x 1 0 2 * H 3^. Only case (3.5a) i s a p p l i c a b l e . Equations (2.16) f o r j then r e s u l t i n the f o l l o w i n g values of B, : B, =2x10-»4pM*Mn 5x10» 3<H<5x10»s (3.9a) B 1=4x1O-«*0M 2H . r L 5x10»s<M<1.6x10»* (3.9b) To e v a l u a t e B we need e s t i m a t e s of p and MJI. He f i r s t c o n s i d e r the case that p has the upper l i m i t 1/10M so that the u n i v e r s e does not become black h o l e dominated.. To e s t i m a t e M/2 we c o n s i d e r the cosmic f l u i d a t the time of b l a c k hole f o r m a t i o n t=M. Since g a l a x i e s have such l a r g e 160 angular momentum, they w i l l be r o t a t i n g with v=1 a t t h e i r edges. T h i s g i v e s an average angular v e l o c i t y of 1/1 (/VI)= 5 x 1 0 - 2 * H - l £ . The l o c a l angular v e l o c i t y on s c a l e s much l e s s than 1 i s l i k e l y much g r e a t e r , but the average alignment of these s m a l l e r eddies w i l l r e s u l t i n the same e f f e c t i v e a n g u l a r v e l o c i t y 1/1 f o r the b l a c k h o l e s . The e f f e c t i v e r o t a t i o n parameter i s then Mn=5x 10~ 2*H 1 / 2. The above estimate f o r Hii, can be a l t e r e d , depending on the e a r l y h i s t o r y of the s c a l e f a c t o r B ( t ) . Since 1 i s p r o p o r t i o n a l t o E, any change i n R from t V 2 b e h a v i o u r w i l l change 1 from the value above, and thus change SI. Host n a t u r a l t o c o n s i d e r i s a matter dominated expansion (by the bl a c k holes) which occurs i f £>1/10M. T h i s has B p r o p o r t i o n a l t o t 2 / 3 . In other models E can i n c r e a s e f a s t e r than t 2 ^ , r e s u l t i n g i n even l a r g e r values of Hn» The pre s e n t magnetic f i e l d , a l l o w i n g f o r galaxy c o l l a p s e , i s given by 3x10-s 9H 3B,, by egn (3 . 8 ) . S u b s t i t u t i n g f o r HA, t h i s r e s u l t s i n : B r=3x10-*'^ M*s* B f =5x10-»*«YH»i* 5x10»3<M<5x10is 5x10»s<H<1.6x10i* (3.10a) (3.10b) 161 IV. DISCUSSION The present magnetic f i e l d of our galaxy (Heiles,1976) and t h a t i n other g a l a x i e s {eg Sandage,Lynds,1963) i s of order 10~ 5 to 1 0 - 6 gauss. We take i t t o be 4x10-* gauss. T h i s i s 1 0 - 6 2 i n n a t u r a l u n i t s * 8 . A l l o w i n g f o r the c o l l a p s e phase of a galaxy, o n l y 1/100th of t h i s f i e l d ( i . e . 4 x l 0 - 8 gauss) i s r e g u i r e d f o r the f i e l d p r e d i c t e d by egn (3.10).. From egn (3.10a) we f i n d t h a t the maximum present f i e l d , with £=1/10M, i s 2 x 1 0-". T h i s i s s h o r t of the r e g u i r e d f i e l d by a f a c t o r of about 10* 3. One means of i n c r e a s i n g the f i e l d i s by i n c r e a s i n g the black h o l e d e n s i t y . The e f f e c t s of a l a r g e black h o l e d e n s i t y are l i s t e d i n Table 2 and d e s c r i b e d i n chapter 3. F i r s t we c o n s i d e r the e f f e c t s of having a l a r g e number of b l a c k holes of s m a l l e r mass M, than those g e n e r a t i n g the magnetic f i e l d (mass M) . The h o l e s of mass M, w i l l have evaporated b e f o r e f i e l d g e n e r a t i o n o c c u r s , so t h e i r net e f f e c t i s to change the s c a l e f a c t o r R by a f a c t o r of (^lOM,)*^. T h i s means t h a t 1 was s m a l l e r by t h i s f a c t o r when the black holes of mass H formed, ( p r o v i d i n g tKHj/jg, 2). Thus MA i s l a r g e r by t h i s f a c t o r : 1< {px 10M, ) *^<5x10s * 8 1 gauss i s 2.33x10~ 5 7 n a t u r a l u n i t s , see appendix C. 162 T h i s i s not a s u f f i c i e n t i n c r e a s e to r e s u l t i n the observed g a l a c t i c f i e l d . Next we c o n s i d e r a l a r g e r d e n s i t y of black h o l e s with mass a . Then Ksi i s i n c r e a s e d by (^10M) 1 / 3, and p i n the formula (3. 10) f o r B i s l a r g e r than 1/10H. The expansion with £>1/10M w i l l be black hole dominated f o r t<100M 3/2 and r a d i a t i o n dominated f o r t>100M 3/2 (si n c e more than h a l f of the mass of the black holes has evaporated and i s i n the form o f r a d i a t i o n ) . Thus we can s t i l l use egn (3.10) which a p p l i e s f o r a r a d i a t i o n dominated expansion, m u l t i p l i e d by (^10M )V 3 ( f o r the i n c r e a s e i n HJL) . From egn (3.10) we have; B p=6x10-i*«p*3M*** 5x10» 3<fl<5x10i5 (4.1a) B p = 10-»* 7^*/ 3B 3sy6 5x10»s<H<1.6x101* (4.1b) f o r jS> 1/10M. A f i e l d of 10-** i s obtained f o r M= 1.6x10** and ^=2x10 - 9. For p=1» the minimum mass t h a t can give B f=10 - 6* i s H=2x10**. Thus with i n c r e a s e d black h o l e d e n s i t y , we can achieve a present day f i e l d which agrees with that observed f o r the galaxy. The e s t i m a t e s above i n d i c a t e t h at a present g a l a c t i c f i e l d of 4 x 1 0 - 6 gauss could have been by generated by n e u t r i n o c u r r e n t s from r o t a t i n g black holes formed i n the e a r l y u n i v e r s e . These black h o l e s evaporate with the product o f average alignment over a g a l a c t i c s i z e s c a l e and r o t a t i o n 163 parameter B J i having a value of about 1 0 _ l s . T h i s value f o r the e f f e c t i v e lifl c o u l d be s i g n i f i c a n t l y l a r g e r i f the black h o l e s were not a l i g n e d randomly except f o r g a l a c t i c r o t a t i o n , as assumed here. The black hole f o r m a t i o n process i n a t u r b u l e n t medium has not been considered but more than enough angular momentum i s present i n the cosmic medium f o r the black holes to have i n d i v i d u a l r o t a t i o n parameters of order u n i t y , as d i s c u s s e d i n the i n t r o d u c t i o n of chapter 3. The r e q u i r e d i n t e n s i t y of black hole f o r m a t i o n ranges from 1 ( f o r H=4x109gm) to 2x10~ 9 ( f o r a=3x10*»gm) , by eqn (4.1). The i n t e n s i t y of black hole f o r m a t i o n , ^ , i s the f r a c t i o n of energy d e n s i t y going to form black holes a t time fl when they form. Thus the b l a c k h o l e s can be a s m a l l p e r t u r b a t i o n when they are formed, but they always dominate the energy d e n s i t y i n the u n i v e r s e , l a t e r d u r i n g t h e i r e v a p o r a t i o n . The times of formation f o r the above range of b l a c k hole masses are 10- 3°sec ( f o r 4x10 9gm), and 8 x 1 0 ~ 2 a s e c ( f o r 3x10*» gm) . T h e i r number d e n s i t y 1 9 a t the time of formation ranges from 9x105 3cm-3 (4x109gm) to 4x10 3«cm- 3 (3x.W>llgn). The reason t h a t these numbers vary over such a l a r g e range i s t h a t the s m a l l e r black h o l e s have such a short l i f e t i m e . T h i s r e s u l t s i n a short time f o r the n e u t r i n o , and thus proton, c u r r e n t s 1 9 See egn (2.7a) of chapter 3 ( i n n a t u r a l u n i t s ) . . 164 t o operate i n ge n e r a t i n g the magnetic f i e l d . The s m a l l black h o l e s r e q u i r e a l a r g e r number d e n s i t y to compensate f o r t h i s e f f e c t . In order to produce the g a l a c t i c f i e l d , the black holes must dominate the energy d e n s i t y of the universe. T h i s i s not i n c o n s i s t e n t with o b s e r v a t i o n s , provided t h a t the emission from the bl a c k holes can t h e r m a l i z e so as not to d i s t o r t the spectrum of the 3 degree microwave background. I f the bl a c k h o l e s evaporate e a r l y enough, before t . ^ , t h i s w i l l occur ( Z e l ' d o v i c h , Sunyaev, 1969). I f the r a t i o of the present d e n s i t y t o c r i t i c a l d e n s i t y , p jo0 , i s 1 0 - 2 , then t + K c r w ^ i s 10s. Then black h o l e s of mass M,, which evaporate before t=10s, can a l t e r the h i s t o r y of the s c a l e f a c t o r R so that t h e e f f e c t i v e asymmetry parameter MJi i s i n c r e a s e d . However the f i e l d g e n e r a t i n g black h o l e s cannot dominate the energy d e n s i t y i n the u n i v e r s e . I t was shown above that black h o l e s o f mass M, c o u l d not i n c r e a s e MJL and thus B s u f f i c i e n t l y . I f f/p* i s 1, then t + ^ f ( i i i s 100 years. Now the c u r r e n t generating black holes of mass fl can dominate, with black hole l i f e t i m e l e s s than t ^ r j i e r m , so t h a t the microwave background i s not d i s t o r t e d . Then eqns (4.1) are a p p l i c a b l e . Therefore i n order t h a t the microwave background not be d i s t o r t e d , the d e n s i t y i n the un i v e r s e must be near the c r i t i c a l d e n s i t y . A more complete treatment would i n c l u d e e f f e c t s such as 165 black hole f o r m a t i o n over a spectrum of masses r a t h e r than a s i n g l e value and would use more d e t a i l e d e s t i m a t e s o f the e n e r g i e s and numbers of n e u t r i n o s emitted by black h o l e s . The next step needed f o r the mechanism i s an understanding i n the p r o p e r t i e s of the very e a r l y u n i v e r s e (at times of order 10 - 2 8 seconds) i n order to p r e d i c t the i n t e n s i t y and r o t a t i o n f o r black hole f o r m a t i o n . V i l e n k i n ( p r i v a t e communication) has suggested c o n s i d e r i n g d i r e c t charged p a r t i c l e e m i s s i o n from r o t a t i n g black h o l e s , which would be asymmetrical due to weak i n t e r a c t i o n s with n e u t r i n o s , or asymmetrical weak decay of p o l a r i z e d p a r t i c l e s emitted by r o t a t i n g black h o l e s . The l a t t e r would e l i m i n a t e the d e t r i m e n t a l e f f e c t s here due to the extremely long mean f r e e path of the n e u t r i n o s , which put an upper l i m i t on the generation time f o r g a l a c t i c s i z e d magnetic f i e l d s . A p r e g a l a c t i c magnetic f i e l d i s not n e c e s s a r y 2 0 to e x p l a i n the present observed g a l a c t i c f i e l d . However, a p r e g a l a c t i c f i e l d seems t o be an e s s e n t i a l i n g r e d i e n t f o r galaxy formation. Wasserman(1978), s t a r t i n g with a cosmic magnetic f i e l d at the time of recombination, such as can be produced by the mechanism presented here, shows t h a t the 20 Parker(1969) has estimated t h a t a m p l i f i c a t i o n of some seed f i e l d by t u r b u l e n t motions i n the galaxy c o u l d c r e a t e a f i e l d o f 10-* gauss i n l e s s than 10 9 years. 166 e x i s t e n c e of g a l a x i e s , g a l a c t i c angular momenta and g a l a c t i c magnetic f i e l d s can a l l be accounted f o r 2 1 . T h i s i s g u i t e s i g n i f i c a n t , c o n s i d e r i n g our l a c k of understanding of the process of galaxy f o r m a t i o n . The i n t e r g a l a c t i c magnetic f i e l d has an upper l i m i t of 1 0 - 2 times the g a l a c t i c magnetic f i e l d . T h i s f a c t o r i s j u s t due t o the c o l l a p s e to form g a l a x i e s , which a m p l i f i e s t h e i r f i e l d by a f a c t o r of 100. Since there i s expected to be an i n i t i a l c o n c e n t r a t i o n o f f i e l d i n r e g i o n s where g a l a x i e s form, the r a t i o of i n t e r g a l a c t i c to g a l a c t i c magnetic f i e l d s t r e n g t h s could be c o n s i d e r a b l y l e s s than 1 0 - 2 . T h i s i s c o n s i s t e n t with the o b s e r v a t i o n s but i s independent of the model of magnetic f i e l d g e n e r a t i o n as long as the f i e l d i s generated before galaxy formation. V. SOKflABY The black h o l e s considered here evaporate d u r i n g the r a d i a t i o n e r a . The emitted n e u t r i n o s produce proton and e l e c t r o n c u r r e n t s . The proton c u r r e n t i s dominant and, being p o s i t i v e l y charged, g i v e s r i s e to magnetic f i e l d s . The mass range of black holes capable of producing the r e q u i r e d 2 1 T h i s i s d i s c u s s e d b r i e f l y i n the l a s t s e c t i o n of chapter 1. 167 magnetic f i e l d i s 4x 109gm<n<3x1(M *gm. The black: h o l e s r e s u l t i n a modified Friedmann expansion (p>2x 10 - 6/(M/Igm)). In order that the microwave background not be d i s t o r t e d , the d e n s i t y i n the un i v e r s e must be near the c r i t i c a l d e n s i t y . The energy d e n s i t y i n black h o l e s f o r the best case i s 1 0 - 9 o f the t o t a l energy d e n s i t y when they form, but t h i s i s TO8 times g r e a t e r than the upper l i m i t f o r black holes not to d i s t u r b the standard model of the universe, a l t e r n a t e l y , the black holes must have g r e a t e r e f f e c t i v e r o t a t i o n parameter than expected from g a l a c t i c r o t a t i o n . T h i s p o s s i b i l i t y i s not p r e d i c t a b l e and i s thus not s t u d i e d f u r t h e r . The b a s i c c o n c l u s i o n obtained here i s t h a t p r i m o r d i a l ( r o t a t i n g ) black h o l e e v a p o r a t i o n s c o u l d have generated the g e n e r a l background magnetic f i e l d observed i n the galaxy. In the next c h a p t e r , the focus of a t t e n t i o n s h i f t s t o a more fundamental aspect o f quantum f i e l d theory i n curved spacetime. The e f f e c t s of p a r t i c l e i n t e r a c t i o n s , f o r a s c a l a r f i e l d , on the black hole e v a p o r a t i o n process are s t u d i e d i n the two dimensional b l a c k hole spacetime. 168 5- ftp SCALAR FIELD IN THE 2-D BLACK HOLE METBIC I. INTRODUCTION Without i n t e r a c t i o n s , a black hole ( e i t h e r 2 or 4 dimensional) emits a l l p a r t i c l e s p e c i e s t h e r m a l l y with temperature 1/(8-rH). T h i s i s the famous Hawking r e s u l t (Hawking: 1974,1975)*. The p r o b a b i l i t y d i s t r i b u t i o n of the number of outgoing p a r t i c l e s i n each mode has been shown i n d e t a i l to obey the thermal d i s t r i b u t i o n law (Parker: 1975; Wald:1975). The Hawking r a d i a t i o n from an evaporating black h o l e i s known to be thermal, f o r the case of f r e e p a r t i c l e e m i s s i o n , e i t h e r when the black hole i s immersed i n a thermal r a d i a t i o n bath of the same temperature, or when there are no incoming p a r t i c l e s . There a number of gues t i o n s that one can r a i s e r e g a r d i n g the nature of the emission from an evaporating b l a c k hole when i n t e r a c t i o n s become important. The gues t i o n addressed here i s whether the emission remains thermal when p a r t i c l e -p a r t i c l e i n t e r a c t i o n s are taken i n t o account. There are a » T h i s i s d i s c u s s e d i n the i n t r o d u c t o r y chapter o f t h i s t h e s i s . In chapter 2, the d e t a i l e d f e a t u r e s of the thermal emission of n e u t r i n o s by a r o t a t i n g black hole are c a l c u l a t e d . 169 number of asp e c t s t o t h i s q u e s t i o n . The f i r s t aspect i s whether the emission remains thermal, when i n t e r a c t i o n s are i n c l u d e d , f o r a b l a c k hole immersed i n a thermal bath. I f the i n t e r a c t i o n s a l t e r the c h a r a c t e r of the emission from being thermal, the black hole would cease to be a thermodynamic o b j e c t . T h i s would be a blow t o black hole p h y s i c s (and thermodynamics), s i n c e work over the past decade 1 has suggested t h a t black h o l e s obey the laws of thermodynamics. I f the emi s s i o n i s not t h e r m a l , the second law of thermodynamics i s v i o l a t e d . The f o l l o w i n g argument demonstrates t h i s ( B i r r e l l ,Davies:1978). Put a black hole of mass M i n an e n c l o s u r e with thermal r a d i a t i o n of temperature such t h a t energy e m i s s i o n and a b s o r p t i o n by the b l a c k hole are balanced. In t h i s steady s i t u a t i o n , the mass and ar e a , hence entropy, of the black hole remain c o n s t a n t . Since the b l a c k hole absorbs high entropy thermal r a d i a t i o n and emits lower entropy nonthermal r a d i a t i o n , the t o t a l entropy o f the system decreases. a l t e r n a t e l y the decreased entropy o f the emitted r a d i a t i o n could be used to run a heat, engine of the second kind. T h i s a l s o v i o l a t e s the second law. The e f f e c t of i n t e r a c t i o n s has been s t u d i e d by Gibbons and Perry (1976,1978) f o r a black hole immersed i n a heat 1 See the i n t r o d u c t o r y chapter o f t h i s t h e s i s . * 170 bath. They extend the o r i g i n a l arguments of Hawking (1974,1975), f o r thermal emission of guanta of f r e e f i e l d s by black h o l e s , to the case o f i n t e r a c t i n g f i e l d s . The f r e e f i e l d Green f u n c t i o n i s used to c o n s t r u c t an i n t e r a c t i n g f i e l d theory. Gibbons and Perry argue t h a t , i n the b l a c k hole spacetime, the f r e e f i e l d Green f u n c t i o n i s i d e n t i c a l to the thermal Green f u n c t i o n . T h e i r arguments s t r o n g l y suggest that t h e black hole emission remains thermal even with i n t e r a c t i o n s . They c o n s i d e r only the case where an e g u i l i b r i u m i s e s t a b l i s h e d between the thermal r a d i a t i o n e n t e r i n g the black hole and the Hawking f l u x l e a v i n g the black h o l e . In t h i s case, d e t a i l e d balance between a l l p o s s i b l e processes and t h e i r i n v e r s e s ensures t h a t the r a d i a t i o n remains thermal. T h i s i s s u f f i c i e n t t o demonstrate t h a t the second law of thermodynamics remains v a l i d f o r systems c o n t a i n i n g black h o l e s . The second aspect t o the g u e s t i o n of the thermal c h a r a c t e r of black hole emission i s f o r a black h o l e immersed i n a vacuum, r a t h e r than i n a thermal bath. Here there are a number of arguments t o suggest that the thermal c h a r a c t e r would not be p r e s e r v e d i n the presence of i n t e r a c t i o n s . In t h r e e s p a t i a l dimensions, the r a d i a t i o n from the black hole expands i n t o empty space. The r a d i a t i o n energy d e n s i t y decreases while the spectrum remains e s s e n t i a l l y unchanged ( i f no i n t e r a c t i o n s are present) , c h a r a c t e r i s t i c of the 171 e m i s s i o n temperature o f the black hole. The s t a t e of maximum entropy has a lower temperature the lower the energy d e n s i t y . I n t e r a c t i o n s would t h e r e f o r e be expected t o cause a decay i n the mean energy of the quanta (toward a lower temperature), by a l l o w i n g the quanta to decay i n t o 2 or more lower energy quanta. For example, i f the black h o l e had a temperature high enough to emit pions, these pions would e v e n t u a l l y decay i n t o 2 or more photons, say. In a thermal bath t h i s e f f e c t i s compensated by photons i n the bath combining to produce pions at p r e c i s e l y the same rate that the pions decay. For a black h o l e i n a spacetime with one s p a t i a l dimension, t h i s d i l u t i o n o f energy d e n s i t y as the r a d i a t i o n t r a v e l s away from the black hole does not take p l a c e . The energy d e n s i t y of an outgoing thermal spectrum of f r e e p a r t i c l e s remains c o n s t a n t . The s p e c t r a l temperature remains the c o r r e c t temperature f o r that energy d e n s i t y . In 2 dimensions (one s p a t i a l , one temporal), there i s nothing i n c o n s i s t e n t i n the outgoing spectrum remaining thermal even with i n t e r a c t i o n s . However, s i n c e the f l u x i n going towards the black hole i s zero f o r a black hole i n vacua, one can reg a r d the i n g o i n g f l u x and the outgoing f l u x as 2 separate thermodynamic systems which i n t e r a c t i o n s c o u l d couple. I f the i n t e r a c t i o n s can allow the outgoing quanta to i n t e r a c t so as t o produce i n g o i n g quanta, there i s no reason to expect the 1 7 2 outgoing f l u x t o remain thermal. For a massless f i e l d i n a f l a t 2 dimensional spacetime, however, such i n t e r a c t i o n s a re f o r b i d d e n by energy and momentum c o n s e r v a t i o n . The outgoing p a r t i c l e s are a l l massless and the sum of t h e i r 4-momenta i s a n u l l v e c t o r . Any combination of n u l l vectors which i n c l u d e s both outgoing and i n g o i n g 4-momenta cannot be n u l l , however. T h e r e f o r e , i n a f l a t 2 dimensional spacetime one would expect a u n i d i r e c t i o n a l thermal f l u x of massless p a r t i c l e s to remain thermal, even with i n t e r a c t i o n s . In the presence of a black h o l e , on the other hand, t h i s reguirement need not h o l d . The presence of the b l a c k hole breaks t r a n s l a t i o n i n v a r i a n c e and thus e l i m i n a t e s the reguirement of momentum c o n s e r v a t i o n . I n p h y s i c a l terms, the g r a v i t a t i o n a l f o r c e s on the p a r t i c l e s mean that the momentum of the p a r t i c l e s themselves i s not conserved. One would t h e r e f o r e expect that the thermal emission of massless f i e l d s from a black hole would be changed by i n t e r a c t i o n s . The g r a v i t a t i o n a l f i e l d together with the i n t e r a c t i o n would be expected t o provide a c o u p l i n g between the outgoing thermal f l u x and the ingoing f l u x at zer o temperature. B i r r e l and Davies (1978) have c a l c u l a t e d the e f f e c t s o f i n t e r a c t i o n s on the r a d i a t i o n from a 2 dimensional black hole with no incoming r a d i a t i o n . The model they use i s the 173 c o n f o r m a l l y i n v a r i a n t massless T h i r r i n g model 3. They f i n d no a l t e r a t i o n of the outgoing r a d i a t i o n from thermal. T h i s i s because the conformal i n v a r i a n c e o f the i n t e r a c t i n g f i e l d f o r the massless T h i r r i n g model ensures momentum c o n s e r v a t i o n , so t h a t outgoing p a r t i c l e s cannot decay i n t o i n g o i n g p a r t i c l e s . I n c l u s i o n , say, of a mass term, would break the conformal i n v a r i a n c e , allow such decays, and probably r e s u l t i n nonthermal r a d i a t i o n from the b l a c k hole. In order to demonstrate the above f e a t u r e s , we c a r r y out the c a l c u l a t i o n s f o r the e f f e c t o f a J i 4 > * i n t e r a c t i o n on the thermal emission of a black hole. We study the £<fr* s e l f i n t e r a c t i o n s i n c e i t i s the most widely s t u d i e d i n t e r a c t i o n f o r curved space c a l c u l a t i o n s . Seasons f o r t h i s i n c l u d e the r e l a t i v e s i m p l i c i t y o f the s c a l a r f i e l d , the need f o r only a s i n g l e f i e l d when a s e l f i n t e r a c t i o n i s c o n s i d e r e d , and the polynomial n a t u r e of the i n t e r a c t i o n p o t e n t i a l * . »£* i s the s i m p l e s t n o n t r i v i a l i n t e r a c t i o n which i s s t a b l e : fl<£2 i s e q u i v a l e n t to a mass term i n the Hamiltonian (which has the form m2<p) , and i s u n s t a b l e 5 . ss T h i r r i n g model i s a model of a s e l f i n t e r a c t i n g fermion f i e l d f i n 2 dimensions. The i n t e r a c t i o n Lagrangian i s 3 («?*;'* V) ( ^ t ) a n d i s thus analogous to the 1<p* i n t e r a c t i o n c o n s i d e r e d here f o r a s c a l a r f i e l d . • The • i n t e r a c t i o n r e f e r s t o the term that appears i n the i n t e r a c t i o n Hamiltonian f o r an i n t e r a c t i n g system, as such i t can be viewed as a p o t e n t i a l energy f u n c t i o n . s The p o t e n t i a l goes to minus i n f i n i t y f o r one of l a r g e p o s i t i v e or n e g a t i v e <£, r e g a r d l e s s of the s i g n of fl, 3 The massle 17 4 The 2 dim e n s i o n a l black hole spacetime i s the same as the standard 4 dimensional (Schwarzschild) black h o l e , except t h a t one s p a t i a l dimension i s present i n s t e a d of t h r e e . The event h o r i z o n at r=2H i s present a l s o f o r the 2-D black h o l e . The 2-D black hole spacetime i s a l s o used here because the s c a l a r f i e l d s o l u t i o n s are much simpler i n 2 dimensions. The observable c a l c u l a t e d here i s the f l u x of p a r t i c l e s per u n i t frequency seen by an observer f a r away and to the f u t u r e of the black h o l e . The t o t a l number of i n t e r a c t i o n produced p a r t i c l e s , per u n i t frequency, i s d i v i d e d by the s p a t i a l volume and then m u l t i p l i e d by the p a r t i c l e v e l o c i t y t o o b t a i n a f l u x . a c a l c u l a t i o n f o r the i n t e r a c t i o n i n f l a t space i s c a r r i e d out i n p a r a l l e l here, f o r comparison with the black hole r e s u l t s . The f l a t space considered (with c o o r d i n a t e s t , r ) i s taken to have a thermal f l u x of s c a l a r p a r t i c l e s moving i n the p o s i t i v e r d i r e c t i o n from the past. The black hole space i s taken to be i n the vacuum s t a t e f o r outgoing p a r t i c l e s , i n the past, an observer i n the f u t u r e , a t l a r g e p o s i t i v e r , then measures the same r e s u l t s ( i . e . that f o r a thermal f l u x ) , when the i n t e r a c t i o n i s switched o f f , f o r e i t h e r the black hole spacetime or the f l a t spacetime. For the c a l c u l a t i o n s we i n c l u d e an i n g o i n g thermal f l u x o f a r b i t r a r y temperature. T h i s demonstrates t h a t when the temperature of the i n g o i n g f l u x i s the same as t h a t of the 175 outgoing f l u x , d e t a i l e d balance a p p l i e s . Then the outgoing r a d i a t i o n remains thermal even with the i n t e r a c t i o n s . In the l i m i t t h a t the temperature of the in g o i n g f l u x goes to zero, the r e s u l t s are i d e n t i c a l t o the case t h a t the vacuum f o r i n g o i n g p a r t i c l e s i s used. The black hole c a l c u l a t i o n with constant fl g i v e s i d e n t i c a l r e s u l t s to the f l a t space c a l c u l a t i o n i f the i n t e r a c t i o n i s chosen t o be s p a t i a l l y dependent i n a p a r t i c u l a r way f o r the l a t t e r . The s p a t i a l dependence of a i n f l a t space then g i v e s i n f o r m a t i o n about where the i n t e r a c t i o n s , which a l t e r the r a d i a t i o n from being thermal, occur i n the black h o l e spacetime. They occur j u s t o u t s i d e t h e event h o r i z o n . These are momentum nonconserving processes which couple the outgoing f l u x to the ingoing f l u x . They do not occur f o r f l a t space i n which momentum i s conserved. Some i n f i n i t i e s may be expected t o a r i s e from the c a l c u l a t i o n s f c r both the black hole and f l a t space c a s e s , as f o r any f i e l d t h e o r y . T h i s i s found to be the case. To remove the i n f i n i t i e s we add a mass term t o the f i e l d e g u a t ion. The mass undergoes an i n f i n i t e r e n o r m a l i z a t i o n to c a n c e l some of the i n f i n i t i e s from the H<t>* c a l c u l a t i o n , k f i n i t e answer can then be obtained by s c a l i n g the i n t e r a c t i o n c o n s t a n t ft. 176 Chapter O u t l i n e An o u t l i n e o f t h i s c h apter i s now given. The f r e e s c a l a r f i e l d i s d e s c r i b e d f i r s t . D i s c r e t e energy mode f u n c t i o n s are used, and the f i e l d operator i s separated i n t o i n g o i n g and outgoing p a r t s . The black hole e v a p o r a t i o n vacuum s t a t e i s d e f i n e d i n the standard manner and forms the i n i t i a l s t a t e f o r the i n t e r a c t i o n c a l c u l a t i o n f o r the black hole c a s e . For the f l a t space case, the i n i t i a l s t a t e has a f l u x of outgoing thermal r a d i a t i o n . Next, the s c a l a r f i e l d i s s t u d i e d using standard p e r t u r b a t i o n theory, t o see how the thermal c h a r a c t e r o f the p a r t i c l e e m i s s i o n i s a l t e r e d by the ^<f>* i n t e r a c t i o n . In s e c t i o n I I I , the S-matrix i s d e f i n e d i n terms of the i n t e r a c t i o n Hamiltonian, and then used t o w r i t e the f i n a l s t a t e i n terms of the i n i t i a l s t a t e . An e x p r e s s i o n f o r the f l u x of outgoing p a r t i c l e s per u n i t energy, when the system i s i n the f i n a l s t a t e , i s found i n s e c t i o n IV. I t i s separated i n t o p a r t s o f d i f f e r e n t o r d e r i n the c o u p l i n g c o n s t a n t fl . The zero order p a r t i s the thermal f l u x of the i n i t i a l s t a t e . In s e c t i o n V, the p r o p e r t i e s of the b l a c k hole e v a p o r a t i o n vacuum are used to w r i t e the black hole e x p r e s s i o n i n terras o f a thermal sum over s t a t e s , j u s t as the f l a t space e x p r e s s i o n i s . The f i r s t order p a r t i s shown to vanish i n s e c t i o n VI. The second order p a r t i s w r i t t e n as a 177 product of amplitudes, which y i e l d s a set of diagrams a s s o c i a t e d with the amplitudes. A l l diagrams which have nonvanishing c o n t r i b u t i o n s to the f l u x are l i s t e d . In s e c t i o n VII, the p r o p e r t i e s of the thermal s t a t e are used t o show t h a t diagrams which i n v o l v e i n t e r a c t i o n s among only outgoing p a r t i c l e s c a n c e l with t h e i r i n v e r s e s . The r a d i a l dependence of the p a r t i c l e p r o d u c t i o n i s examined i n s e c t i o n V I I I . The f l a t space c a l c u l a t i o n g i v e s i d e n t i c a l r e s u l t s to the black hole c a l c u l a t i o n with ^ con s t a n t , i f one takes fl t o be a p a r t i c u l a r f u n c t i o n of r f o r the f l a t space case. In s e c t i o n IX, the. s p a t i a l i n t e g r a t i o n s and the sum over the thermal s e t of s t a t e s are c a r r i e d out. In s e c t i o n X, diagrams i n v o l v i n g i n going p a r t i c l e s are ev a l u a t e d . T h i s r e s u l t s i n i n f r a r e d and u l t r a v i o l e t d ivergences. The i n f r a r e d divergences are due to a combination of the masslessness of the s c a l a r f i e l d and the 2 dimensional nature of the spacetime. In s e c t i o n XI, r e n o r m a l i z a t i o n of some of the divergences of s e c t i o n X i s c a r r i e d out. The r e s u l t s are d i s c u s s e d i n s e c t i o n XII. I I . THE PB EE SC&L&B FIELD The massless s c a l a r f i e l d operator $ s a t i s f i e s the s c a l a r f i e l d e q u a t i o n : n ^ O A R u bjg*"F3*4/W) 3=o (2.1) The Schwarzschild metric ( i . e . , the pa r t e x t e r i o r t o the event h o r i z o n a t r=2H) f o r a black hole i n 2 dimensions i s de s c r i b e d by the i n t e r v a l d s 2 : ds 2= (1-2R/r) dt 2-(1-2!1/r)-idr2=(l-2H/r) dudv (2.2) r and t are the S c h w a r z s c h i l d c o o r d i n a t e s , and u and v are the n u l l c o o r d i n a t e s f o r the Schwarzschild metric of eqn (2.2): v=t-r* u=t+r* r*=r+2M In (r/2H-1) (2.3) The c a l c u l a t i o n i s c a r r i e d out i n p a r a l l e l f o r f l a t space, which has the m e t r i c : ds2=dt 2-dr 2=dxdy (2.2f) x=t-r and y=t+r a r e t h e n u l l c o o r d i n a t e s f o r f l a t space. |7<? Equations p e r t a i n i n g to the f l a t space c a l c u l a t i o n w i l l be f l a g g e d with an f , as i n eqn (2.2f) above. In the n u l l c o o r d i n a t e s , the f r e e f i e l d e q uation (2.1) t a k e s the simple form: <a v4 = 0 ^ ^ = 0 (2.4) f o r the black hole and f l a t space cases, r e s p e c t i v e l y . Thus the s o l u t i o n s of (2.1) separate i n the v a r i a b l e s d i r e c t l y . T h i s y i e l d s : 4=f(u)+g(v) <^ = f ( y > * 9 ( x ) < 2 ' 5 ) The f ' s r e p r e s e n t i n g o i n g waves and the g»s r e p r e s e n t outgoing waves. T h i s i s because the n u l l c o o r d i n a t e s u and y are constant along l i n e s t+r* and t+r, r e s p e c t i v e l y , whereas v and x are constant along t - r * and t - r , r e s p e c t i v e l y . To provide an i n f r a r e d c u t o f f and to provide a d i s c r e t e r a t h e r than an i n f i n i t e set of modes, we r e q u i r e the s c a l a r f i e l d t o be p e r i o d i c on the boundaries shown i n f i g . 16 ( i n appendix A, compare t h i s with f i g u r e 1b) K Ingoing modes are r e q u i r e d to be p e r i o d i c on v=*_L, outgoing modes on u=*L. The i n t e r i o r s to the boundaries are described by the range of * Pig.1b i s a p p l i c a b l e to the 2D b l a c k hole m e t r i c . The r e g i o n s o u t s i d e I and I I (not shown i n f i g . l b ) d i f f e r f o r the Kerr and 2D black hole m e t r i c s . Fig.1b a p p l i e s only a l o n g the symmetry a x i s (0=0) o f the Kerr* m e t r i c , which i s a l s o 2 d i m e n s i o n a l . 180 n u l l c o o r d i n a t e s u,v and x,y g i v e n by: -L<u<L -L<x<L f o r i n g o i n g waves -Kv<L -L<y<L f o r outgoing waves (2.6) I n the usual approach, the wave f u n c t i o n s are Bade p e r i o d i c i n the s p a t i a l v a r i a b l e r . S i n c e s p a t i a l t r a n s l a t i o n i s not a symmetry of the b l a c k hole m e t r i c , t h i s i s i m p o s s i b l e here. T h i s r e s u l t s i n d i s c r e t e modes f o r the s o l u t i o n s o f egn (2.1) and puts a lower l i m i t on the frequency w. The l a t t e r f e a t u r e i s important i n t r e a t i n g the i n f r a r e d (low freguency) d i v e r g e n c e s which occur i n the c a l c u l a t i o n s . The c o n s t a n t , L, w i l l be allowed t o approach i n f i n i t y a f t e r the c a l c u l a t i o n s a r e completed. As L approaches i n f i n i t y the two r e g i o n s expand t o i n c l u d e the e n t i r e spacetime, i n p a r t i c u l a r the e n t i r e p a s t h o r i z o n (D=0). However, even f o r f i n i t e L, the s c a l a r f i e l d mode s o l u t i o n s s t i l l extend over the e n t i r e s p a c e t i a e * . The f i e l d o p e r a t o r ^ i s expanded i n normal modes'. These are assumed to be p e r i o d i c i n t h e range given by eqn (2.6) o f u and v, or of x and y. The standard f i e l d o perator expansion has <p as a sum o f p o s i t i v e frequency modes, <Pn^ » times a n n i h i l a t i o n o p e r a t o r s , a^, p l u s neqative frequency modes, j ^ . , times c r e a t i o n o p e r a t o r s , a ^ : * I-e., the diagram of f i g . .fb of Appendix A, where the boundaries u,v=*L are i g n o r e d . . • T h i s standard procedure i s d e s c r i b e d i n any t e x t on quantum f i e l d theory (e.g. Bjorken and D r e l l , 1965). 181 4=?<& + an*</U. + > (2 .7 ) In f l a t space, the d i v i s i o n between p o s i t i v e and negative frequency i s the same f o r a l l i n e r t i a l observers d e s p i t e the d i f f e r e n c e i n t i n e c o o r d i n a t e s . I . e, f , (r, ) exp (iw, t ( ) , wj >0, f o r one observer, can always be w r i t t e n as a f o u r i e r sum (or i n t e g r a l ) of f 2 ( r £ ) exp ( i w t t j , f o r a second observer with time c o o r d i n a t e t t , with w^X). T h i s i s not true i n curved spaced: d i f f e r e n t d e f i n i t i o n s of p o s i t i v e freguency are not e g u i v a l e n t . They r e s u l t i n d i f f e r e n t d e f i n i t i o n s * 0 of what c o n s t i t u t e s a p a r t i c l e . The observable we c o n s i d e r i s the f l u x per u n i t freguency of outgoing p a r t i c l e s in* r e g i o n I o f the b l a c k hole spacetime (see f i g . l b ) . Thus i s separated i n t o an in g o i n g p a r t cf>^ and outgoing p a r t cfc^ # f o r convenience i n l a t e r c a l c u l a t i o n s : < M ~ * < £ * x (2-8) The ingoing and outgoing p a r t s are def i n e d i n terms of the n u l l c o o r d i n a t e s u,v and x,y. T h i s s e p a r a t i o n works because of the s e p a r a t i o n of v a r i a b l e s (eqn (2.5)) f o r the f r e e f i e l d 1 nor f o r a c c e l e r a t e d o b s e r v e r s i n f l a t space, see e.g. Onruh,1976. * T h i s t o p i c i s d i s c u s s e d i n the i n t r o d u c t o r y chapter o f t h i s t h e s i s . For a review of t h i s t o p i c see Parker,1976. 182 e q u a t i o n (2.1) . F o r t h e b l a c k h o l e c a s e , i n b o t h r e g i o n s I a n d I I , t h e i n g o i n g modes o r i g i n a t e n e a r p a s t n u l l i n f i n i t y ^ " ( s e e f i g . V b ) , w h e r e t h e s p a c e t i m e i s a s y m p t o t i c a l l y f l a t . O b s e r v e r s a t r e s t n e a r p a s t n u l l i n f i n i t y , i n r e g i o n I , t h e n d e f i n e p o s i t i v e f r e g u e n c y i n g o i n g modes by t h e s t a n d a r d c o m p l e x e x p o n e n t i a l s w i t h w>0: ^ = e x p ( - i w u ) //2Lw w=ntr/L n = l,2,3... (2 .9) T h e s e m o d e s a r e d e f i n e d o n l y i n r e g i o n I a n d v a n i s h i n r e g i o n I I . S i m i l a r modes o f i d e n t i c a l f u n c t i o n a l f o r m , w h i c h we l a b e l cy^ , a r e d e f i n e d i n r e g i o n I I a n d v a n i s h i n r e g i o n I . T h e d e f i n i t i o n o f 2 s e t s o f modes i s n e c e s s a r y t o d i s t i n g u i s h p o i n t s w i t h t h e s a m e * * u , v c o o r d i n a t e s . T h e o b s e r v a b l e t h a t we s t u d y i s i n r e g i o n I o n l y . H o w e v e r , t h e r e g i o n I I o r ' h a t 1 m o d e s a r e i m p o r t a n t i n t h e d e f i n i t i o n o f t h e v a c u u m s t a t e , w h i c h p l a y s a c e n t r a l r o l e i n t h e c a l c u l a t i o n s . F o r f l a t s p a c e , we c h o o s e p o s i t i v e f r e q u e n c y i n g o i n g modes t h a t a n i n e r t i a l o b s e r v e r * * w o u l d u s e . T h e s e a r e g i v e n b y : * • T h e same c o o r d i n a t e s u , v c o v e r b o t h r e g i o n s o f t h e b l a c k h o l e s p a c e t i m e . a l t h o u g h we a r e i n t e r e s t e d i n o b s e r v e r s a t r e s t , t o c o m p a r e w i t h t h o s e n e a r £T i n t h e b l a c k h o l e s p a c e t i m e . 183 £^=exp(-iwy) /J"2Lw w=im/L n=1,2,3... (2.9f) These modes (eqns (2.9) and ( 2 . 9 f ) ) , and the modes of eqns (2.11b) and (2.11f) below, are normalized under the s c a l a r f i e l d i n n e r product. We wr i t e the i n n e r product of two s o l u t i o n s F,G as (F,G) : (F,G)=(-i/2) J d x P g T g°^[ (^F*) G-F*(^G) ] (2.10) g i s the spacetime metric and q i s i t s determinant. T h i s i s the two dimensional analogue of the f o u r dimensional s c a l a r product given by- eqn (9) of chapter 1. Sin c e i n g o i n g and outgoing waves (egn (2.5)) are always o r t h o g o n a l , one can choose x°=constant as v (or x) and u (or y) co n s t a n t s u r f a c e s . In the f i r s t case, the in n e r product reduces t o : (f, , f J = (-i/2) f du[ ( d f * / o ' u ) f l - f * ( d f z / d u ) ] (2.10a) To y i e l d c o r r e c t l y normalized modes, e.g. (<J^#$m)= $AMi r t n e i n t e g r a l i n eqn (2.10) f o r the i n n e r product must be l i m i t e d t o the range of r * giv e n by egn (2.6), even though the modes extend over a l l r * . T h i s d i s c r e p a n c y vanishes as L approaches i n f i n i t y . For purely i n g o i n g waves, the i n t e g r a l i n eqn (2.10a) i s over the range -L<u<L. For p u r e l y outgoing waves, the analogous formula t o eqn (2.10a) i s obtained by r e p l a c i n g u by v and the i n g o i n g f waves by the outgoing g waves of egn 18* (2.5). The i n t e g r a l then has the range -L<v<L t o r e s u l t i n c o r r e c t n o r m a l i z a t i o n . P o s i t i v e freguency outgoing modes f o r the b l a c k hole spacetime are g i v e n by**: exp(-iwV)/(Tijw 1 w>0 (2.11a) V=-4Hexp(-V/4H) i s the a f f i n e parameter on the past h o r i z o n . The modes of egn (2.11a) are d e f i n e d over both r e g i o n s I and I I of the black hole spacetime. , E q u i v a l e n t l y , one can use the modes i n terms of v g i v e n by: ^=(exp(2-rT?l¥)v^+exp(-2TTnwi?« ) / ^ i n h (4^FHW) w=m-tr/L m= + 1,+2#... (2.11b) A. ^ ( v ) and f^(v) a r e outgoing modes, nonzero only i n r e g i o n s I and I I , r e s p e c t i v e l y , e q u i v a l e n t * * t o the in g o i n g modes ^ ( u ) and ($„(u) °f e < 3 n (2.9). The p e r i o d i c i t y reguirement a t the boundaries o f the r e g i o n of egn (2.6) and f i g . 16 i s imposed f o r the modes (2.11b) to o b t a i n a d i s c r e t e l y l a b e l l e d s e t of modes, w here t a k e s on both p o s i t i v e and negative v a l u e s . The modes (2.11b) can be shown to be p o s i t i v e freguency * 3 T h i s i s d e s c r i b e d i n s e c t i o n IV of chapter 2 and i n Onruh,1976. ** The e x p l i c i t form of ^ ( v ) i s given i n eqn (4.1). 185 with r e s p e c t to the c o o r d i n a t e V, as the modes (2.11a) are, by F o u r i e r a n a l y z i n g i s found only t o c o n t a i n components exp (-iwV) with w>0. S i m i l a r l y , the e x p r e s s i o n (2.11b) with and interchanged, i s found t o be negative freguency i n the V sense f o r a l l w, i . e . , c o n t a i n i n g only components with w<0. Consider f i g . 1 of Appendix A: the d i r e c t i o n of c o o r d i n a t e time t i s i n o p p o s i t e d i r e c t i o n i n r e g i o n I I t o t h a t i n r e g i o n I . A p o s i t i v e frequency mode i n the V sense, (2.11a) or eqn (2.11b), c o n t a i n s mainly p o s i t i v e v freguency modes and n e g a t i v e v freguency modes T l ^ , with e x p o n e n t i a l l y s m a l l parts of negative v freguency modes ^-n A and p o s i t i v e v frequency modes H ^ . The modes (2.11a) or (2.11b) are p o s i t i v e frequency as seen by an observer a s y m p t o t i c a l l y f o l l o w i n g a n u l l geodesic i n t o the black h o l e near the past h o r i z o n l s . Onruh (1976) has shown t h a t the vacuum s t a t e , with no p a r t i c l e s d e s c r i b e d by the modes (2.11a), i s e g u i v a l e n t 1 6 to the vacuum s t a t e used by Hawking (1875). Hawking»s vacuum s t a t e has no p a r t i c l e s i n the past f o r the spacetime i n which the black hole forms by c c l l a p s e of normal matter. The above d e f i n i t i o n of p o s i t i v e » s Along a l i n e o f i n c r e a s i n g V and constant U. See f i g u r e 1b of Appendix A. 4 6 These vacuum s t a t e s are e q u i v a l e n t only i n the sense that they r e s u l t i n the same e f f e c t s f a r i n the f u t u r e of the c o l l a p s e to form the black h o l e ( f o r the Hawking vacuum) or of the c r o s s o v e r p o i n t of the h o r i z o n s ( f o r the Onruh vacuum, see f i g . lb) . 18 6 frequency a l s o r e s u l t s i n the Hawking process of thermal p a r t i c l e emission from a black hole, as seen be an observer f a r away from and i n the f u t u r e of the black h o l e . For f l a t space, p o s i t i v e frequency outgoing modes are given by outgoing plane waves with w>0: % =exp (-iwx) / riLw 1 w=n-rr/L n=1,2,3... (2. 11f) IV For the black hole spacetime, the p o s i t i v e frequency modes are giv e n by eqns (2.9) and (2.11b). The i n g o i n g and outgoing p a r t s o f the f i e l d o p e r a t o r cj> then have the expansions: ^ = I ( «§.*•» * * » C2. 12) 6 ^ = ^ < & V * $ * b l ) (2.13) From t h i s p o i n t on TL <]*<Qt eg, i s used to i n d i c a t e 2T <t) * y n and ^ i s used t o i n d i c a t e The o p e r a t o r s a n,b d e f i n e the vacuum s t a t e J0> which has no p a r t i c l e s incoming from near past n u l l i n f i n i t y nor p a r t i c l e s outgoing from near the past h o r i z o n : a. |0>=a I 0>=b i 0>=0 f o r a l l m,n (2. 14) 187 For the f l a t space c a l c u l a t i o n one has: (2.12f) (2. 13f) The f l a t space vacuum |0) i s d e f i n e d by: d n|0)=e n|0)=0 f o r a l l n (2. 14f) The s t a t e |0) has no ingoing or outgoing p a r t i c l e s as seen by any i n e r t i a ! observer i n f l a t space. The above formalism d e s c r i b e s the f r e e s c a l a r f i e l d . The next s e c t i o n c o n s i d e r s the i n c l u s i o n of i n t e r a c t i o n s . P e r t u r b a t i o n theory i s used i n the i n t e r a c t i o n p i c t u r e of guantum mechanics. He c a l c u l a t e the out s t a t e |out> i n terms of the i n s t a t e |in>. The i n s t a t e f o r the black hole spacetime i s taken as the vacuum |0>. In the i n t e r a c t i o n p i c t u r e , the o p e r a t o r s evolve i n time a c c o r d i n g to the f r e e f i e l d equations. The s t a t e ||£(t)> evo l v e s a c c o r d i n g to the S matrix S (-*»,t) . The S matrix i s de f i n e d i n terms of the I I I . CALCULATION OF THE OPT STATE HITH THE acji 1_INTEB&CTION 18S i n t e r a c t i o n Hamiltonian H ^ t ) as f o l l o w s : H J(t)=£lr(V4)$* (3.1.) S(-»,t)=P e x p ( - i ] H ^ t ' j d t ' ) (3.2) lV-(t)>=S (-«,t) |in> (3.3) P i s the time o r d e r i n g operator 1"*. The i n t e r a c t i o n r e g i o n over which the i n t e g r a l s o f egns (3.1) and (3.2) are t o be e v a l u a t e d , i s taken t o be -K<t<K and 2M<r<K, f o r both the black hole and f l a t space c a l c u l a t i o n s . The S matrix i s expanded i n powers of -fl : S(-ov=o)= 1 - ( i V D $ d r d t # r , t ) • (3.4) -(a 2/32)£drdtdr»dt •P(cJ>(r,t) *<£(r»,t«) •) Sin c e <f> i s s e l f a d j o i n t , the a d j o i n t of the S matrix i s found to be: S 1 " ( - c o/o)= 1 * ( i a / 4 ) f d r d t ( ^ r , t ) • (3.5) -(^2/32) £drdtdr«dt»P (£(r,t) ^ r ' r t ' ) « ) *** For a p r e c i s e d e f i n i t i o n see eg Kaempffer (1965) 189 P i s t h e t i n e a n t i - o r d e r i n g o p e r a t o r 1 * . T h e o u t s t a t e i s g i v e n by e g n (3.3) w i t h t s e t t o i n f i n i t y . T h e e x p e c t a t i o n v a l u e , i n t h e o u t s t a t e , o f any o b s e r v a b l e c a n t h e n be e v a l u a t e d w i t h t h e i n t e r a c t i o n i n c l u d e d . I n t h e r e m a i n d e r o f t h e c h a p t e r , t h e f l u x o f o u t g o i n g p a r t i c l e s p e r u n i t f r e q u e n c y i s t h e o b s e r v a b l e c o n s i d e r e d . I V . E X P R E S S I O N S FOR T H E FLOX 0? OUTGOING P A R T I C L E S , AND ZERO ORDER C A L C U L A T I O N I n t h i s s e c t i o n , e x p r e s s i o n s a r e d e r i v e d f o r t h e f l u x o f o u t g o i n g p a r t i c l e s p e r u n i t f r e q u e n c y , f o r b o t h t h e b l a c k h o l e a n d f l a t s p a c e t i m e s . T h e i n t e r a c t i o n i s i n c l u d e d b y u s e o f t h e S - m a t r i x . N e a r f u t u r e n u l l i n f i n i t y i n r e g i o n I ( see f i g . 1), o u t g o i n g p a r t i c l e s a r e d e f i n e d i n t e r a s o f p o s i t i v e f r e q u e n c y a o d e s o f t h e s t a n d a r d f o r m : ^ = e x p ( - i w v ) / / Z L w w=mr/L n=1,2,3... (4.1) H o d e s TJI o f i d e n t i c a l f u n c t i o n a l f o r m a r e d e f i n e d f o r r e g i o n >« "P i s t h e same a s P b u t p u t s e a r l i e r t i m e s t o t h e l e f t r a t h e r t h a n t o t h e r i g h t . 190 I I o f t h e black hole spacetime. We d e f i n e an out vacuum a n n i h i l a t e d by c K ) c \ o p e r a t o r s . The outgoing p a r t of the f i e l d o p erator has the expansion i n terms of the modes ¥^ and (egn(4.1)) : The number ope r a t o r f o r outgoing p a r t i c l e s of mode n i n r e g i o n I has the form: c + c n (4.3) ( e ^ e ^ f o r f l a t s pace). The vacuum with no outgoing p a r t i c l e s near f u t u r e i n f i n i t y i s w r i t t e n as |0) (the same n o t a t i o n as f o r f l a t space) and s a t i s f i e s c n|0)=0. T h i s vacuum i s not the same as the vacuum |0> f o r no p a r t i c l e s i n the d i s t a n t past o f the b l a c k hole. From egns (2.13) and (4.2) one f i n d s the t r a n s f o r m a t i o n between b and c o p e r a t o r s : c N=(exp(2Trami/L) b^+exp (-2irnmr/L) bl*) /J2sinh (4wHmr/L) 0.4) The out s t a t e e v o l v e s v i a the S matrix from the i n s t a t e vacuum J0>. In t h i s s t a t e , the number of outgoing p a r t i c l e s per u n i t freguency range, i n r e g i o n I , i s given by: dN(w)/dw=<0|S'V<».*>)cfti" c^S (-»,.) |0>L/ir ^9l T h i s i s j u s t t h e n u m b e r o f p a r t i c l e s i n m o d e n t i m e s t h e d e n s i t y o f s t a t e s f a c t o r L / f r . T h e c o n v e r s i o n f r o m p a r t i c l e s p e r m o d e t o p a r t i c l e s p e r u n i t f r e q u e n c y i n t e r v a l dw i n v o l v e s L/TT b e c a u s e w e q u a l s n-n/L a n d cfcs i s t h e n u m b e r o p e r a t o r f o r a s i n g l e m o d e (dn=1). d N / d w i s d i v i d e d b y t h e s p a c e v o l u m e 1 * o f L t o g i v e a n u m b e r d e n s i t y , t h e n m u l t i p l i e d b y t h e s c a l a r p a r t i c l e v e l o c i t y , 1 , t o g i v e t h e f l u x o f o u t g o i n g p a r t i c l e s p e r u n i t f r e q u e n c y , d F ( w ) / d w . T h i s i s g i v e n b y : dF(w)/dw=<Ojs" t"{-*,oo)c n + c^S(-to,«i) |0>/TT (4.5) I n t h e f l a t s p a c e c a l c u l a t i o n , t h e i n s t a t e i s c h o s e n n o t t o b e t h e v a c u u m , b u t r a t h e r t o b e a p r o d u c t o f t h e v a c u u m f o r i n g o i n g m o d e s a n d a m i x e d s t a t e d e s c r i b e d b y a t h e r m a l d e n s i t y m a t r i x f o r o u t g o i n g m o d e s . I n t h e a b s e n c e o f t h e 3^* i n t e r a c t i o n , o n e t h e n h a s a n o u t g o i n g t h e r m a l f l u x o f s c a l a r p a r t i c l e s n e a r f u t u r e n u l l i n f i n i t y , e x a c t l y a s f o r t h e e v a p o r a t i n g b l a c k h o l e . T h i s w i l l b e d e m o n s t r a t e d s h o r t l y . O n e c a n w r i t e t h e i n s t a t e d e n s i t y m a t r i x ^ f o r f l a t s p a c e a s : »• L i s t h e c o r r e c t v a l u e t o u s e f o r t h e s p a t i a l v o l u m e h e r e : t h e s c a l a r m o d e f u n c t i o n s a r e n o r m a l i z e d o v e r 2 L i n t h e c o o r d i n a t e r * ( b y 1//2Lw) , w h i c h i s e q u i v a l e n t t o L ( f o r L>>2H) i n t h e c o o r d i n a t e r . 192 P = 2 T T 2 exp(-fikw m) Jk m) (k^| (4.6) y/p, i s the temperature of the outgoing thermal f l u x ; Z i s a n o r m a l i z a t i o n c o n s t a n t : Z= ( 1 - e x P ( - p j ) (4.7) |k^) i s the s t a t e with no i n g o i n g p a r t i c l e s and with k outgoing p a r t i c l e s i n mode m, each with energy wm. {k} stands f o r a s e t of k v a l u e s : k, p a r t i c l e s i n the energy s t a t e with m= 1 (w=u/L) , k ^ p a r t i c l e s in the energy s t a t e with m=2 (w=2tr/L) , e t c . The sum i n egn (6.2) i s over a l l s e t s of k values. For each s e t {k}= (k,,k z,...},• the product over a l l m,k p a i r s (m= 1 to i n f i n i t y ) of (1 -exp vn) ) exp (-Px<rtwr»v) I W ( V I i s taken. The e x p e c t a t i o n value o f the number o p e r a t o r f o r outgoing p a r t i c l e s i n mode n i n f l a t space i s given by the t r a c e : Tr {pSre^en S). The f l u x per u n i t freguency i s g i v e n by t h i s , m u l t i p l i e d by the d e n s i t y of s t a t e s factor, L/TT, times 1/L ( v e l o c i t y / v o l u m e ) . To take the t r a c e , we sum over a complete s e t of s t a t e s : Tr y>A) = X ( i l p A J i ) He choose the s t a t e s of the complete set t o be products of 193 s i n g l e energy I k ^ ) s t a t e s : | i ) = T T | k m ) . Then the f l u x i s given by: dF +(w)/dw=Tr ( p S 1 e A t e , v S ) / l ^ 'E: [ T T < 1-exp <-AW,J ) exp (- fikw^) ( k j j ST{-eora^ eJ~eA S [ Tf | k ) (4. 5f) dF(w)/dw i s evaluated t o ze r o , f i r s t and second order i n ^. The z e r o order c o n t r i b u t i o n i s : dF (w) to)/dw=<0 | c n ^ c A | 0>/TT = 1/(exp (8nHmr/L)-1)/ir (4.8) T h i s i s a thermal d i s t r i b u t i o n o f p a r t i c l e s with e n e r g i e s w and temperature 1/(8TM)# and i s j u s t the Hawking r e s u l t f o r 2 - D black hole e v a p o r a t i o n of f r e e massless s c a l a r p a r t i c l e s . The z e r o order f l u x f o r f l a t space i s : dF^ , ( w)^J/ d w = (1-exp (-^ w) ) Z k e x p ^ k w ) / T T IT (1-exp i-&vm)) exp (-Akw„,)J = ( e x p ( ^ w ) - 1 ) - i / 7 r (4.8f) T h i s i s a thermal d i s t r i b u t i o n i n w with temperature 1^?. The q u a n t i t y i n c u r l y brackets on the second l i n e of (4.8f) sums t o u n i t y . I t i s equal t o : 194 V. THE BLACK HOLE FLUX AS A THERBAL SUM OVER STATES Here we put the e x p r e s s i o n (4.5), f o r the f l u x of outgoing p a r t i c l e s i n the black hole spacetime, i n the form of a thermal sum. I t i s then i n the same form as t h e f l a t space e x p r e s s i o n , eqn ( 4 . 5 f ) . , He use the f i e l d o perator expansion egn (4.2) f o r • *e express the vacuum J 0> i n terms of s t a t e s f o r outgoing p a r t i c l e s near f u t u r e n u l l i n f i n i t y , i e , f o r p a r t i c l e s d e f i n e d by the c op e r a t o r s . The vacuum s t a t e d e f i n e d i n terms of the b o p e r a t o r s , |0>, can be expressed i n terms of t h a t d e f i n e d i n terms of the c o p e r a t o r s , JO), ty (Unruh, 1976) : |0>=1Texp(exp(-Aw m/2) c+c*) 10) /Z (5.1) *.-| ' -+- A + The c ^ and c„ , r e f e r to the c r e a t i o n o p e r a t o r s f o r outgoing p a r t i c l e s d e f i n e d i n r e g i o n I and i n r e g i o n I I , r e s p e c t i v e l y , of the a n a l y t i c e x t e n s i o n of the Schwarzschild m e t r i c 2 0 . Equation (5.1) can be derived through use of eqns (2.13) and (4.2) f o r ^  . b ^ l t , ^ ) and b_n= ($_ A,<k* ) ( « > 0 ) , are 2 0 see f i g u r e lb i n Appendix A. 195 obtained i n terms of the o p e r a t o r s c„ , c ^ a n d , The form of |0> i n terms of |0) i s then deduced from the r e l a t i o n s b J0>=b |0>=0, expressed i n terms of the c's. Both |0) and \0> vacua are d e f i n e d over the a n a l y t i c a l l y extended m e t r i c , which i n c l u d e s r e g i o n I and I I . J0>, f o r outgoing waves o n l y , i s based on p o s i t i v e freguency with r e s p e c t t o K r u s k a l time T. Constant T s u r f a c e s are h o r i z o n t a l l i n e s i n f i g . 1 b . However JO) i s based on S c h w a r z s c h i l d time . t . Constant t s u r f a c e s are the dotted l i n e s a l l pa s s i n g through the p o i n t j o i n i n g r e g i o n s I and I I i n f i g . l b . Thus t i n c r e a s e s i n the opposite d i r e c t i o n t o T i n r e g i o n I I . The r e s u l t i n g d i f f e r e n c e i n p o s i t i v e freguency d e f i n i t i o n s f o r the |0> and |0) vacua means that one vacuum s t a t e w i l l c o n t a i n p a r t i c l e s as seen by an observer who uses the other s t a t e as h i s vacuum. The i n t r o d u c t o r y chapter o f t h i s t h e s i s d e s c r i b e s t h i s e f f e c t . E g u a t i o n (5.1) expresses the |0> vacuum as c o n t a i n i n g a thermal spectrum of p a r t i c l e p a i r s , one member i n r e g i o n I, and one i n r e g i o n I I . T h i s can be seen by expanding the e x p o n e n t i a l and o p e r a t i n g with powers o f c^"c+ to obtain 0, 1, 2 e t c . , p a r t i c l e p a i r s t a t e s from the |0) vacuum. The n o r m a l i z a t i o n constant Z i s found from s e t t i n g <0|0>=1: 2 2 = Tf 0-e*p(-flw,J ) " 1 (5.2) 196 Now we s u b s t i t u t e f o r 10> i n egn (4.5) using eqn (5.1). He l e t | k^) stand f o r the s t a t e 2 * with k outgoing p a r t i c l e s of mode in i n r e g i o n I and k outgoing p a r t i c l e s of mode m i n region I I . The p a r t i c l e s are d e f i n e d i n terms of the vacuum JO) f o r s c a l a r p a r t i c l e s i n the black hole spacetime, s i m i l a r t o the p r e v i o u s d e f i n i t i o n o f \k m) f o r f l a t spacetime. Then the f a c t o r i n eqn (5. 1) can be w r i t t e n : e x F ( e x p ( - p ^ / 2 ) c ^ &*) 10) = 2 exp(-^kw^/2) | k^) (5.3) The vacuum e x p e c t a t i o n value i n eqn (4.5) i s then g i v e n by: df (w) /dw= = ( V Z 2 ) [ T J ' Z exp(-Akw^/2) (k K | j S + c ^ c A S x[TT ^ exp(-£Jw p / 2 ) | j p ) J/TT (5.4a) =[TT^ 11-exp (-/iw^) )exp(-Skw^) (k^| ]S rc+c„S xcTT'Z j k , j 3/TT = ^ {Tl^ ( 1- exp (-^wj Jexpj-yskw^) (k jjs+c^cA s^ JTj kjj/n (5.4b) The o f f d i a g o n a l elements of egn (5.4a) are removed by the o r t h o g o n a l i t y of the d i f f e r e n t outgoing modes i n r e g i o n I I . Eqn (5.4) and eqn (4. 5f) are i d e n t i c a l i n form except f o r the s u b s t i t u t i o n of c f o r e. 2 * T h i s s t a t e i m p l i c i t l y has no in g o i n g p a r t i c l e s 197 From now on we use the f o l l o w i n g n o t a t i o n f o r the thermal sum over s t a t e s : ^ ( i f . . . H)=^TT(1-exp ( - ^ J ) e x p ( - y 5 k w m ^ [ l [ ( k ^ . . | j f l k m ) | (5.5) The black hole p a r t i c l e f l u x then takes the form: dF(w)/dw=(1/ T r)^ p ^ M s t c ^ S ^ ) (5.6) When we e v a l u a t e the second order p a r t i c l e p r o d u c t i o n , we use the form (5.6) f o r both b l a c k hole and f l a t spacetimes, with the corresponding s u b s t i t u t i o n s f o r the v a r i o u s symbols. For example, and I k^) are d i f f e r e n t f o r the f l a t space and b l a c k hole space c a s e s . The f i e l d o p e r a t o r e x p r e s s i o n s f o r ^ i n the black hole and f l a t space cases look the same- the o n l y d i f f e r e n c e i s the appearance of r * f o r the black hole r a t h e r than r i n the complex e x p o n e n t i a l (see egns (2.6), (2.6f) and egns (4. 1) , (2. 7f)) . 199 VI. DIAGRAMS FOR PROCESSES CONTRIBUTING TO THE OUTGOING F_LUJt. In the l a s t s e c t i o n the black hole f l u x was shown to take the form of thermal sum. Here we use the p r o p e r t i e s of the S matrix t o reduce the number of terms we have to c o n s i d e r i n e v a l u a t i n g the f l u x f o r e i t h e r the b l a c k hole spacetime or f l a t spacetime. F i r s t we w r i t e the S matrix of eqns (3 .4) and (3.5) as: s = i + s i s , n 2 s 1 s ^ i + a s ^ + ^ s * (6.1) F o r any s t a t e |«*) with u n i t norm, i t s time evolved s t a t e S|*0 w i l l a l s o have u n i t norm: One can d i r e c t l y v e r i f y t h i s using egns (3 .4) and (3.5), which g i v e : 1=(*|S +SU) = 1 + M*|S, + 3*100+** («\s*sx +Si+S*|eO (6.2) S + S. +S +S (6.3) To o b t a i n t h i s ve have used the r e l a t i o n : jdtdt»[ P(4>*4>« •) +P(4*^ *) ]=2 Jdtdt *W* 199 In the thermal sum, a l l s t a t e s Jtf) are of the form 7T I**,). T h e r e f o r e they are e i g e n s t a t e s of the number o p e r a t o r : cjjc^ lot) lot) where n^, i s a r e a l number. T h i s statement i s not t r u e f o r a ge n e r a l s t a t e ( f o r example, l|?) = l ° ) •! n^) ) . Using t h i s we can w r i t e , f o r example: ( D ^ s f c ^ l o O =(<X\c^S+M (6.4) Now we s u b s t i t u t e f o r S and i n egn (5.6) f o r t h e f l u x o f outgoing p a r t i c l e s : (<*l S+cJc^S I eg = ( (A cjv„ loC) + c / c * (S, •S,+) |<X.) +AZ (*|S+cJc wS 4 • (S^+sJ) eZc»,\oL) +o(a3) where we have used eqn (6.4). The f i r s t order p a r t vanishes because S, +S^=0. The second order part can be r e w r i t t e n using the second r e l a t i o n of egn (6. 3) : df<2)(w)/dw=A2'5:p< )cUlsr[c^ciu,Sl JloQA" (6.5) where S, i s given by: S, = (-i/4)|drdt4>* (6.6) 20 0 For the a n a l y t i c a l l y extended b l a c k hole spacetime, d) c o n s i s t s of a p a r t 4 ^ i n r e g i o n I (which i s zero i n region II) and a pa r t 4 ^ i n r e g i o n I I (which i s zer o i n r e g i o n I ) . The c o p e r a t o r s appearing i n the commutator i n eqn (6.5) are i n r e g i o n I o n l y and thus commute with "c and c + . As a r e s u l t of the zero o v e r l a p of the part of <fi i n r e g i o n I and the part of 4> i n r e g i o n I I , the commutator i n eqn (6.5) c o n t a i n s only terms of the former and not of the l a t t e r . One can thus i g n o r e any occurences of modes and of c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s with ' h a t s 1 ( i . e . f o r re g i o n II) i n e v a l u a t i n g the commutator. In c o n s i d e r i n g the e v a l u a t i o n of the second order f l u x we w i l l f i n d i t u s e f u l to i n s e r t a complete s e t of s t a t e s |^ >) , as f o l l o w s : ^ ( o q s f j f ) (p ICcJc^S, ]|pg (6.7) T h i s allows one to w r i t e the f l u x as a sum of products of amplitudes f o r a number of processes, each o f which has a d i r e c t p h y s i c a l i n t e r p r e t a t i o n i n terms of diagrams. Eqn (6.7) i n c l u d e s products o f f i r s t order with f i r s t order amplitudes and products of z e r o order with second order amplitudes. The l a t t e r were converted, using eqns (6.1) to (6.6), t o the form of a product of amplitudes which are both f i r s t order. Thus a l l diagrams that we c o n s i d e r are f i r s t 201 o r d e r . To see what diagrams we need to c o n s i d e r , we w r i t e (see egn ( 1 . 2 ) , with • <f>^) : Then we make the expansion: Every term has products of 4 o p e r a t o r s : cjj~or c w times X j X ^ where x i s cne of cT,c,a*~ or a. For example, <&T<£3 c o n t a i n s c^pi+c^c^, c ^ a ^ a ^ , e t c . Consider any terms c o n t a i n i n g <f-w hich a l s o c o n t a i n s a c ^ from one of the <f>* s i n egn ( 6 . 9 ) . I t w i l l be c a n c e l l e d by another term with i n the p o s i t i o n of the 4 and a c*" from a <f> i n the p o s i t i o n of of the f i r s t term. T h i s i s because 4>* i s always of the opposite s i g n of <&j*. For example, the term 4 ^ 4 ° ^ ^ o f e ( ± n ( 6»9) w i l l be c a n c e l l e d by the term -c^"Ht? <P The net r e s u l t of the above d i s c u s s i o n i s that a l l terms of c ^ t u ^ 4 ] loO c o n t a i n i n g c « ^ « o p a i r s v a n i s h . Diagrammatically t h i s means t h a t : 202 we choose the of the number (f> I c+c»4 * -4«c +c w | *) =4NM (/ft i <£4 \OL ) where 4N„ i s the number of p a r t i c l e s o f energy w i n s t a t e |£) minus the number i n s t a t e [e>c) • Thus i f the number of p a r t i c l e s of energy w i s the same, i n s t a t e s \oi) and J£), t h e r e i s no change i n the p a r t i c l e f l u x . With [fL) chosen as an e i g e n s t a t e of the number o p e r a t o r , we can r e w r i t e eqn (6.7) as: ^ i j d r d t ( p \ & \ a C ) I 2 / Ife (6.7a) The s p l i t i n t o a product of amplitudes with a s s o c i a t e d diagrams i s more obvious f o r eqn (6.7a) than f o r (6.7). For drawing diagrams we use the f o l l o w i n g c o n v e n t i o n s . A l i n e with an arrow p o i n t i n g towards the vertex i s a p a r t i c l e e n t e r i n g an i n t e r a c t i o n , a s s o c i a t e d with c,^, or a,<$( from q)*. A l i n e with an arrow p o i n t i n g out i s a c r e a t e d p a r t i c l e l e a v i n g the i n t e r a c t i o n vertex, a s s o c i a t e d with a clHf* or a^C^ from q>*. A l i n e with an arrow p o i n t i n g to the r i g h t (towards l a r g e r r values) i s an outgoing p a r t i c l e (c, or C"]HV*) » a l i n e with an arrow p o i n t i n y to the l e f t i s an v a n i s h . T h i s can be seen d i r e c t l y i f i n t e r m e d i a t e s t a t e s \p\ to be e i g e n f u n c t i o n s o p e r a t o r , f o r then one has: 2 0 3 i n g o i n g p a r t i c l e (a, <(), or a [*<{)*) . For f u r t h e r c l a r i f i c a t i o n , i n diagrams with both i n g o i n g and outgoing p a r t i c l e s the i n g o i n g p a r t i c l e l i n e s are drawn above the v e r t e x and outgoing p a r t i c l e l i n e s are drawn below the v e r t e x . Some diagrams c o n t a i n a l o o p and 2 e x t e r n a l l i n e s . The l o o p i s from 0 , 0 * 4 * ^ , or a, a+<$*$t p a i r s i n , which have a p a r t i c l e c r e a t e d and destroyed at the same p o i n t . The diagrams here are d i f f e r e n t than those c o n v e n t i o n a l l y used f o r Feynman diagrams. Since there are no a n t i p a r t i c l e s , there i s no c o n f l i c t with using the arrows here, r a t h e r than l e f t to r i g h t , to i n d i c a t e the time d i r e c t i o n . Here we use l e f t to r i g h t t o i n d i c a t e i n c r e a s i n g r . For example, the diagram: vJj r e p r e s e n t s an outgoing p a r t i c l e of energy w decaying i n t o 2 outgoing p a r t i c l e s , energies w, and w^ , and 1 i n g o i n g p a r t i c l e of energy w^ . The p r e v i o u s 2 diagrams r e p r e s e n t an outgoing p a r t i c l e of energy w undergoing a s e l f - i n t e r a c t i o n (the loop) and an e l a s t i c s c a t t e r i n g with another outgoing p a r t i c l e . Mow we c o n s i d e r the e f f e c t of the i n t e g r a t i o n over time i n S, (egn ( 6 . 6 ) ) . Both black hole and f l a t spacetimes are time t r a n s l a t i o n i n v a r i a n t . Thus the t i n t e g r a t i o n expresses c o n s e r v a t i o n o f energy f o r any i n t e r a c t i o n s . For example, one 20 4 term of <p* i n v o l v i n g only outgoing p a r t i c l e s (^s) has: Jdt H d V 4 ? H ? "=> J a t e x p ( - i ( w - w ( - w J i - « J ) t ) = 2 L & ^ w ^ J (6.10) The l i m i t s on the t i n t e g r a t i o n are taken as -L<t<L s i n c e our f r e q u e n c i e s are d i s c r e t e (w=mr/L). Eqn (6.10) r e q u i r e s the sums of e n e r g i e s of the c r e a t e d p a r t i c l e s to be egua l to w f o r the decay: W > > u/ a Conservation of energy has the f o l l o w i n g important i m p l i c a t i o n . Diagrams f o r (^|[ c$z^, ]{cL) with any c"*c or a* a p a i r s f o r e x t e r n a l l i n e s v a n i s h . T h i s f o l l o w s from the e a r l i e r argument f o r no c ^ c ^ p a i r s , because the c^* v e r t e x has 4 e x t e r n a l l i n e s ( i f ther<? are no loops) , and energy i s conserved. We l i s t the diagrams which i n v o l v e outgoing p a r t i c l e s only f i r s t : A l l other diagrams i n v o l v i n g o n l y outgoing p a r t i c l e s v a n i s h , by our pre v i o u s arguments. 20 5 For processes i n v o l v i n g i n going p a r t i c l e s , the nonvanishing diagrams are: w The only diagram with a loop t h a t i s nonvanishing i s : 1 0 Tn the above, I have assumed t h a t there i s no i n g o i n g f l u x A l l of the above diagrams have/^N^Jal (see eqn (6.7a)). Diagrams which i n v o l v e 2 or 3 outqoing p a r t i c l e s of energy w, before or a f t e r the i n t e r a c t i o n vertax, can be c o n s i d e r e d a subtype of the diagrams alr e a d y given. For example, diagram no. 4 could have 2 or 3 of the l i n e s f o r outgoing p a r t i c l e s r e p r e s e n t p a r t i c l e s o f energy w, r a t h e r than j u s t 1. Diagrams with 4N =±2 or £3 have been c a l c u l a t e d . In the l i m i t L approaches i n f i n i t y , the former give a f i n i t e c o n t r i b u t i o n t o dF^/dw, and the l a t t e r v a n i s h . However, diagrams with AN^=± 1 d i v e r g e . L a t e r we w i l l s c a l e flz as 1/L, so that 4 N W = ± 1 diagrams become f i n i t e . T h i s causes the A l l w = ± 2 diagrams t o vanish. Because of t h i s we ma'ke no f u r t h e r mention of diagrams with 4N„ = *2 or ±3. I f there i s an i n g o i n g f l u x , then the i n v e r s e processes f o r the above 10 diagrams occur, j u s t as the i n v e r s e (T«=0) . 206 processes f o r the i n t e r a c t i o n s i n v o l v i n g only outgoing p a r t i c l e s occur. There are a d d i t i o n a l diagrams i n v o l v i n g i n g o i n g p a r t i c l e s e n t e r i n g and l e a v i n g the v e r t e x . The diagrams f o r the i n v e r s e processes can be o b t a i n e d by r e f l e c t i n g l e f t - r i g h t through the v e r t i c a l plane c o n t a i n i n g the vertex p o i n t , then r e v e r s i n g a l l arrows (so t h a t an i n g o i n g p a r t i c l e remains an ingoing p a r t i c l e a f t e r r e f l e c t i o n ) . To e v a l u a t e the outgoing f l u x produced by the i n t e r a c t i o n we proceed as f o l l o w s . He e v a l u a t e eqn ( 6 .5 ) f o r a l l the nonvanishing diagrams j u s t l i s t e d . Except f o r the s i n g l e loop diagram, no c*c or a*a p a i r s occur i n ($1 [ cJ°w»S,] |u) . T h e r e f o r e the 4 o p e r a t o r s i n the terms of [ c^c^,S,] can be put i n normal ordered form ( c r e a t i o n o p e r a t o r s to the l e f t ) . In a d d i t i o n , each of the o p e r a t o r s mu£t. match up with an operator i n , so t h a t T + •+ S, [ c j c ^ , S ] j a) i s nonvanishing. The terms of S, can a l s o be put i n normal ordered form (except f o r the case cf the loop diagram) . The t and t ' i n t e g r a t i o n s can then be c a r r i e d out as i n eqn ( 6 . 1 0 ) . The t* i n t e g r a n d i s the complex conjugate of the t i n t e g r a n d s i n c e each c l o l 4' w(r,t) i s p a i r e d with c (r • ,t') , e t c . The r e s u l t i s ^ L ^ ^ I + I + J f o r the example of eqn ( 6 . 1 0 ) . H e r e , C u . , i s the Kronecker d e l t a f o r w and w,+w +w,. The r and r ' i n t e g r a t i o n s and thermal sum over s t a t e s are c a r r i e d out l a t e r . F i n a l l y the frequency sums f o r the unconstrained e x t e r n a l l i n e s (or f o r the locy? i n the loop diagraa) are e v a l u a t e d . In the next s e c t i o n we c o n s i d e r i n t e r a c t i o n s i n v o l v i n g outgoing p a r t i c l e s o n l y . 207 VI I . DETAILED BALANCE FOB PDBELY OUTGOING1NTEBACTIQNS In t h i s s e c t i o n , we c o n s i d e r the diagrams i n v o l v i n g only outgoing p a r t i c l e s . The t h r e e diagrams and t h e i r i n v e r s e s were l i s t e d i n the p r e v i o u s s e c t i o n . He demonstrate t h a t f o r i n t e r a c t i o n s i n v o l v i n g only outgoing p a r t i c l e s , the p r i n c i p l e of d e t a i l e d balance a p p l i e s . That i s , f o r every i n t e r a c t i o n o f any given type, the thermal p o p u l a t i o n of s t a t e s i n the outgoing f l u x ensures that the i n v e r s e i n t e r a c t i o n o c c u r s at p r e c i s e l y the same r a t e . The net r e s u l t i s t h a t the i n t e r a c t i o n has no e f f e c t on the thermal outgoing p a r t i c l e f l u x . To demonstrate t h a t d e t a i l e d balance holds, we c o n s i d e r the f i r s t o f the 3 p a i r s of diagrams f o r purely outgoing i n t e r a c t i o n s . T h i s p a i r i s : He e v a l u a t e t h e i r c o n t r i b u t i o n to dF(z)(w)/dw. By eqn ( 6 . 5 ) t h i s i s given t y : - A M T ^ p ^ M S ^ ^ c f c ^ c + C u M ( - i / 4 j d r d t ^ ^ * ^ , ) • ^ ^ ? P J * | S > I c + c l C l c N) ( - i / u f d r d t * * * Y i > 3 ) (7.1) The f a c t o r of 4 before the summation s i g n i s from there being 208 4 p o s i t i o n s i n egn (6.9) where or 4 j , c a n occur. The only p a r t s of S/ t h a t s u r v i v e i n egn (7.1) are those of the form: u c j c ^ c , ( i ^ I d r ' d t ' ^ H ? Hf %) ana 4c J c^c , + c w ( i / 4 jar • at • cf• * if!* < ^ * £ ) (7.2) f o r the f i r s t ana secona terms of egn (7.1), r e s p . The f a c t o r s of 4 take i n t o account terms l i k e c, c^c^c^ which are put i n normal ordered form. Here *fj* stands f o r t^-(r ,,t*)« As d i s c u s s e d e a r l i e r the t and t' i n t e g r a t i o n s give c o n s e r v a t i o n o f energy: 4 L 2 £ U . The r , r ' i n t e g r a l s are the same f o r both terms. T h i s w i l l be d i s c u s s e d i n more d e t a i l l a t e r . , Here we use the n o t a t i o n t h a t J k^ .) means the s t a t e with k, p a r t i c l e s i n energy l e v e l w„,. A k without a s u b s c r i p t r e f e r s t o the energy of o b s e r v a t i o n , w. The only s t a t e s which do not. c o n t r i b u t e a m u l t i p l i c a t i v e f a c t o r of unity i n the thermal sum are o f the f o r m 2 2 |«*L) = | k, k,, kz, ks ) . Here we c o n s i d e r only the case w,,w , and w3 a r e a l l d i s t i n c t . Hhen they are not, say w,=wl# then the a p p r o p r i a t e s t a t e i s IoO = Ikk, k3). The arguments f o r the case o f n o n d i s t i n c t w^'s p a r a l l e l those given here. The p r o b a b i l i t y f a c t o r p* then has the form: 2 2 |c*) can be w r i t t e n as | kk, k^... k- ) where j ranges to i n f i n i t y . However, a l l but a few of the s t a t e s |kj) tr a c e o u t : (l-expf-fw^) ) "2exp(-^kw. ) <kj|kj)=1. 209 p6(= (1-exp (-^w)) (1-exp (-/Sw()) (1-exp (-^wz)) (1-exp {-fvs )) exp(-p(kw+k, w l+k aw 2*kjW_ ?)) (7.3) T h e r e f o r e the terms that s u r v i v e i n eqn (7. 1) are i d e n t i c a l but f o r the f a c t o r : "5.3 j3&Lj?* P ( ~* ( k , ' * k ' w ' * k ^ * k J w 3 > { ( k k . k - j c j c ^ c . c ^ c f c ^ c . ^ j k k . k J c 5 ) -(kk, k ^ k ^ l c T c ^ c J c ^ c ^ c ^ ^ l k k , k^k^)} (7.4) The e x p e c t a t i o n v a l u e s i n the c u r l y b r a c k e t s i n egn (7.4) are eva l u a t e d t o y i e l d : fMk,*1) ()c^1) ( k 2 + 1)-(k*1)k,k^k 3} (7.5) To show that egn (7.4) vanishes, we compare the f i r s t term with k , k l # k x , k 3 a g a i n s t the second terms i n which we change k t o k'=k-1, k, to k,«=k (*1, k x to k.J=k x+1, k 3 to k 3 ' = k 3 * 1 . The terms with k=0 i n the f i r s t l i n e have no c o u n t e r p a r t i n the second l i n e , nor do terms with k 1 , , k ^ 1 or k 3 , = 0 i n the second l i n e have c o u n t e r p a r t s i n the f i r s t l i n e . T h i s i s because k' and k,, k t and k 3 are non-negative i n t e g e r s . However a l l these terms without c o u n t e r p a r t s v a n i s h . The remaining p a i r e d terms r e s u l t i n the e x p r e s s i o n 2 1 0 ( e q u a l t o eqn (7.4)): x f 1-exp (^(w-w, -wx-w3 ))} (7.6) However by c o n s e r v a t i o n o f energy, w=w, +w^+Wj , t h i s v a n i s h e s i d e n t i c a l l y . The arguments f o r t h e o t h e r 2 p a i r s of diagrams f o r i n t e r a c t i o n s i n v o l v i n g o n l y o u t g o i n g p a r t i c l e s are i d e n t i c a l . The p r i n c i p l e of d e t a i l e d b a l a n c e h o l d s f o r p u r e l y o u t g o i n g i n t e r a c t i o n s . The c o n t r i b u t i o n by the i n t e r a c t i o n t o the o u t g o i n g f l u x i s t h e n c o n t a i n e d i n the s e t of 10 diagrams o f the p r e v i o u s s e c t i o n , which i n v o l v e one or more i n g o i n g p a r t i c l e s . F o r a no n z e r o t e m p e r a t u r e f o r t h e i n g o i n g f l u x the i n v e r s e s o f t h e s e 10, p l u s 4 more diagrams , must be i n c l u d e d . V I I I . COMPARISON OF T H E R A D I A L DEPENDENCE_0F_BLACK_HQLE_AND F L A T S P A C E P A f i T I C L E _ P R O D U C T I O N B e f o r e e v a l u a t i n g t h e c o n t r i b u t i o n t o the f l u x by diagrams w i t h i n g o i n g p a r t i c l e s , we compare the r a d i a l dependence o f the second o r d e r p a r t i c l e f l u x e s f o r b l a c k h o l e and f l a t space c a s e s . T h i s g i v e s an i d e a o f where the i n t e r a c t i o n s o c c u r . The second o r d e r o u t g o i n g p a r t i c l e f l u x 2 1 1 a t f r e q u e n c y w i s g i v e n by e q n ( 6.5): dF(w)6> /dw = [ d r d r ' J d t d t « ( * 2 / 1 6T) • t h e t a n d r d e p e n d e n c e o f t h e i n t e g r a n d o f ( 8 . 1 ) i$ e n t i r e l y i n t h e P a n d F f a c t o r s ( s e e e q n ( 9 . 1 ) b e l o w ) . T h u s t h e d e p e n d e n c e i s o f t h e f o r m : e x p ( iw ( t - t •) + i w z ) (8.2) w h e r e w a n d w a r e s o m e s u m o f f r e q u e n c i e s s u c h a s w + w 3 - w , - w x . z i s g i v e n b y r * - r * ' f o r t h e b l a c k h o l e c a l c u l a t i o n , a n d b y r - r ' f o r t h e f l a t s p a c e c a l c u l a t i o n . w a n d w a r e n o t n e c e s s a r i l y t h e s a m e , s i n c e t h e r e l a t i o n s : v = t - r * a n d u = t * r * , r e s u l t i n d i f f e r e n t c o m b i n a t i o n s o f t h e w * s f o r t h e r a n d t d e p e n d e n c e s o f t h e t e r m s * ' o f e q n s ( 3 . 1 ) . T h e t i m e d e p e n d e n c e i s i d e n t i c a l f o r t h e b l a c k h o l e a n d f l a t s p a c e c a l c u l a t i o n s . T h e o n l y d i f f e r e n c e i s i n t h e r a d i a l d e p e n d e n c e i n t h e r e p l a c e m e n t o f r * b y r . F i r s t , we t a k e t h e i n t e r a c t i o n t o o c c u r i n a f i n i t e s p a t i a l r e g i o n , i . e . fl n o n z e r o f o r B < r < B + a ( B , a c o n s t a n t s ) . * • F o r e x a m p l e , c o n s i d e r : P, ?x = e x p ( i (w, ( t - t 7 ) - i (w, -w a ) z ) / ( 2 L ) 2 w , w x . P a n d F a r e d e f i n e d b y e q n ( 9 . 1 ) . 212 By choosing R>>2fl, the d i f f e r e n c e between r and r * i s n e g l i g i b l e , so t h a t the r e s u l t f o r the black hole, dF(w)<?l/dw, i s i d e n t i c a l to t h a t f o r f l a t space, dF^ (w)(23/dw. , Thus the i n t e r a c t i o n , i f l i m i t e d t o a r e g i o n f a r away from the black h o l e , cannot t e l l the d i f f e r e n c e between a thermal f l u x from i n f i n i t y i n f l a t space and t h a t from a black hole®. As the i n t e r a c t i o n r e g i o n i s brought c l o s e r t o the black h o l e , by reducing R (with a s m a l l ) , the d i f f e r e n c e between dF(w)(2Vdw and dF^(w)t2)/dw i n c r e a s e s . Next, we c o n s i d e r the cases where tha i n t e r a c t i o n r e g i o n , with A=constant, encompasses the r e g i o n 2M<r<K i n the black hole space, and 2M<r<K or -K/2<r<K/2 i n the f l a t space. A t h i r d f l a t space case we c o n s i d e r i s fl a f u n c t i o n of r . For the black h o l e case the r i n t e g r a l can be r e w r i t t e n as: J d r [ . . . ] = l d r * ( 1 - 2 M / r ) [ . . . ] (8.3) The r ' i n t e g r a l can be w r i t t e n s i m i l a r l y . In egn (8.3) we have w r i t t e n r ( r * ) = r f o r the i n v e r s e f u n c t i o n of egn (2.3) , r * ( r ) . The e x p r e s s i o n f o r the p a r t i c l e p r o d u c t i o n i n f l a t space i s i d e n t i c a l t o eqn (9.1), but with z giv e n by r - r ' r a t h e r 1 5 We are comparing a thermal f l u x , with temperature T, with a black hole of mass fl, such t h a t y? = 8rM=1/T. 213 than r * - r * ' . C a l l the r e s u l t i n g e x p r e s s i o n egn ( 8 . I f ) . He r e l a b e l the f l a t space r , which i s a dummy v a r i a b l e of i n t e g r a t i o n i n eqn (8. 1 f ) , by r * . The f l a t space second order p a r t i c l e f l u x i s then: dE^ (w)tVdw= J d t d t ' [ d r * d r * ' A(r*) /|(r*«) /16TT x Z P o c M 4 ' *Cc*fc» '4>* 3K) (8.4) where z i s now given by r * - r * * . The i n t e g r a n d , a p a r t from the s p a t i a l dependence of a , i s independent of r except f o r an o s c i l l a t o r y phase from exp (iw (r*»-r*)) terms. He f i r s t c o n s i d e r the f l a t space cases with c o n s t a n t fl f o r 2H<r*<K or -K/2<r*<K/2. Except f o r the boundaries where fl goes t o 0, the i n t e g r a l of the o s c i l l a t o r y terms does not c o n t r i b u t e to the second order f l u x . As K approaches i n f i n i t y , t he i n t e g r a l f o r the second case v a n i s h e s , but not f o r the f i r s t . T h i s w i l l be demonstrated i n s e c t i o n IX. Now we show that i f we c o r r e c t l y choose the s p a t i a l dependence of fl i n the f l a t space c a l c u l a t i o n , the second order f l u x i s i d e n t i c a l to t h a t f o r the black hole case. Instead of fl=constant f o r 2H<r*<K or -K/2<r*<K/2, we take a continuous f u n c t i o n o f r * : *(r*)= (1-2S/r (r*) )*, 2.^<C(r*}<Y< (8.5a) an where r(r*) is defined by dr (r*) / d r * = 1-2M/r (r*) . We now have: if 4 (w)CzVdw= Jdtdt'jdr*dr*« 6TT) (1-2fl/r) (1-2M/r») x I p ^ W ^ ' t c X ^ M K ) (8.5b) with r, r' functions of r* and r * 1 . F o r large r* the factor 1-2H/r approaches unity. The integrand of eqn (8.5b) is then purely oscillatory and makes no contribution to the second order flux for large r*. F o r large negative r* one has r approach 2M so that the eVtetteft^ ie interaction vanishes also for large negative r* or r*'. These results apply to both black hole and flat space cases since eqn (8.5b) is identical to egn (8.1). One difference in interpretation is that a is a function of r* (egn (8.5a) ) for the flat space case, but a constant for the black hole case.The second difference is that r* i s the ordinary radial coordinate for flat space but is the tortoise coordinate r+2Mln (r/2M-1) for the black hole space, r* approaching negative infinity corresponds to approaching the horizon. We now interpret this to say that the interaction sees no particles right at the horizon. The interaction acts as though the particle creation is gradually turned on outside the horizon. * 015 IX. INTEGRATION OVER THE INTERACTION REGION t AND SUM OVER THERMAL STATES In the next s e c t i o n we e v a l u a t e the c o n t r i b u t i o n to the f l u x by the i n t e r a c t i o n from nonvanishing diagrams. That i s , from those i n v o l v i n g i n g o i n g p a r t i c l e s . In t h i s s e c t i o n we c o n s i d e r the e f f e c t s o f the i n t e g r a t i o n over the i n t e r a c t i o n r e g i o n and the sum over thermal s t a t e s . From the d i s c u s s i o n i n s e c t i o n VI, we are able to write the o p e r a t o r s i n normal ordered form f o r each of the 2 amplitudes of egn (6.7). T h i s means t h a t the wavefunctions always occur i n p a i r s i n the e x p r e s s i o n , egn (6.5) , f o r the outgoing f l u x . We use the n o t a t i o n : P,=4*(r,t) v,(r',t») = exp(iw, ( t - t • -r**r*«) ) /2Lw, F,=4*(r,t) $ ( r V t »)=exp(iw, ( t - t • +r*-r*«) ) /2Lw, (9.1) For the c a l c u l a t i o n f o r f l a t space, r e p l a c e r * by r . We have a l r e a d y evaluated the t,t» i n t e g r a l s i n s e c t i o n VI. T h i s gave c o n s e r v a t i o n o f energy. In w r i t i n g out the r , r ' i n t e g r a l s , we use the f o l l o w i n g n o t a t i o n : 0 216 H<w) = jdrdr« exp(iwz) (9.2) In egn (9.2), z i s r*-r*» f o r the black hole case, and r - r ' f o r the f l a t space case.. When we e v a l u a t e the second order p a r t i c l e f l u x per u n i t frequency, dF (w)(z)/dw, we l e a v e the s i z e of the i n t e r a c t i o n r e g i o n (the l i m i t s on the i n t e g r a l s i n eqn (9.2)) a r b i t r a r y . Independent of frequency, one has H(w)=H(-w) H(w)*=H(w) (9.3) That i s , H i s symmetric and r e a l . We now c o n s i d e r the e v a l u a t i o n of eqn (9.2). We could choose the range of i n t e g r a t i o n f o r r f o r the black h o l e and f l a t space c a l c u l a t i o n s so t h a t the i n t e r a c t i o n i s e f f e c t i v e over the f u l l s p a t i a l extent i n both cases: 2H<r,r'<K, K l a r g e , f o r the black hole, -K/2<r,r•<K/2, K l a r g e , f o r f l a t space. A second p o s s i b i l i t y i s to choose the i n t e r a c t i o n i n f l a t space to be e f f e c t i v e f o r r>2H. Then one has 3 nonzero f o r : 2H<r,r»<K, K l a r g e , f o r black hole 2H<r,r«<K, K l a r g e , f o r f l a t space (9.4) The other p o s s i b i l i t y we c o n s i d e r i s t h a t mentioned i n s e c t i o n V I I I , with $ the f u n c t i o n of the f l a t space r * given 217 by egn (8. 5a) . For the black hole spacetime, the i n t e r a c t i o n i s taken t o be non-zero o n l y i n the r e g i o n 2B<r<K w i t h i n r e g i o n I (and zero throughout r e g i o n I I ) . T h i s i s i l l u s t r a t e d i n f i g u r e 16, i n appendix A. The value of K i s independent of the value of L. For w=0, one has H(w)=K 2 f o r the black hole case and the f l a t space cases. The r,r» i n t e g r a l vanishes f o r the f l a t space case i n which the i n t e r a c t i o n extends over a l l r (K approaches i n f i n i t y ) , except f o r w=0. One has: J drdr »exp (iw (r «-r)) = 4 s i n 2 (wK/2)/w2 (=K2 f o r w=0) jdrdr'exp (iw (r«-r) ) = 0 f o r w^ O (9.5a) However, when the i n t e r a c t i o n i s l i m i t e d to the range of egn (9 . 4 ) above f o r f l a t space, the r i n t e g r a l i s nonzero: f J drdr'exp(iw (r»-r ) ) = 4 s i n 2(w(K/2-H))/w 2 (= (K-2H) 2 f o r w=0) 2.K )drdr'exp (iw (r«-r)) =1/w2 f o r w*0 (9.5b) (9.5b) i s most e a s i l y obtained by a contour i n t e g r a l i n the 218 complex r plane**. Thus we see t h a t the presence of a boundary on the i n t e r a c t i o n r e g i o n has gre a t s i g n i f i c a n c e . When t h e r e i s no boundary, the p a r t i c l e p r o duction by the i n t e r a c t i o n ( f o r wj^ O terms) van i s h e s . For the black hole case, there i s n e c e s s a r i l y a boundary, and the r e s u l t of the r a d i a l i n t e g r a t i o n i s s i m i l a r to eqn (9.5b), f o r low f r e g u e n c i e s . For the black hole case, we i n t e g r a t e using the r c o o r d i n a t e r a t h e r than r * as i n s e c t i o n V I I I . In the l i m i t K approaches i n f i n i t y , H (w) i s g i v e n by: Ofc J drdr'exp (iw (r*«-r*) ) = | [ dr (1-2M/r) z M : " e x p (iwr) J * = (/S/2w) (2sinh (^w/4) )-» y*=8irM (9.6) T h i s i s eval u a t e d by p u t t i n g the r i n t e g r a l i n a form which y i e l d s the Gamma f u n c t i o n (e. g. Abramnowitz and Stegun, 1964) , which then leads d i r e c t l y to the r e s u l t of eqn (9.6). The d i f f e r e n c e between the r e s u l t f o r f i n i t e K (K>>2M) and K approaches i n f i n i t y i n eqn (9.6) i s n e g l i g i b l e . In the l i m i t t h a t the b l a c k hole mass approaches zero, eqn (9.6) approaches 1/w*, i . e . , the same value as the f l a t space r i n t e g r a l , eqn (9.5b). For the f l a t space case with A the f u n c t i o n of r * given 2 6 Here we l e t K approach i n f i n i t y b efore i n t e g r a t i o n to o b t a i n the l a t t e r r e s u l t . 2.19 by eqn (8.5a), the r a d i a l i n t e g r a l has the same value as the, black hole r a d i a l i n t e g r a l , eqn (9.6). T h i s i s apparent from eqn (8. 3). M l 3 r a d i a l i n t e g r a l s , eqns (9.5a), (9.5b) and (9.6), can be w r i t t e n i n the form: with x=r* f o r the black h o l e case and x=r f o r the f l a t space cases. I f 3 were a constant f o r a l l x, then the i n t e g r a l vanishes f o r wj'O (eqn (9.5a) second l i n e ) . T h i s expresses momentum c o n s e r v a t i o n , s i n c e w i s the d i f f e r e n c e i n momentum of p a r t i c l e s e n t e r i n g and l e a v i n g an i n t e r a c t i o n . I f 3 i s not const a n t f o r a l l x, i . e . a f u n c t i o n of x, the i n t e g r a l i s ^ i n general^ nonvanishing. In t h i s way; momentum nonconserving decays are a s s o c i a t e d with changing fl. For the f l a t space case with 5)(r) chosen to gi v e the same r e s u l t as the black hole case (eqn (9.6) u s i n g eqn (8.3)), % changes c o n t i n u o u s l y , but most r a p i d l y near r=2M. Therefore the f l u x i s a l t e r e d from thermal mainly by i n t e r a c t i o n s near r=2H. In the black hole case t h i s t r a n s l a t e s i n t o near r*=2M, j u s t o u t s i d e the event h o r i z o n at r=2M. The only d i f f e r e n c e between the black hole and f l a t space c a l c u l a t i o n s i s i n the H(w) f a c t o r . With K l a r g e , and dxdx' fl(x) A U M exp (iw (x-x') ) * 2 2 0 w^O, egns (9.5) and (9.6) g i v e : H(w ) = ( / V 2 w ) ( 2 s i n h ( pw/4) ) - i f o r black h o l e , f l a t space wifc 3(r) H(w)=0 f o r f l a t space without boundary H(w)=1/w2 f o r f l a t space with boundary at r=2M. Fo r low freguency ( p r « 1 ) , one has H(w)=l/w 2 f o r the black h o l e space, as w e l l as f o r f l a t space with i n t e r a c t i o n boundary at r=2M. w^<<! i s the same as a wavelength of 2TT/W>> 16tr 2a. T h i s wavelength i s much l a r g e r than the l o c a l (on s c a l e 2H) g r a v i t a t i o n a l f i e l d of the black hole. The low freguency modes probe the l a r g e s c a l e of the spacetime and see only the presence of the boundary on the i n t e r a c t i o n r e g i o n . The high freguency modes, i n the black hole space, are s e n s i t i v e to the presence of the event h o r i z o n and the g r a v i t a t i o n a l f i e l d of the black h o l e . I f there i s no boundary on the i n t e r a c t i o n r e g i o n i n f l a t space, H(w) vanishes (w^O). As mentioned i n the i n t r o d u c t i o n , we expect the p a r t i c l e f l u x at i n f i n i t y , due t o the i n t e r a c t i o n , to vanish i n t h i s case. T h i s i s because energy and momentum c o n s e r v a t i o n prevent outgoing p a r t i c l e s t o decay i n t o i n g o i n g p a r t i c l e s , l e a v i n g the energy d e n s i t y i n the outgoing f l u x c o n s t a n t . The i n t e r a c t i o n s among outgoing p a r t i c l e s o n l y then does not a f f e c t the outgoing thermal f l u x s i n c e i t i s the s t a t e of maximum entropy. 221 Sum Over Thermal S t a t e s The next s t e p i n the e v a l u a t i o n of dF{w)t2Vdw i s the summation over the thermal s t a t e s v i a k and w . In the sum over k we need to c o n s i d e r e v a l u a t i n g G (w) , d e f i n e d by: G - <w) = Xk Jexp(-Akw) = (-1/w a/dAf* § exp(-*kw) (9.7) Host terms c o n t r i b u t i n g to the outgoing f l u x have on l y a s i n g l e power of k. However some terms, e.g. the terms of eqn (7.1) with 2 or more wj_»s the same, w i l l have p a r t s l i k e k (k-1). The sum f o r j=0 and the d e r i v a t i v e s are e a s i l y e v a l u a t e d , y i e l d i n g : Gt.= (1-exp(-^w) )-» G, = (1-exp (-p w) )-*exp (-/i w) (9.7a) G^=G, +2 (1-exp (-pw) )-'exp(-2pw) G 3 = G,+6G, z Only up t o Gy i s needed s i n c e the l a r g e s t power of k appearing i n eqn (6. 5) i s k 3 . T h i s f o l l o w s from the arguments of s e c t i o n 6 t h a t no c^c^ p a i r s occur i n (^ 1 £ cjc» ,<p* J|«) and from c o n s e r v a t i o n of energy. The only diagrams t h a t w i l l have k 3 are those with 3 p a r t i c l e s combining i n t o one when the 3 p a r t i c l e s a l l have the same energy. The complete l i s t of f a c t o r s that occur i s : 1, k, k+1, k ( k - 1 ) , (k+1)(k*2), k ( k - 1 ) ( k - 2 ) , and (k* 1) (k+2) (k*3) . He have 222 the r e l a t i o n s ; 1-exp (-£w) 1-exp (-«w) 1-exp (-^ w) G, =<exp(4w)-1)-i=g (G, • G<3)=g*1=exp(^w) g (G. x-G f )=2g2 (Ga_*3G,+2G0)=2exp (2^w) g* (Gj-3Gaj»-2G, )=6g 3 (G 3+bG^+6Gt *3G0) =6exp(3^w) g 3 1-exp (-/2w) 1-exp (-pw) 1-exp (-/Jw) 1-exp (9.8) g i s d e f i n e d by the second r e l a t i o n above. X. EVALUATION OF DIAGRAMS WITH INGOING PARTICLES. In t h i s s e c t i o n we e v a l u a t e the diagrams with i n g o i n g p a r t i c l e s l i s t e d i n s e c t i o n VI. These 10 diagrams a r e the o n l y ones t h a t c o n t r i b u t e to the outgoing f l u x when t h e r e are no i n g o i n g p a r t i c l e s i n the i n i t i a l s t a t e . We showed i n s e c t i o n VII t h a t the p r i n c i p l e of d e t a i l e d balance h e l d f o r diagrams with o n l y outgoing p a r t i c l e s , so that they make no net c o n t r i b u t i o n to the f l u x . 223 No Decays To Ingoing P a r t i c l e s For_The_Case_Qf Momentum Co n s e r v a t i o n Before s t a r t i n g the c a l c u l a t i o n s , ue show t h a t the second order f l u x vanishes f o r the case t h a t the i n t e r a c t i o n r e g i o n extends over a l l of f l a t space. The r dependence of the wavefunctions f o r i n g o i n g and outgoing p a r t i c l e s , egn (9.1), are of o p p o s i t e s i g n . The f r e g u e n c i e s f o r the r and t dependence of the r , r ' , t , t ' i n t e g r a n d f o r the second order f l u x (e.g., see egn (8.1)) are then always d i f f e r e n t f o r diagrams which i n v o l v e both i n g o i n g and outgoing p a r t i c l e s , a l l of the 10 diagrams are o f t h i s t y pe. The t , t * i n t e g r a t i o n g i v e s c o n s e r v a t i o n of energy (J*=0 i n egn (8.2)). T h i s i m p l i e s t h a t w i s nonzero f o r the r , r ' dependence. By egn (8.5a), the r,r» i n t e g r a l vanishes when the i n t e r a c t i o n extends over a l l space. T h i s i s the e x p r e s s i o n of momentum c o n s e r v a t i o n -massless outgoing p a r t i c l e s cannot decay i n t o ingoing p a r t i c l e s and conserve momentum. T h e r e f o r e , f o r the case that the i n t e r a c t i o n extends over a l l space, with fl=constant, momentum c o n s e r v a t i o n i m p l i e s t h a t the i n t e r a c t i o n has no e f f e c t on the thermal outgoing f l u x . 224 I n c l u s i o n Of A Thermal Ingoing F l u x We next c o n s i d e r the cases when^H(w) i s " nonzero. T h i s a p p l i e s to the cases l i s t e d i n egn ( 9 . 4 ) : the black h o l e case or the f l a t space cases with boundary on the i n t e r a c t i o n r e g i o n or with # a f u n c t i o n of r . To e v a l u a t e t h e diagrams f o r the i n t e r a c t i o n produced f l u x , we take the i n g o i n g f l u x to be thermal at temperature T». A f t e r the c a l c u l a t i o n s we can l e t T• go to zero If • goes t o i n f i n i t y ) . T h i s shows t h a t , f o r T'=T, the p r i n c i p l e of d e t a i l e d balance i s again e f f e c t i v e , so that t h e r e i s no a l t e r a t i o n of the thermal f l u x by the i n t e r a c t i o n . T h e r e f o r e , i n a d d i t i o n to the 10 diagrams of s e c t i o n VI, we must a l s o e v a l u a t e t h e i r i n v e r s e s . A l s o , diagrams with i n g o i n g p a r t i c l e s e n t e r i n g and l e a v i n g the i n t e r a c t i o n vertex must be e v a l u a t e d . The f o l l o w i n g 4 diagrams and t h e i r i n v e r s e s i n c l u d e a l l such processes: i l 12 22 5 Diagrams with i n t e r a c t i o n s among in g o i n g p a r t i c l e s only have no e f f e c t on the number of outgoing p a r t i c l e s of freguency w. He present the c a l c u l a t i o n s i n d e t a i l f o r one of the f i r s t 9 diagrams. The r e s u l t s f o r the f i r s t 9 diagrams and the l a s t 4 are s i m i l a r . The t e n t h diagram, with the l o o p , i s c a l c u l a t e d s e p a r a t e l y . L a t e r we show that the loop diagram, which i s the only one t o c o n t a i n u l t r a v i o l e t d i v e r g e n c e s , can be removed by a mass r e n o r m a l i z a t i o n . E v a l u a t i o n Of Diagram No. 1 Diagram no. 1 i s now e v a l u a t e d i n d e t a i l . For a non-zero temperature f o r the ingoing f l u x , we c o n s i d e r the i n v e r s e processes as w e l l : ^3 w The f i r s t diagram above r e p r e s e n t s the decay of an outgoing p a r t i c l e of frequency w i n t o 2 outgoing p a r t i c l e s and one i n g o i n g p a r t i c l e . T h i s decreases the f l u x t h a t an observer would see compared to a p u r e l y thermal f l u x . The c o n t r i b u t i o n s to the second order f l u x , eqn (6.5), from these 2 diagrams i s : - j d r d t d r ' d t • fa*/16Tr) 5P* X <<X| [ 4 x3 x2c+c, C i a 74x 3cfc?4 c t tP , P a F 7 P* ( c # v t> Ay 226 -4x3x2c +c'£a +/c ) t t4x3cjp, c ^ a ^ P ^ P ^ F * ]|«,) (10.1) The f a c t o r s of 4x3 and 4x3x2 a r e from the number of terms of [ c ^ C y . , ^ 4 ] and <f>'4 that apply to the diagram of i n t e r e s t . P and F are d e f i n e d by egn (9.1). Two types of s t a t e s J*.) are of i n t e r e s t f o r egn (10.1). When w, and w^  are d i s t i n c t | et) i s J k k j k ^ k j ) , where Jkj) has k ? i n g o i n g p a r t i c l e s i n the i n g o i n g s t a t e with energy w^ . When w, and wa are the same, |K) i s | k k , k 3 ) . We consider the two cases s e p a r a t e l y , then add them l a t e r . For w, and wx d i s t i n c t , one has: P « = (1-exp (-(&w) ) (1-exp(-0W, )) (1-exp (-p,) ) (1-exp(yw s) ) exp (- |g(kw + k, w, •k a _ w j - £ ' k 3 w 3 ) (10.2) 1/^ 5* i s the temperature of the ing o i n g f l u x . The t , t ' i n t e g r a t i o n g i v e s c o n s e r v a t i o n of energy i n the form of 4L* • T n e r f r ' i n t e g r a t i o n i s by d e f i n i t i o n (eqn (9.2)) H (w+w?-w, - wz ) . T h e r e f o r e the second order f l u x reduces t o : (-A 2/16TT) (24x12x4L 2/16L 4) {"^  P C^M*. *1) (k^+1) (k^ + 1) - (k + 1) k t k Lkz ] ) S t o j W « H (w + ws-w , -w^ ) Aw,w^ ? (10.3) Here ^ i s L s i n c e the sum over a l l other s t a t e s g i v e s a m u l t i p l i c a t i v e f a c t o r of u n i t y , as remarked 227 p r e v i o u s l y . No* we t r a n s f o r m the second term i n (10.3) changing k t o k'=k-1, k t to k.^k,*!, k^ to k^» = k x*1, k 3 to k 3' = k 3*1. The p a r t of (10.3) i n c u r l y brackets then becomes: , Z P A k ( k l * 1 ) ( k ^ 1 ) (k,+ 1)[1-exp(/g(w-wl-w,_)-^'wi) ] (10.4) By c o n s e r v a t i o n of energy, the l a s t part of egn (10.4) i s given by: [ l-expft^-yS') * s) ]. He perform the thermal sum over s t a t e s , using egns (9.7) and (9.8) so that k becomes g(w), k, + 1 becomes g(w,)+1 etc. . The c o n t r i b u t i o n to the f l u x i s then: (-9A 2/2irL2)g(w)/w ! E (g(w,) *1) (g(w i)+1) (g* (w?)+1) H(2wj){ 1-exp w3) ]/wtwxw3 (10.5) One has g ,(»)=(exp(^fw)-1)-», from egn (9. 8). The c a l c u l a t i o n f o r the case that the i n g o i n g s t a t e i s the vacuum |0in) i s i d e n t i c a l except that the absence of the i n v e r s e process means the e x p o n e n t i a l term i n sguare b r a c k e t s i n egn (10.5) i s absent, as we \l t k e ^'Cw^. The second order c o n t r i b u t i o n i s d i v e r g e n t . To e v a l u a t e i t we convert the freguency sums to i n t e g r a l s and go over t o t h e continuum l i m i t , L approaches i n f i n i t y . The c o r r e c t 228 procedure f o r t h i s i s : A X i s r e p l a c e d by d^w (10.6) A i s an u l t r a v i o l e t c u t o f f , used to c h a r a c t e r i z e any u l t r a v i o l e t d i v e r g e n c e s . I n f r a r e d d ivergences are c h a r a c t e r i z e d by L approaches i n f i n i t y . For w approaches 0, g (M) approaches 1/^ Sw. The second order f l u x f o r the diagram no. 1 and i t s i n v e r s e , egn ( 1 0 . 5 ) , becomes: (-9>*/2TT*) <g(w)/w)[ ((g« (w) + 1)/w) (L/^TT) 2 H (2w) [ 1-exp ( ( f - ^ •) w) ] -2((g(w) + 1)/w) ( L / ^ ) ( 1 / p ' ) l n L A K Z ( f - f ) • I, ] (10.7) The f i r s t term i s the l e a d i n g divergence from the l i m i t t h a t W j , ! ^ both approach 0(as-rr/L), the second term i s from w, ,wa approaches 0,* or w,0> and the t h i r d term i s the remainder from the f i r s t two terms. The t h i r d term of egn (10.7) i s given by: - T , - T \ ( i o . 8 ) 229 where T, and T x a r e the terms i n square brackets on the f i r s t and second l i n e s of eqn (10.7). I, i s i t s e l f d i v e r g e n t , even though the l e a d i n g d i v e r g e n c e s have been s u b t r a c t e d out (T, and T x ) . I f one expands a l l the terms i n egn (10.8) i n T a y l o r s e r i e s about w, = w^=0, then the w, and w z i n t e g r a l s together have the form: (A ,L + B, InL+C, ) (A J.+B^lnL+C^) where A, B, and C are f u n c t i o n s of w. Only the L 2 p a r t i s w r i t t e n out e x p l i c i t l y i n eqn (10.7) (as T , ) . S i m i l a r l y the L l n L p a r t o f the i n t e g r a l (from w,+wa approaches w) i s w r i t t e n out e x p l i c i t l y i n eqn (10.7) as T v. The f u l l r e s u l t f o r w,*wx approaches w has the form: (A (L*B, lnH-C.) ( B ^ l n L * C z ) . Now we e v a l u a t e the c o n t r i b u t i o n from eqn (10.1) f o r diagram nO. 1 and i t s i n v e r s e f o r the case t h a t the fr e q u e n c i e s w, and w^  are the same. S i n c e the 2 outgoing p a r t i c l e s l e a v i n g the i n t e r a c t o n vertex now have the same energy, the s t a t i s t i c s are d i f f e r e n t t h a t i f w, and w^  were not e q u a l . In p a r t i c u l a r one has: p^ = (1-exp(-^?w) ) (1-exp (-^ w, )) (1-exp (-^«w3)) 230 exp (-^(kw+k,*, ) - ^ ' k ^ ) (10 .9) Performing the t , t ' and r , r ' i n t e g r a t i o n s reduces the second order f l u x c o n t r i b u t i o n eqn (10.1) t o : (-9->*/4TT) { | p ^ ( k ( k ( * 2 ) (k, +1) (k 3+1)-(k+1) (k,-1 )k ,k s ) x ^ J f l ( w + 1 ' r 2 « l ) A < ' ( z « 3 (10. 10) We transform the second term of eqn . (10.10) by tr a n s f o r m i n g k to k* = k-1, k, to k'=k,*2, k^ to k^=k 3+1. The part o f eqn (10.10) i n c u r l y b r a c k e t s then takes the form: 2: p^k(k l*2) (k,+1) ( k 3 * 1 ) [ 1-exp (/?(w-2w| )-£'« 3) 3 (10. 11) By c o n s e r v a t i o n of energy, the l a s t p a r t o f eqn (10.11) i s given by [ 1-exp (kf'fi ') H j ) ]• The thermal sum over s t a t e s (see eqns ( 9 . 7 ) and (9.8)) i s c a r r i e d out so th a t eqn (10.10) becomes: ( -942/4TTL2) (g/w) ^  ((g« (w§) *1)/w )(f(«, ) A , 2 ) ^ 2 ^ w , H < 2 w j ) i : 1-exp ( y - ^« )^ ) ] (10. 12) with f (w) =2g 2exp (2^w) . For v approaches 0, f(w) approaches 2/<f*) 2. The c o n t r i b u t i o n t o the p a r t i c l e f l u x , eqn (10. 12) , f o r 231 diagram no. 1 with w, and w z egual, i s a l s o d i v e r g e n t . He go over t o the continuum l i m i t using egn (10.6), and f i n d t h a t egn (10.12) becomes: (-932/4^1,) (g/w)f ((g'+1)/w) (21/3/3^ 27T 1)H (2w)f 1-exp((fi-p ») w) } • I,] (10.13) The f i r s t term of egn (10.13) r e s u l t s from (w-w 5)/2 approaches 0 i n (10.12) and the second term i s the remainder: (10.14) where T, i s the f i r s t term i n sguare brackets i n egn (10.13). The term from w? approaches 0 (which gave T x i n egns (10.7) and (10.8) ) vanishes as (InL) /L when m u l t i p l i e d by the T/L f a c t o r i n f r o n t of (10.13).. The complete c o n t r i b u t i o n to the second order outgoing f l u x from diagram no. 1 and i t s i n v e r s e i s given by the sum of egns (10.7) and (10.13). The c o n t r i b u t i o n from »,=w x of eqn (10.13) i s e g u i v a l e n t t o m u l t i p l y i n g the L 2 d i v e r g e n t p a r t of egn (10.7) by a f a c t o r of (H2/3) and adding It/2v.L t o I , . The net r e s u l t i s that diagram no. 1 has an L 2 divergence, a K 2 L l n L divergence, and a complicated i n t e g r a l t h a t c o n t a i n s the lower order p a r t s of the L 2 and K 2 L l n L 232 d i v e r g e n c e s . L a t e r we w i l l show t h a t a mass r e n o r m a l i z a t i o n can remove the L 2 divergence. F u r t h e r , when a l l diagrams are i n c l u d e d , a l l the K 2 L l n L divergences and the L l n L d i v e r g e n c e s from the i n t e g r a l s sum to zero, l e a v i n g KL, L, and lower order terms. T h i s holds true f o r the case of an i n g o i n g thermal f l u x and f o r the case o f an ingoing vacuum. The K terms a l l come from diagrams which have one i n g o i n g p a r t i c l e , energy w3, l e a v i n g the vertex, where w2 gees to 0. An example i s : VJ3< I n the l i m i t w3 goes to 0, the amount of momentum nonconservation becomes a r b i t r a r i l y s m a l l . Thus these processes can occur a r b i t r a r i l y f a r away from the black hole, and diverge as the s i z e of the i n t e r a c t i o n r e g i o n , K. One can a l s o do the c a l c u l a t i o n s with i n t e r a c t i o n r e g i o n with s i z e L i n s t e a d of K, but then the o r i g i n of the divergence i s not so e a s i l y seen. The c a l c u l a t i o n s f o r the remaining 8 of the 9 f i r s t graphs and t h e i r i n v e r s e s are s i m i l a r , as well as f o r the l a s t 4 graphs (and t h e i r i n v e r s e s ) l i s t e d a t the beginning of t h i s s e c t i o n . Next we d i s c u s s the r e s u l t s of the c a l c u l a t i o n s f o r the 13 graphs ( a l l but the loop graph no. 10). 233 R e s u l t Of E v a l u a t i o n Of Graphs 1 To 9 And 11 T o l U The r e s u l t s of the c a l c u l a t i o n s are not presented here i n d e t a i l f o r a l l of the graphs. For graph no. 1, the r e s u l t s a re given by egn (10.7) plus^ eqn (10.13). In g e n e r a l there are 3, and only 3, types of divergent terms. These are: L 2 terms, K 2 L l n L terms, and complicated i n t e g r a l s with lower order d i v e r g e n t and f i n i t e p a r t s from the L 2 and K 2 L l n L terms. The L 2 terms r e s u l t from graphs 1,2,3,4,7,8,11,12,13, and 14 (each summed with i t s i n v e r s e graph) . A l l of these terms have the same s i g n . They are a l l p r o p o r t i o n a l t o : L 2 (g(w)/w) ((g« (w) •!) /w) H(2w) (1-exp ((f-fl •) «)) (10. 15) L a t e r , we w i l l see t h a t , because of t h i s , they can be removed by a mass r e n o r m a l i z a t i o n . The L 2 divergence a r i s e s from emission or a b s o r p t i o n of a p a i r of a r b i t r a r i l y low freguency s c a l a r p a r t i c l e s . For example, f o r graph no. 1 the L 2 divergence a r i s e s from w, and w^  approaches 0: 234 Since the L 2 d i v e r g e n c e i s r e n o r m a l i z a b l e , we can omit i t from f a r t h e r d i s c u s s i o n . The K 2 L l n L terms occur i n the c a l c u l a t i o n of graphs 1,2,5,6,7 and 9. These d i v e r g e n t terms occur when the energy of the emitted i n g o i n g p a r t i c l e , or p a r t i c l e s , goes t o 0. He expect these graphs t o di v e r g e as the s i z e of the i n t e r a c t i o n r e g i o n becomes l a r g e , s i n c e they i n v o l v e an a r b i t r a r i l y s m a l l amount of momentum nonconservation (H(a*g), w3 goes to 0 ) . As a r e s u l t , they can occur anywhere throughout space rather than j u s t near the black hole (or where 3 changes, f o r f l a t space) . The K 2 l l n L terms are a l l p r o p o r t i o n a l t o : (AVTT3) (g(«)/w) <(g(w)+i)/w) ( ( f ^ ) ^ ' ) (L/n) i n (LM) K 2 (10. 16) They a r i s e from diagrams which, because of em i s s i o n o r a b s o r p t i o n of low energy s c a l a r p a r t i c l e s , are a r b i t r a r i l y c l o s e t o the diagrams with c£cM p a i r s . These l a t t e r diagrams were argued t o v a n i s h i d e n t i c a l l y i n s e c t i o n VI. F o r example, c o n s i d e r no. 1 above with w ,w approaches 0. In t h i s l i m i t , the diagram i s l i k e : 235 which vanishes because of the p a r t i c l e of energy w. e n t e r i n g and l e a v i n g the vertex. One might expect t h a t p rocesses with e m i s s i o n c a n c e l processes with a b s o r p t i o n of i n f r a r e d s c a l a r p a r t i c l e s . T h i s i s the case here as the K 2 L l n L c o n t r i b u t i o n s from diagrams 9,6 and 5 are of opposite s i g n , and of c o r r e c t m u l t i p l e s of the f a c t o r i n egn (10.16), t o c a n c e l with those from diagrams 1,2 and 7, r e s p e c t i v e l y . T h i s c a n c e l l a t i o n does not occur f o r the next lower order divergence from these diagrams. The divergence from w3 approaches 0 i n v o l v e s the f o l l o w i n g i n t e g r a l : T h i s can be d e r i v e d from egn (10.5) f o r graph no. 1 (for example) by expanding (g 1 ("?)+1)/*j # <g («») • 1)/*, ( * i t h w,=w-the divergence comes from w3 approaches 0, the f i r s t l i n e of egn (9.5b) can be used f o r fi (2w?) f o r a l l 3 cases o f i n t e r e s t (black hole and f l a t space with 3=constant f o r 2M<r<K, and f l a t space with A(r) chosen t o d u p l i c a t e the b l a c k hole r e s u l t ) .W e take K l a r g e but f i n i t e (K<<L as L approaches i n f i n i t y ) . The f i r s t 2 terms of the above i n t e g r a l are: dWj/w^ H(2w i) {1+AWj • BWj 2 + . • •) i n w Since 236 Hhen we sum diagrams 1 and 9, 2 and 6, and 5 and 7, we f i n d t h at the constant term i n the w3 power s e r i e s always vanishes. For example, the c o n t r i b u t i o n from diagrams 9 plus J£ 3 ( g + 1 ) g , ^ ( 9 V + ')H(2« 3)l 1-exp ((/-/?•) vx) JS 1+1JW^/WW, T h i s can be reduced t o : 2 g expC^w) g^ C g ^ + l) H(2w 3)[ 1-exp { • ) w3) J/wwtw3 xexp [ g (w-w^ + w, ) exp (A w3/2) /(«-«.,. +w,) -g (w-w^-w3) exp /2) /(w-w^-w^ ) ] T h i s can be expanded i n a power s e r i e s around w^ , w3 e q u a l t o 0 and the frequency sums converted t o i n t e g r a l s . The q u a n t i t y i n square b r a c k e t s i s of the form f (w-wx) (w-j+aw-^+bw^s*. ) where f i s some f u n c t i o n . Because the w? f a c t o r i s always p r e s e n t , the only i n t e g r a l of H(2w3) t h a t i s present f o r w^  goes to 0 i s the second o f the above 2. The remaining 1: and t h e i r i n v e r s e s , i s p r o p o r t i o n a l t o : - "21 g(g, + i ) ( g a + D (g^»+1) H (2w3) [ i - e x P ( ( / - / 237 i n t e g r a n d f o r the i n t e g r a l i s of the form ( V « r 2 ) (1 + A^«-Ew x*+...) which y i e l d s L, InL, and f i n i t e p a r t s . The l e a d i n g divergence i s then KL, from the i n f r a r e d processes t h a t occur a t a l l r. Now we d i s c u s s the Lln L d i v e r g e n t p a r t s of the i n t e g r a l s ( l i k e I, of egn (10 . 8 ) ) which are from graphs with a f i n i t e amount of momentum nonconservation. As mentioned e a r l i e r these a l l sum to 0 as did the tfllnL terms above. These divergences have the same o r i g i n as the L 2 d i v e r g e n c e s - from e m i s s i o n of an i n f r a r e d p a i r of s c a l a r p a r t i c l e s . They are c a l c u l a t e d by expanding the v a r i o u s f a c t o r s i n the freguency i n t e g r a l s (see I, of egn (10 . 8 ) ) i n T a y l o r s e r i e s about w,=w4=0. The v a r i o u s expansions r e g u i r e d are given by: 9 = V | » * , 2-V2w, (g, + 1)/wl=1/pwv 2+1/2w, H (2w-2w, )=H(2w) -w,dH (2w)/dw + .. . ( 1-exp ( l f - p •) (w-w, ) ) ) = (1-exp ((p-/3 •)«))• (f-p ') exp ((/?-/" ) w) w, + ... (gM»-»») *1)/(w-wv ) = (( g ' + 1)/w) ( U if'/ (exp (^«w)-1) • 1/w) w, • ...) (10. 17) and o t h e r s d i r e c t l y o b t a i n a b l e from these. The L l n L d i v e r g e n t p a r t s a r i s e from a product of s e v e r a l z e r o order f a c t o r s with one f i r s t order f a c t o r ^ and are r e l a t i v e l y s t r a i g h t f o r w a r d to compute. Diagrams with g,/w, 238 f a c t o r s ( f o r a n n i h i l a t e d p a r t i c l e s of energy w,) have terms t h a t c a n c e l with terms from (gi + 1)/w f a c t o r s ( f o r c r e a t e d p a r t i c l e s ) , s i n c e the f i r s t order p a r t s , -1/2w, and l/2w (, ar e o p p o s i t e i n s i g n . Diagrams with, f o r example, w+w, i n s t e a d of w-w, i n the l a s t 3 f a c t o r s of eqns (10. 17) occur j u s t the c o r r e c t number of times to c a n c e l those with w-w,. The sum of the LlnL divergences from the i n t e g r a l s f o r a l l diagrams i s 0. The Nonzero C o n t r i b u t i o n To The Flux F i n a l l y we c o n s i d e r the remaining lower order ( i n L) terms of the i n t e g r a l s . These terms have L, KL, ( I n L ) 2 , InL, KlnL, K and f i n i t e p a r t s . They do not have the frequency dependence of eqn (10.15) which can be renormalized, nor do they sum t o 0. These terms a r e a l l s i m i l a r to t h a t f o r diagram no. 1. For diagram no. 1, the net remaining p a r t s of the c o n t r i b u t i o n t o the second order f l u x a r e : 239 (-9A2/2TT3) (g/w) [ I, - (L/^rr) l n (L/n) {(g' + 1) /w H(2w) ([ 1-exp {{/ly*) v) } x[ 1 + 2( («V/9/(exp( (8«w)-1)+1/S w) ] •2 »)/£) exp Uf-fi •) w)) -((g'+1)/w) (2/^ g )CaH(2w)/dw)( 1-exp ((/?-/$•) w) J • l\/2irL3 (10.18) I, and I, are given by eqns (10.7), (10.8) and (10.13) (10.14), The r e s u l t above i s only the c o n t r i b u t i o n from diagram no. 1. The c o n t r i b u t i o n s from diagrams 2 t o 9 and 11 t o 14 are s i m i l a r i n form. In eqn (10.18), the r e n o r m a l i z a b l e L 2 divergence i s removed, and the K 2 L l n L and the L l n L terms, which sum t o zero with those from other diagrams, are e x p l i c i t l y s u b t r a c t e d out. These remaining terms c o n t a i n the e f f e c t s of the i n t e r a c t i o n s i n reducing the outgoing p a r t i c l e f l u x a t frequency w from i t s thermal v a l u e . The reason t h a t these terms are i n f r a r e d divergent i s t h a t the c a l c u l a t i o n i s 2 dim e n s i o n a l combined with the ze r o mass of the s c a l a r f i e l d . Zero Temperature l i m i t He now c o n s i d e r the e f f e c t s o f ^' approaches i n f i n i t y (the temperature of the i n g o i n g f l u x goes to 0 ) . The dependence of the r e s u l t s on i s apparent from eqn (10.15) 240 f o r the L 2 divergence, eqn (10.16) f o r the K 2 L l n L d i v e r g e n c e , and eqn (10.18) f o r graph no. 1 (with K 2 L l n L and L 2 d i v e r g e n c e s removed). For the L 2 d i v e r g e n t part and f o r the i n t e g r a l s ( l i k e I ( ) we have g* approach 0 and (1-exp((^>-p')w)) approach 1 f o r l a r g e when w i s f i n i t e . When w approaches 0, we c o n s i d e r t h i n g s more c l o s e l y using diagram no. 1 as example. I f we had an ingoing vacuum | 0 i n ) , then egn (10.5) would be changed by removal of the f a c t o r [ 1-exp ({p-p •) w3 ) ]. In a d d i t i o n , the i n g o i n g p a r t i c l e l e a v i n g the vertex has 1/w? a s s o c i a t e d with i t r a t h e r than (g' (w ) + 1 ) / M 3 . The e x t r a f a c t o r of g»= (exp(^w)-1 ) - i i n the l a t t e r case, takes i n t o account s t i m u l a t e d emission by s c a l a r bosons i n the i n g o i n g thermal f l u x . The net f a c t o r f o r the case of the ingoing vacuum i s H(2w 3 ) / W j . With the i n g o i n g thermal f l u x , the net f a c t o r i s H (2w 2) (g» (w s)+1)/w 3 (1 -exp ( [f-p ') w 3)) . For w3 goes to 0 t h i s becomes H(2w2) ( ^ ' - ^ ) w ^ . Thus the degree of divergence of the i n d i v i d u a l diagrams i s not a f f e c t e d by having an i n g o i n g thermal f l u x . The K 2 I l n L terms from a l l diagrams sum to z e r o f o r the case that we i n c l u d e the i n g o i n g f l u x at temperature 1/^ >' and a l s o f o r the case t h a t t h e r e i s no i n g o i n g f l u x . Thus there i s no e s s e n t i a l d i f f e r e n c e between the r e s u l t s f o r ^» f i n i t e and ^ ' approaches i n f i n i t y . However i f one has the c o n t r i b u t i o n s to the second order f l u x , 241 from a l l diagrams, vanishes. T h i s means t h a t d e t a i l e d balance f o r diagrams i n v o l v i n g both i n g o i n g and outgoing p a r t i c l e s o c c u r s . Next we e v a l u a t e the only remaining graph, the loop graph, no. 10. E v a l u a t i o n Of The Loop Graph Here we b r i e f l y e v a l u a t e the loop graph and i t s i n v e r s e : The change from the e v a l u a t i o n of the p r e v i o u s graphs i s that here we are i n t e r e s t e d i n the pa r t o f which c o u p l e s to a s t a t e with one in g o i n g and one outgoing p a r t i c l e . P r e v i o u s l y a l l diagrams had 4 e x t e r n a l l e g s so t h a t only normal ordered p a r t s of (e.g : 4><&c3 • <P^ where :: i n d i c a t e s normal ordering) were of concern. The part of 4>* t h a t c o n t a i n s <^<^122:<<P*aO •<f*1}) = M l ; 1 2 X V L « ( We i n s e r t t h i s i n t o the e x p r e s s i o n f o r the second order outgoing f l u x , egn (6 . 5 ) . The r e s u l t i s : (-36i»2 / T T 3 ) (ln/l+lnL/ff) * (g/w) ( (g • +1) /w)H (2w)[ 1-exp Uf-p%) w) ] 242 (10.19) T h i s r e s u l t f o r the loop diagram i s troublesome because of the u l t r a v i o l e t divergence t h a t occurs. A i s the upper freguency c u t o f f i n t r o d u c e d i n egn (10.6). However, a mass r e n o r m a l i z a t i c n can remove the divergence of eqn (10.19). In the next s e c t i o n , we c o n s i d e r the e f f e c t s o f adding a mass term of the form Sm2^ to the i n t e r a c t i o n Hamiltonian. T h i s i s done with the purpose of removing the di v e r g e n c e s which occur to second order i n the c a l c u l a t i o n f o r the outgoing f l u x from the black h o l e . I I . RENORMALIZATION OF THE DIVERGENCES TO SECOND ORDER A common f e a t u r e of f i e l d t h e o r i e s i s the occurence of di v e r g e n c e s or i n f i n i t i e s i n any c a l c u l a t i o n s which are beyond f i r s t order i n the c o u p l i n g c o n s t a n t . For quantum e l e c t r o d y n a m i c s i n f l a t spacetime, standard methods to remove these d i v e r g e n c e s are w e l l known. The diverge n c e s can be removed to a l l orders by r e n o r m a l i z a t i o n of the e l e c t r o n mass(m) and charge (e). The unrenormalized or bare v a l u e s of m and e which appear i n the f i e l d eguations f o r a f r e e e l e c t r o n absorb the i n f i n i t i e s to y i e l d the p h y s i c a l values of m and e of an e l e c t r o m a g n e t i c a l l y i n t e r a c t i n g e l e c t r o n . 243 The divergences which are removed are a l l u l t r a v i o l e t d i v e r g e n c e s and are due t o loop type graphs. I n f r a r e d d i v e r g e n c e s , i n a 4 dimensional spacetime, always sum to 0 when a l l r e l e v a n t graphs are summed. He expect t o be a b l e to r e n o r m a l i z e the u l t r a v i o l e t d i v e r g e n c e s , but not n e c e s s a r i l y t h e i n f r a r e d d i v e r g e n c e s . Bunch, Panangaden and Parker(1979); Bunch and Panangaden (1979) show t h a t a l l second order processes o c c u r i n g i n A4>* f i e l d theory i n a 4 dimensional curved spacetime are r e n o r m a l i z a b l e . T h i s i s achieved by i n c l u d i n g a mass term and a r b i t r a r y c o u p l i n g t o the s c a l a r c u r v a t u r e i n the s c a l a r f i e l d e quation. They a l s o g e n e r a l i z e the E i n s t e i n f i e l d e g u a t i o n s to i n c l u d e a c o s m o l o g i c a l constant and terms q u a d r a t i c i n the c u r v a t u r e t e n s o r . The v a r i o u s c o n s t a n t s undergo i n f i n i t e r e n o r m a l i z a t i o n s to remove the i n f i n i t i e s i n the theory (up to second order i n 3) due to Feynman-type loop diagrams. Here we have r e s t r i c t e d our c a l c u l a t i o n t o a 2 d i m e n s i o n a l black hole spacetime or f l a t spacetime. The r e n o r m a l i z a t i o n procedure i s expected to be much s i m p l e r i n our case. There i s no guarantee that i t w i l l work s i n c e the number of parameters that we have a v a i l a b l e t o r e n o r m a l i z e i s much more r e s t r i c t e d . However only one term, from the loop graph, was found t o be u l t r a v i o l e t d i v e r g e n t . Here we c o n s i d e r the i n c l u s i o n of a mass term to r e n o r m a l i z e the 244 divergences which were found i n the p r e v i o u s s e c t i o n . Mass Renormalization A mass term o f the form hSm2<f>2 i s added to the i n t e r a c t i o n Hamiltonian to y i e l d : H x= ^ d r ( V 4 ) <4*+4Sm24>2) (11-1) The mass term i s p e r t u r b a t i v e ( p r o p o r t i o n a l t o ??) , so i t does not a f f e c t the f r e e f i e l d s o l u t i o n s . The e f f e c t o f t h i s mass term i s t o r e p l a c e r£* by «£**4 Sm2<£2. T h i s a l s o a p p l i e s to the e x p r e s s i o n s f o r the S matrix (egns (3.4) and (3.5)) and f o r the second order p a r t i c l e f l u x (egn (6. 5 ) ) . The diagrams f o r %4>* and iSm^2 terms are compared below: -X — The arguments of s e c t i o n VI s t i l l apply so t h a t the va n i s h i n g of the f i r s t o rder f l u x i s not a f f e c t e d by the a d d i t i o n of the mass term. The second order c a l c u l a t i o n shares many of the f e a t u r e s of s e c t i o n s VI and IX. Since S, of egn (6.5) now has 4**4om 24 2 r the p a r t i c l e f l u x now c o n t a i n s the 4 terms: 24 5 • 4&m2((*|<£«2[cic k,,c^3l*0 + eU<P**[ c+bu,,** 31*) > • (4gffl2)2 { ^ ^ ^ [ C ^ Q o , ^ 3K) (11.2) The l a s t 3 terms of eqn (11.2) are e v a l u a t e d h e r e , the f i r s t term having been e v a l u a t e d i n s e c t i o n s VII and X. The d e t a i l s of t h i s e v a l u a t i o n are not presented here. The c o n t r i b u t i o n t c the p a r t i c l e f l u x from the mass term i n v o l v e s the l a s t 3 terms of egn (11.2) times (A 2/16TT) , i n t e g r a t e d over t , r , t * , r * and summed over s t a t e s . The thermal sums are performed using egns (9.7) and (9.8).. The r , t , r , , t l i n t e g r a t i o n s are then c a r r i e d out to give c o n s e r v a t i o n of energy and H(w), defined by eqn (9.2). In the l i m i t of l a r g e L and A , t h i s y i e l d s : (-12fm2^2Ar2)g(w) (g'(w) +1) H (2w) / w 2 X ( l - e x p ( ^s-^3') w)) -4 (Sin2) 2(AZAK) g (w) (g* (w)+1) H ( 2v)/» 2x (1-exp ((/-£•)*)) (11.3c) A l l of the c o n t r i b u t i o n s from the mass term are p r o p o r t i o n a l t c : x (lnfLArVlnA) (11.3a) same as (11.3a) (11 .3b) g (v) (g« (v)*1)H(2v) /v*x (1-exp ( { f - p •) w)) 246 Th e r e f o r e they have the same frequency dependence as the u l t r a v i o l e t d i v e r q e n t term and a l l of the L 2 d i v e r g e n t terms which c o n t r i b u t e t o dF te/dw from the 3^>4 i n t e r a c t i o n . £m 2 can then be chosen t o c a n c e l a l l of the div e r g e n c e s of the %<j+ c a l c u l a t i o n which are a l s o p r o p o r t i o n a l to the above f a c t o r . The d i v e r g e n t behaviour of the 3<£* c a l c u l a t i o n i s changed as f o l l o w s by using the terms from the mass c a l c u l a t i o n . He w r i t e the r e s u l t f o r the f l u x of the 2<f>+ c a l c u l a t i o n as: Aq(w) (g« («) *1) H(2w)/w 2x(1-exp( ( ^ ' ) «) ) + ^ B . f . (w) (11.5) and w r i t e the r e s u l t of the im 2^ 2 c a l c u l a t i o n as: £n 2C g(w) (g' (w) +1) H(2w)/w 2x( 1-exp ( w) ) (11.6) where a, B , and C are di v e r g e n t c o n s t a n t s depending on L and A. For example, s e t t i n g Sm2=-A/C gi v e s a net r e s u l t f o r the f l u x o f : ^B-f^Cv) (11.7) T h i s i s j u s t the sum of eqns (11.5) and (11.6). T h i s p a r t i c u l a r c h o i c e of Sm2 removes a l l of the terms p r o p o r t i o n a l t o eqn (11.4). Another p o s s i b l e c h o i c e i s one 247 t h a t would remove only the u l t r a v i o l e t d i v e r g e n t terms. However the c h o i c e which r e s u l t s i n eqn (11.7) removes a l s o the worst i n f r a r e d divergences ( L 2 ) . Egn (11.7) i s c o n s i d e r a b l e improvement over egn (11.5) f o r the 3<^* c a l c u l a t i o n without the mass term. We are unable to remove a l l of the i n f r a r e d d i v e r g e n c e s , i . e . L and KL terms and InL type terms. Insuf f i c i e n c y Of Renormalization In 2 D For H a s s l e s s a ^ * a mass r e n o r m a l i z a t i o n by i t s e l f i s not s u f f i c i e n t t o remove the i n f r a r e d divergences to second order f o r the 2 dimensional black hole or f l a t spacetimes. To see what a l t e r n a t e r e n o r m a l i z a t i o n procedures c o u l d be used, wa f i r s t l o c k at r e n o r m a l i z a t i o n f o r other cases. We c o n s i d e r i n curved spacetime as s t u d i e d by Bunch, Panangaden, and Parker (1979) and Bunch and Panangaden (1979), h e r e a f t e r r e f e r r e d t o as BPP and BP. Two problems a r i s e over and above the problems of r e n o r m a l i z a t i o n i n quantum e l e c t r o d y n a m i c s (which have been s o l v e d ) . F i r s t , 2 7 For any f i e l d theory i n f l a t space, a l l vacuum to vacuum processes (diagrams without e x t e r n a l l i n e s ) can be i g n o r e d . S t a b i l i t y of the vacuum i m p l i e s that amplitudes f o r a l l such processes must sum to y i e l d an unobservable phase f a c t o r of exp(iql). In curved space the nonuniqueness of the vacuum i m p l i e s t h at these processes are r e a l p h y s i c a l p r o c e s s e s , and thus must be c o n s i d e r e d . 24 8 vacuum to vacuum processes must be r e n o r m a l i z e d 2 7 . Second, there i s nc proof that divergences t o a l l orders i n the c o u p l i n g constant a r i s e from a s m a l l number of subdiagrams 2 8 so t h a t each o r d e r must be renormalized s e p a r a t e l y . BPP and BP work i n 4 dimensions. BPP show that a l l f i r s t order d i v e r g e n c e s can be removed by normal o r d e r i n g with r e s p e c t to an a d i a b a t i c vacuum s t a t e or by r e n o r m a l i z i n g f , the c o u p l i n g to the c u r v a t u r e , and m, the mass of the s c a l a r f i e l d . To second order, BP show that m, § and ^ are renormalized to make 2 and 4 p a r t i c l e amplitudes f i n i t e , whereas vacuum to vacuum amplitudes must be renormalized by the c o u p l i n g c o n s t a n t s i n the E i n s t e i n f i e l d e g u a t i o n s ( i . e . i n the g r a v i t a t i o n a l p a r t of the Lagrangian). The q u e s t i o n now i s what does t h i s say about r e n o r m a l i z i n g t o second order f o r a 2 dimensional spacetime? Since 2 dimensional spacetimes are c o n f o r m a l l y f l a t , a d i a b a t i c vacuum s t a t e s can be d e f i n e d , so t h a t f i r s t order r e n o r m a l i z a t i o n by normal o r d e r i n g should work. However, the c a l c u l a t i o n here i n c l u d e s second order processes. In 2 dimensions there are no E i n s t e i n e g u a t i o n s , so that there are no a s s o c i a t e d c o u p l i n g c o n s t a n t s t o ren o r m a l i z e the vacuum to vacuum processes. I f the theory 2 8 For quantum el e c t r o d y n a m i c s , have been shown t o a r i s e diagrams. divergences to a l l orders from 3 elementary d i v e r g e n t 2 4 9 were r e n o r m a l i z a b l e i n 2 dimensions, then r e n o r m a l i z i n g m,^ , and 9 should be s u f f i c i e n t t o remove a l l divergences. So f a r we have only c o n s i d e r e d a f i r s t order mass term 3 Sm2^. The r e s u l t of t h i s was that the f i r s t order c o n t r i b u t i o n to the f l u x vanished, and the second order c o n t r i b u t i o n i s g i v e n by eqns (11.3). A second o r d e r mass term "A2S^m2 has a second order c o n t r i b u t i o n i d e n t i c a l to the f i r s t order c o n t r i b u t i o n from 3om2, and thus vanishes. S i m i l a r l y , a second order c o u p l i n g c o n s t a n t ^ 2 4*/4 has the same c o n t r i b u t i o n s as the f i r s t order term and a l s o v a n i s h e s . F i n a l l y , s i n c e our divergences were the same f o r the f l a t space or black hole c a l c u l a t i o n s , r e n o r m a l i z a t i o n of the c o u p l i n g t o the c u r v a t u r e , which vanishes i n f l a t space, cannot help remove the d i v e r g e n c e s . The c o n c l u s i o n i s t h a t the remaining i n f r a r e d d i v e r g e n c e s are nonrenormalizable. However, massless /?^ 4 s c a l a r f i e l d t h e o r y i s nonrenormalizable 2'' i n 2 dimensions, so t h i s i s not s u r p r i s i n g . Both the f r e e and i n t e r a c t i n g • assies.? s c a l a r f i e l d s i n 2 dimensions have t h e i r 2 * The n o n r e n o r m a l i z a b i l i t y can be seen by the f o l l o w i n g power coun t i n g arguments. One e v a l u a t e s a g e n e r a l diagram with V v e r t i c e s , E e x t e r n a l l i n e s , and I i n t e r n a l l i n e s f o r the massive 9,^ + s c a l a r f i e l d i n 2 dimensions. One c a l c u l a t e s the m (mass) dependence of the i n t e g r a l s and f i n d s they go as m _ 2 V, independent of E (or I ) , and t h e r e f o r e d i v e r g e as m goes to zero. Since the i n t e g r a l s blow up as m goes to 0 independent of E, one cannot add counterterms to the Lagrangian (of the form 4 E ) which c a n c e l the d i v e r g e n c e s . I thank Drs. L. Bosen, J . Feldman f o r c o n v e r s a t i o n s on t h i s p o i n t . * 250 mathematical d i f f i c u l t i e s (Hightman, 1964; K l e i b e r , 1967; Hakanishi, 1S78) . The divergences f o r the i n t e r a c t i n g massless s c a l a r f i e l d are, i n p a r t , s i m i l a r i n o r i g i n to the c o l i n e a r d i v e r g e n c e s t h a t occur i n massless quantum e l e c t r o d y n a m i c s i n 4 dimensions. However, the theorem which assures t h a t the c o l i n e a r divergences sum to zero breaks down f o r massless i R 2D. Because 2 dimensions i s r a t h e r u n p h y s i c a l , the c a l c u l a t e d r e s u l t s of t h i s chapter should be i n t e r p r e t e d with c a u t i o n . S c a l i n g Of 3 To Obtain F i n i t e R e s u l t s Since r e n o r m a l i z a t i o n i s unable to give f i n i t e r e s u l t s f o r the a l t e r a t i o n of the f l u x from thermal, we c o n s i d e r an a l t e r n a t i v e . The remaining divergence i s L, t h e r e f o r e s c a l i n g the i n t e r a c t i o n c o n s t a n t "A as 1/JT1 gives f i n i t e r e s u l t s . T h i s i s somewhat a r b i t r a r y , and a l s o removes the ( I n L ) 2 , InL and f i n i t e terms of the unsealed c a l c u l a t i o n . To i l l u s t r a t e t h a t ^ s c a l i n g has some p h y s i c a l b a s i s , we look a t a simple process f o r which the r e s u l t s are expected to be f i n i t e and nonzero. He c a l c u l a t e the p r o b a b i l i t y f o r t h e process and f i n d t h a t the i n t e r a c t i o n constant fl , f o r massless ^ * theory i n 2 dimensions, must s c a l e as 1//1?. K i s t h e i n t e r a c t i o n r e g i o n s i z e . I f we had n a i v e l y taken K and L to be the same, we i n c o r r e c t l y would have o b t a i n e d the 251 d e s i r e d r e s u l t t h a t fl2 s c a l e s as 1/L. The process c o n s i d e r e d i s the s c a t t e r i n g of 2 p a r t i c l e s o f g iven e n e r g i e s , wt and w^ , i n t o 2 other p a r t i c l e s with e n e r g i e s w3 and w^  : y/v ~ s . ^ ^ ^ * S We w r i t e the S matrix as 1 + aS, + -a 2S z+... so that the f i r s t o r der amplitude f o r the process i s lot) , where \f) i s the s t a t e with the p a r t i c l e s of e n e r g i e s w5 and ws , and \m) i s the i n i t i a l s t a t e with the p a r t i c l e s of e n e r g i e s w, and w-^. The p r o b a b i l i t y f o r the process i s then: (o4AS,+ |^) (/slas, |<*) = ( V 2 L ) • jdtdt • j d r d r ' (*2/16) e i p ( i ( « 3 * ^ - w , - ¥ l ) (t-t»)-i (w,*ww -wk-w,) ( r - r •))/w , v tw 3 vH * U 2/16) Sw.^.^/w* w l», x K 2 / ( 2 L ) 2 (11.8) Here we have used J exp (-iwx) dx=2L£t M i C , . The p r o b a b i l i t y goes as the sguare of the i n t e r a c t i o n r e g i o n s i z e K 2, r a t h e r than as K, because the process i s coherent over the whole r e g i o n . To o b t a i n a r a t e , we d i v i d e by the i n t e r a c t i o n time (2L) and volume (K). We must a l s o d i v i d e by the n o r m a l i z a t i o n o f t h e incoming f l u x e s : J,°=(i/2) (Hf^VUI*)*, ) = w ^ = 1 / 2 L J0=1/2L (11.9) 252 The product of a l l these f a c t o r s i s 2L/K. F i n a l l y we convert the s c a t t e r e d f l u x e s to un i t frequency range dw from a s i n g l e mode. The mode sp a c i n g i s n/L, so dn=1 i s r e p l a c e d by (L/ir)dw 7: B=U*/16) (1/w, wj (K/2L) (L/TX) dw? / (w % (w, • w., - w^  ) (11.10) where R stands f o r the t r a n s i t i o n r a t e per u n i t volume per u n i t incoming f l u x . Because of c o n s e r v a t i o n of energy, we only need to c o n v e r t one of w? and w^ . We could c o n v e r t both, but then the c o n v e r s i o n from Kronecker d e l t a t o D i r a c d e l t a f u n c t i o n would c a n c e l the second f a c t o r of L and gi v e the same net r e s u l t . S i n c e t h i s must be a f i n i t e number, independent of K or L, we have a * K = f i n i t e , o r : ^ A c Z / K 1 (11.11) "A0 may s t i l l be a f u n c t i o n of r . I f we i n t e g r a t e over f i n a l s t a t e s f o r the above p r o c e s s , the egn (11.10) gives an a d d i t i o n a l InL div e r g e n c e . Furthermore, i f there was a background thermal f l u x o f p a r t i c l e s , s t i m u l a t e d emission according to the Bose E i n s t e i n f a c t o r (exp (|?w.j) - 1 ) - 1 changes the InL to an L d i v e r g e n c e . The o t h e r e f f e c t o f the thermal f l u x , as we saw i n the d i s c u s s i o n a f t e r egn (10.16), i s to lower the dependence on the s i z e o f 253 the i n t e r a c t i o n r e g i o n from K 2 t o K (or from K to 1, per unit i n t e r a c t i o n volume). T h i s i s because the amplitude no longer i s coherent- the K 2 terms are c a n c e l l e d by terms from other processes which occur i n the presence of the thermal f l u x , l e a v i n g only K terms. I t i s seen that the type of divergence obtained depends i n d e t a i l on the process under c o n s i d e r a t i o n . The s c a l i n g of A with the n o r m a l i z a t i o n volume L i s r e g u i r e d i n order to o b t a i n f i n i t e answers f o r our mass ren o r m a l i z e d c a l c u l a t i o n with massless -a<*>4 theory i n 2 dimensions. The i n f r a r e d problem then no lo n g e r i s p r e s e n t , s i n c e 1/L from -a2 makes the KL and L terms f i n i t e and the InL type terms v a n i s h . Because of the r a t h e r ad hoc nature of t h i s procedure, the value of these f i n i t e terms w i l l not be given e x p l i c i t l y . XII. DISCOSSION R e s u l t s The e f f e c t of a i n t e r a c t i o n on the r a d i a t i o n from a 2-D black hole has been examined up to second order i n the c o u p l i n g constant 7l. T h i s was c o n t r a s t e d with the r e s u l t s f o r a 3<£ 4 i n t e r a c t i o n i n a f l a t space c o n t a i n i n g a thermal f l u x coming i n from n e g a t i v e r . 254 The e x p e c t a t i o n s d e s c r i b e d i n the i n t r o d u c t i o n t o t h i s c h a p t e r were borne out. I t was seen that f o r the f l a t space with no boundary on the i n t e r a c t i o n r e g i o n , the p a r t i c l e f l u x remains thermal. T h i s i s because energy and momentum c o n s e r v a t i o n does not allow the outgoing f l u x of massless p a r t i c l e s to i n t e r a c t with the in g o i n g f l u x a t d i f f e r e n t temperature. I n t e r a c t i o n s among outgoing p a r t i c l e s preserve the thermal c h a r a c t e r of the r a d i a t i o n because of d e t a i l e d balance. Every p o s s i b l e decay process i s p r e c i s e l y compensated by i t s i n v e r s e . T h i s only occurs because the outgoing f l u x of p a r t i c l e s has a thermal spectrum. P u t t i n g a boundary on the i n t e r a c t i o n r e g i o n i n f l a t space or making 'X a f u n c t i o n of r r e l a x e s the momentum c o n s e r v a t i o n reguirement so t h a t the outgoing f l u x can decay i n t o i n g o i n g p a r t i c l e s . The presence of the black hole i n the b l a c k hole spacetime has the same e f f e c t . T h i s r e s u l t s i n a nonthermal f l u x . However i f the temperature of the i n g o i n g f l u x i s the same as t h a t of the outgoing f l u x , d e t a i l e d balance again occurs so t h a t the f l u x remains thermal. The f i r s t order p a r t i c l e f l u x vanishes i d e n t i c a l l y i n th e black hole space and i n f l a t space. The second o r d e r f l u x f o r the black hole was shown, i n s e c t i o n V I I I , f o r an i n t e r a c t i o n r e g i o n l i m i t e d to being f a r from the black hole, t o behave i d e n t i c a l l y to the second order f l u x f o r f l a t space. 25 5 Momentum nonconserving p r o c e s s e s which occur to second order are of 2 t y p e s . One type occurs only near the black h o l e and has a f i n i t e amount of momentum nonconservation. T h i s has L, ( I n L ) 2 , InL, and f i n i t e c o n t r i b u t i o n s to the second order f l u x i n v o l v i n g H (w). The second type i n v o l v e s e m i s s i o n of an a r b i t r a r i l y low energy ( i n f r a r e d ) i n g o i n g s c a l a r p a r t i c l e and has a r b i t r a r i l y s m a l l momentum nonconservation. T h i s l a t t e r type can occur everywhere and c o n t r i b u t e s t o the f l u x as K times L, InL, and f i n i t e terms, where K i s the s i z e o f the i n t e r a c t i o n r e g i o n . The processes we are p r i m a r i l y i n t e r e s t e d in are those of the f i r s t type. The d i f f e r e n c e i n the ex p r e s s i o n s f o r the f l u x f o r the black hole and the d i f f e r e n t f l a t space cases l i e s e n t i r e l y i n the r a d i a l i n t e g r a l H (w). T h i s shows the e f f e c t of the g r a v i t a t i o n a l f i e l d o f the black h o l e . F o r l a r g e K, one has H(w)=1/w2 f o r f l a t space with boundary, H (w)= (£/2w) <2sinhpw//4)-» f o r b l a c k hole space or f l a t space with A(r) chosen to d u p l i c a t e the black hole r e s u l t s . Since sinhx>x f o r a l l x>0, the d e v i a t i o n from thermal e m i s s i o n i s g r e a t e r f o r f l a t space with boundary than f o r the black hole, f o r a l l f r e q u e n c i e s . F o r low f r e q u e n c i e s (^w/4«1), the d e v i a t i o n from thermal f o r the black hole i s the same as f o r f l a t space with boundary. pw/4<<1 i s the same as the requirement that the wavelenqth be q r e a t e r than the s i z e o f the black h o l e : A>4T 2M. In t h i s case, the s c a l a r waves are 256 not s e n s i t i v e to the g r a v i t a t i o n a l f i e l d of the black hole, bot o n l y t o the boundary a t r=2!l« By t a k i n g fl=1-2M/r(r*) f o r f l a t space, i t was shown that the r a d i a l i n t e g r a l i s i d e n t i c a l to t h a t f o r the black hole case. r * i s the normal r a d i a l c o o r d i n a t e i n f l a t space and r (r*) i s the i n v e r s e f u n c t i o n o f r*=r*2Mln(r/2M-1). 3 i s zero f o r r*<<0 and u n i t y f o r r*>>2H. The momentum nonconserving i n t e r a c t i o n s which cause the spectrum to be nonthermal occur mainly i n the r e g i o n where i s changing r a p i d l y , near r*=2H. For the black hole case i t i s not the changing $ ( s i n c e fl i s constant) , but the g r a v i t a t i o n a l f i e l d that a l l o w s the momentum nonconserving decays near r*=2M. Here r * i s the t o r t o i s e c o o r d i n a t e o f egn (2. 3) . Thus the o r i g i n o f the nonthermal p a r t i c l e f l u x i n the black hole space i s j u s t o u t s i d e the event h o r i z o n of the black h o l e . T h i s i s not s u r p r i s i n g s i n c e the f r e e p a r t i c l e p r o d u c t i o n by a bl a c k hole i s a l s o concentrated near the ho r i z o n (Davies, F u l l i n g : 1976; Onruh:1977). One may i n t e r p r e t the absence of momentum nonconserving decays very near the h o r i z o n , r*<<0, as due to the absence of c r e a t e d outgoing p a r t i c l e s so c l o s e to the h o r i z o n . The nonvanishing r e s u l t s obtained here suggest t h a t f o r the 2D black hole spacetime and the 2D f l a t spacetimes which allow momentum nonconserving i n t e r a c t i o n s , the r a d i a t i o n become nonthermal when i n t e r a c t i o n s are i n c l u d e d . 257 Two a d d i t i o n a l f e a t u r e s a r e present when we c o n s i d e r the e f f e c t of p a r t i c l e i n t e r a c t i o n s on the r a d i a t i o n from a 4 dimensional black h o l e . F i r s t , the r a d i a t i o n energy d e n s i t y d i l u t e s as 1/r 2 as the r a d i a t i o n spreads away from the black h o l e . Second, the spectrum, without i n t e r a c t i o n s , i s a l t e r e d by b a c k s c a t t e r i n g 3 0 o f f the g r a v i t a t i o n a l f i e l d of the black h o l e . T h i s does not occur i n 2 dimensions. T h i s second e f f e c t d e p l e t e s the low energy end of the r a d i a t i o n spectrum. T h e r e f o r e , even without i n t e r a c t i o n s , the r a d i a t i o n from a 4D black hole does not have a thermal spectrum. I n t e r a c t i o n s would be expected t o allow decays of the outgoing r a d i a t i o n i n t o ingoing r a d i a t i o n , f u r t h e r d e p l e t i n g the spectrum. In g e n e r a l , nonthermal emission by a black h o l e i s expected when i n t e r a c t i o n s are i n c l u d e d , unless the black hole i s immersed i n a thermal bath of temperature 1/8-irH. The I n f r a r e d Problem The unrenormalized second order r e s u l t obtained here f o r the p a r t i c l e f l u x due t o the A<p* i n t e r a c t i o n i s i n f r a r e d and u l t r a v i o l e t d i v e r g e n t . T h i s a p p l i e s to a l l cases except the 3 0 T h i s i s d i s c u s s e d i n chapter 2 f o r n e u t r i n o s emitted by a r o t a t i n g black hole. F i g . 2a presents |B-| 2, the escape p r o b a b i l i t y vs. energy f o r a n e u t r i n o wave, and the thermal F e r m i - D i r a c f a c t o r . The emitted spectrum i s p r o p o r t i o n a l t o the product of the 2 f a c t o r s . 258 f l a t space case where momentum c o n s e r v a t i o n causes the second order f l u x to vanish. F i n i t e r e s u l t s are not to be expected from a quantum f i e l d theory without c a r e f u l a t t e n t i o n t o r e g u l a r i z a t i o n and r e n o r m a l i z a t i o n . R e g u l a r i z a t i o n i s the process of p a r a m e t r i z i n g the divergences so they can be c a l c u l a t e d d e f i n i t e l y . T h i s was done here i n terms of the parameters L and /[ . Renormalization i s the removal of the d i v e r g e n c e s . T h i s i s u s u a l l y accomplished by i n t r o d u c i n g e x t r a p e r t u r b a t i v e terms i n t o the f i e l d Lagrangian, and thus i n t o the i n t e r a c t i o n Hamiltonian. These extra terms g i v e r i s e t o e x p r e s s i o n s which can c a n c e l the divergences. One important aspect of 2 dimensional models i s that waves do not decrease i n amplitude as they propagate. I n f r a r e d (long wavelength) d i v e r g e n c e s are, as a consequence, worse than they would be in a 4 dimensional spacetime. Massless a<f>* theory i s r e n o r m a l i z a b l e to second order i n 4 dimensions, but i s nonrenormalizable i n 2 dimensions. Nonrenormalizable means that i t i s not p o s s i b l e to c a n c e l a l l divergences by adding a f i n i t e number of e x t r a terms to the L a g r a n g i a n 3 1 . For a massive s c a l a r f i e l d the i n f r a r e d divergences would net be present s i n c e the mass p r o v i d e s a low frequency (or energy) c u t o f f . Here we c o n s i d e r e d a massless f i e l d s i n c e we r e g u i r e d exact s o l u t i o n s to the f i e l d 3 1 See footnote a t end of s e c t i o n XI. 259 equations f o r the black hole spacetime i n order t o perform an e x p l i c i t c a l c u l a t i o n . In s e c t i o n XI, a r e n o r m a l i z a t i o n was c a r r i e d out. Since massless 34* i s nonrenormalizable i n 2 dimensions, one can only hope t h a t no unremovable divergences occur i n second o r d e r . I t was found t h a t by r e n o r m a l i z i n g the mass of the s c a l a r f i e l d , we were a b l e to remove the u l t r a v i o l e t divergences and the worst i n f r a r e d d i v e r g e n c e s . The remaining i n f r a r e d divergences are nonrenormalizable. However they can be made f i n i t e i f we s c a l e A 2 as 1/L f o r massless 34* i n 2 dimensions. I n f r a r e d divergences are expected f o r massless i n 2 dimensions. They are made worse i n the presence of the thermal boson f l u x because the s t i m u l a t e d emission d i v e r g e s at low f r e q u e n c i e s . I t i s not c l e a r whether the L 2 i n f r a r e d divergence* which was renormalized to 0 by the mass r e n o r m a l i z a t i o n , should be removed. The s c a l i n g of A2 by 1/L i s j u s t i f i e d only because i t r e s u l t s i n a f i n i t e answer. & more s a t i s f a c t o r y approach would be t o use a massive f i e l d to avoid the i n f r a r e d d i f f i c u l t i e s . In t h i s case the correspondence between the black hole c a l c u l a t i o n and the f l a t space c a l c u l a t i o n with fl (r) no longer h o l d s . The i n f r a r e d problem f o r a 4 D b l a c k hole should p r e s e n t l e s s problems than i n 2D, but the c a l c u l a t i o n s are f a r more d i f f i c u l t . F u r t h e r i n v e s t i g a t i o n along these l i n e s i s needed 260 i n order to o b t a i n f i n i t e unambiguous r e s u l t s f o r the nonthermal f l u x from a black h o l e . Belated Work Here the work of others on the t h e r m a l i t y of black hole emission i s mentioned. Gass and Dresden(1979) c o n s i d e r an i n t e r a c t i n g s c a l a r f i e l d i n the f o u r dimensional Schwarzschild m e t r i c . They f i n d t h a t they cannot r u l e out non-thermal Hawking r a d i a t i o n and a l s o f i n d nonzero 2 and 3 p a r t i c l e c o r r e l a t i o n f u n c t i o n s . They d e f i n e thermal i n terms of a n a l y t i c i t y of the Green f u n c t i o n 3 2 . F o r quantum elect r o d y n a m i c s , Gass and Dresden show t h a t the presence of a black hole a l l o w s otherwise f o r b i d d e n decays. These are decays which i n v o l v e momentum and energy exchange with the black hole. Some p r e v i o u s i n v e s t i g a t o r s have argued f o r the t h e r m a l i t y of black hole emission when i n t e r a c t i o n s are i n c l u d e d . Gibbons and Perry (1976) argue that a b l a c k hole remains i n e q u i l i b r i u m with a heat bath of the same temperature as the black hole even i n the presence o f i n t e r a c t i o n s . D e t a i l e d balance with the thermal incoming r a d i a t i o n i s e s s e n t i a l to maintain the thermal spectrum f o r 3 2 Black holes and thermal Green f u n c t i o n s are d i s c u s s e d i n Gibbons and Perry (1978). 261 t h e r a d i a t i o n from the black h o l e . B i r r e l and Davies (1978) have shown t h a t p a r t i c l e e mission from a 2-D black h o l e immersed i n a vacuum i s ther m a l f o r the massless T h i r r i n g model. However t h i s i s because the massless T h i r r i n g model i s e x a c t l y s o l u b l e and i s e q u i v a l e n t t o a f r e e boson f i e l d . The conformal i n v a r i a n c e of the i n t e r a c t i n g f i e l d i m p l i e s momentum c o n s e r v a t i o n . Momentum nonconserving decays are then not allowed. The outgoing f l u x i s not able to i n t e r a c t w i t h the i n g o i n g f l u x a t zero temperature, and t h e r e f o r e remains thermal. Others have c o n s i d e r e d general techniques f o r c o n s t r u c t i o n c f a r e n o r m a l i z a b l e theory f o r a<4* i n curved spacetime. The s t u d i e s are concerned with 4 di m e n s i o n a l spacetimes. The nonrenormalizable massless ^ * t h e o r y i n 2 di m e n s i o n a l spacetimes i s not of as much p h y s i c a l i n t e r e s t . B i r r e l l (1979) works with mosentum space techniques, which are p a r t i c u l a r l y u s e f u l f o r c a l c u l a t i n g the energy momentum t e n s o r , t o re n o r m a l i z e $4* f i e l d theory i n curved spacetime. Bunch and Parker (1979), s i m i l a r l y , work with a momentum space r e p r e s e n t a t i o n o f the Feynman propagator and demonstrate renor m a l i z a b i l i t y of %4>* f i e l d theory i n an a r b i t r a r y spacetime. Less d i r e c t l y r e l a t e d , but with a c o n c r e t e 26 2 c a l c u l a t i o n , B i r r e l l and Davies (1979) c o n s i d e r j\j>* p a r t i c l e c r e a t i o n f o r a B o b e r t s o n - i a l k e r u n i v e r s e . However g e n e r a l l y speaking, l i t t l e work has been done on i n t e r a c t i n g f i e l d t h e o r i e s , i n c l u d i n g fl^*, i n curved spacetime. 263 BIBLIOGRAPHY References. Chapter 1. Bardeen J , C a r t e r B, Hawking S: Com Hath Phys 31,162(1963} Bekenstein J : Phys Rev D7,2333 (1973) ; D9,3292 (1974) ; D12,3077 (1975) B i r r e l l N, Davies P: »ConforBal Symmetry Bre a k i n g And Co s m o l o g i c a l P a r t i c l e C r e a t i o n In ty Theory 1 P r e p r i n t Dept. Of Hath., K i n g f s C o l l e g e , London, A p r i l 1979. B i r r e l l N, Ford L: Annals Of P h y s i c s 122, 1 (1979) Davies P: Hep Prog Phys 4 1, 1313 (1978) Dewitt B: Phys Rep 19,295(1975) Ford L: Ph. D. 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Jones B: Rev Mod Phys 48, 107 (1976) Leahy D, Onruh W- Phys Rev D19,3509(1979) Misner C, Thorne K, Wheeler J : " G r a v i t a t i o n " , H H Freeman, New York (1970) Nanopolous D, Weinberg S- Phys Rev D 20, 2484 (1979) Omnes R- Phys Rep 3,1 (1972) Page D- Phys Rev D13,198(1976) Page D- Phys Rev D14,3260(1976) 267 Ryan M, Shepley L: "Homogeneous R e l a t i v i s t i c Cosmologies" P r i n c e t o n S e r i e s In P h y s i c s (1975) S c i u l l i F- "AIP Conf. Proc. #22- N e u t r i n o s " American I n s t i t u t e Of P h y s i c s (1974) Steigman G- Ann Rev Astron Astrophys 14,339(1976) V i l e n k i n A- Phys Rev L e t t 41, 1575(1978) Weinberg S- " G r a v i t a t i o n And Cosmology", Wiley(1972) Weinberg S: Phys Rev L e t t 42, 850(1979) Yoshimura M: Phys Rev L e t t 41, 281 (1978) References. Chapter 4. B a i e r l e i n R: M N R A S 184,843 (1978) Carr B: Ap J 201, 1 (1975); Ap J 206, 1 (1 976) G a i l l a r d M: Nature 279,585(1979) Ha r r i s o n E: P R L e t t 30, 188 (1973a) ; M N fi A S 165,185(1973b) ; Ann Rev Astron Ap 11*155(1973c) H e i l e s Cc Ann Rev Astron Ap 14,1(1976) H e i t l e r W: "Quantum Theory Of R a d i a t i o n " Oxford(1954) Jones B: Rev Mod Phys 48, 107(1976) Leahy D, Onruh W: Phys Rev D 19,3509(1979) Misner C, Thorne K, Wheeler J : " G r a v i t a t i o n " , W H Freeman, New Ycrk(1970) Page D, Hawking S: Ap. J . 206, 1 (1976) Parker E: Ap. J . 157, 1 129 ( 1969) Peebles J : Astrophys Space S c i 1 1 ,443 (1971) Rees M, Reinhardt H: Astron Ap 19,189(1972) 268 Sandage A, Lynds C: Science 137,1005(1963) S c i u l l i F: "AIP Conf. P r o c #22- N e u t r i n o s " American I n s t i t u t e Of P h y s i c s (1974) S p i t z e r L: " D i f f u s e Hatter In Space" I n t e r s c i e n c e P u b l i s h e r s (1968) Y i l e n k i n A: P B L e t t 41,1575(1978) Wasserman I: Ap. J . 224,337(1978) Z e l ' d o v i c h Y, Sunyaev B: Astrophys Space S c i 4, 301 (1969) References. Chapter 5. Abramnowitz H, Stegun I : "Handbook Of H a t h e n a t i c a l F u n c t i o n s " , NBS(1964) B i r r e l l N, Davies P: Phys Rev D 18,4408(1978) B i r r e l l N: •Homentum Space B e n o r m a l i z a t i o n Of In Curved Spacetime' P r e p r i n t Dept. o f Hath., King's C o l l e g e , London, Har. 1979 B i r r e l l N, Davies P: 'Conformal Syaaetry Breaking And C o s a o l o g i c a l P a r t i c l e C r e a t i o n In Theory' P r e p r i n t Dept. Of Hath., King's C o l l e g e , London, A p r i l 1979. B i r r e l l N, Ford L: Annals Of P h y s i c s 122, 1 (1979) Bjorken J , D r e l l S:' R e l a t i v i s t i c Quantum F i e l d s ' HcGraw H i l l ( 1965) Bunch T, Panangaden P, Parker L:' On B e n o r m a l i z a t i o n Of F i e l d Theory In curved Spacetime I ' P r e p r i n t Dept. Of P h y s i c s , Univ of Wisconsin, Hilwaukee 1979 Bunch T, Panangaden P:' On Ben o r m a l i z a t i o n Of ~?xf>* F i e l d Theory In Curved Spacetime I I ' P r e p r i n t Dept. Of P h y s i c s , Oniv Of Wisconsin, Hilwaukee 1979 269 Bunch T, Parker L:• The Feynman propagator In Curved Spacetime: A Momentum Space R e p r e s e n t a t i o n ' P r e p r i n t Dept. of P h y s i c s , Univ Of Wisconsin, Milwaukee, 1979. Davies P, F u l l i n g S: Phys Rev D13,2720 (1976) Gass fi. Dresden M:» The I n f l u e n c e Of I n t e r a c t i o n s On The Thermal Character Of Hawking R a d i a t i o n s - p r e p r i n t I n s t i t u t e F o r T h e o r e t i c a l P h y s i c s , S t a t e U n i v e r s i t y Of New York At Stony Brook (1979) Gibbons G, Perry M: Phys Rev L e t t 36,985(1976) ; Proc Rcy Soc A358,467(1978) Hawking S: Nature 248,3 0(1974); Comm Math Phys 43,199 (1975) Kaempffer F: "Concepts In Quantum Mechanics", Academic Press (1965) K l a i b e r , B: " L e c t u r e s In T h e o r e t i c a l P h y s i c s - Boulder L e c t u r e s 1967" Gordon And Breach, New York (1968) Nakanishi N: "Two Dimensional Quantum F i e l d T h e o r i e s I n v o l v i n g Massless P a r t i c l e s " RIMS 252, Research I n s t i t u t e For Mathematical Sciences, Kyoto U n i v e r s i t y , Japan (1978) Parker L: Phys Rev D12, 1519 (1975) Parker L: 'The Production Of Etoentary P a r t i c l e s By Strong G r a v i t a t i o n a l F i e l d s * In Proceedings of The Symposium On The Asymptotic P r o p e r t i e s Of Spacetime- Plenum P u b l . Corp. New York (1976) Schweber S: "An I n t r o d u c t i o n To R e l a t i v i s t i c Quantum F i e l d Theory", Harper And Row,(1961) Unruh W: Phys Rev D14,870( 1976): Phys Rev D15,365 (1977) Wald R: Comm Hath Phys 45, 9 (1975) Wightman A: "Cargese Le c t u r e s In T h e o r e t i c a l P h y s i c s 1964" ed n Levy, Gordon And Breach, New York (1967) 270 References. Appendices Abramnowitz M, Stegun I : "Handbook Of Mathematical F u n c t i o n s " , NBS(1964) Fr a u e n f e l d e r H , Henley E - "Subatomic P h y s i c s " P r e n t i c e -Hall(1974) G a i l l a r d M- Nature 279,585(1979) Page D- Phys Rev D 16,2402(1977) S c i u l l i F- "AIP Conf. Proc. #22- Neutrinos" American I n s t i t u t e Of P h y s i c s (1974) 274 APPENDIX A.,FIGDBES F i g 1a. Diagrams of c o l l a p s e t o form a bl a c k h o l e : Opper- Penrose diagram. Lower- diagram with t c o o r d i n a t e v e r t i c a l and ingoing n u l l rays a t 45 degrees. , F i g . 1b. Kerr m e t r i c topology along symmetry a x i s . N u l l l i n e s (constant u,0 or v,V) are a t p l u s or minus 45 degrees. F i g . 2a. Ferm i - D i r a c f a c t o r and t r a n s m i s s i o n p r o b a b i l i t y | B—| 2 f o r r o t a t i o n parameter o£ = .8 and angular mode (l,m) = (3/2,3/2) vs. freguency Bw. F i g . 2b. S i n g l e mode (l,a) = (3/2,3/2) n e u t r i n o plus a n t i n e u t r i n o number l o s s r a t e spectrum f o r <X=.8. F i g . 3a-e. Neutrino p l u s a n t i n e u t r i n o number l o s s r a t e spectrum f o r the dominant angular modes f o r <=.3, i e (l,m) = (1/2,1/2), (1/2,-1/2), (3/2,3/2), (3/2,1/2) and (5/2,5/2). F i g . 4a-f. Neutrino p l u s a n t i n e u t r i n o number l o s s r a t e spectrum f o r the dominant an g u l a r modes f o r #=.999, i e (l,m) = (1/2,1/2), (3/2,3/2), (5/2,5/2), (7/2,7/2), (9/2,9/2) and (11/2,11/2). F i g . 5a. Neutrino p l u s a n t i n e u t r i n o number l o s s r a t e spectrum (summed over angular modes) f o r r o t a t i n g b l a c k h o l e s with o/.= 0, .3, .5, .7, .8, .9, .99 and .999. F i g . 5b. Power spectrum (energy l o s s from a r o t a t i n g b l a c k hole v i a n e u t r i n o s and a n t i n e u t r i n o s vs. frequency Bw) f o r r o t a t i o n parameters o, .8, and .999. F i g . 6a-b. The anqular e i g e n f u n c t i o n S, (1, m,aw,©) vs. p o l a r angle © f o r 1=3/2, m=*_3/2, M/2 and aw=0, .4, 1. The second e i g e n f u n c t i o n i s given by S a(l,m,aw,6)= (-1) S, (l,m,aw,n-©) . F i g . 6c-e. The angular e i g e n f u n c t i o n S, (1, m,aw,fr) vs. p o l a r angle & f o r 1=5/2, m=±5/2, *3/2, +1/2 and aw values i n d i c a t e d . F i g . 7. Net number l o s s r a t e (neutrinos minus a n t i n e u t r i n o s ) vs. p o l a r angle 6 f o r r o t a t i o n parameters <*=. 1, .3, . 8 and .999. .r 272 F i g . 8a. Net number l o s s r a t e ( i n t e g r a t e d over freguency Mw) v s . p o l a r angle B- f o r each of the dominant (l,m) modes f o r oc = . 1. F i g . 8b. Net number l o s s r a t e ( i n t e g r a t e d over freguency Mw) vs. p o l a r angle & f o r each of the dominant (1,m) modes f o r «(=. 5. F i g . 8c. Net number l o s s r a t e ( i n t e g r a t e d over freguency Mw) vs. p o l a r angle 8 f o r each of the dominant (l,m) modes f o r QC=.999. F i g . 9. Neutrino p l u s a n t i n e u t r i n o number l o s s rate ( i n t e g r a t e d over frequency Mw and summed over angular modes) vs.. p o l a r angle & f o r r o t a t i o n parameters i n d i c a t e d . F i g . 10. Emitted power i n n e u t r i n o s plus a n t i n e u t r i n o s vs. p o l a r angle ©-for r o t a t i o n parameters o( = 0, .8, .99 and .999. F i g . 11. Angular momentum l o s s r a t e v i a n e u t r i n o and a n t i n e u t r i n o e m i s s i o n vs. p o l a r angle 6 f o r r o t a t i o n parameters o(=. 3, .8 and .999. F i g . 12. Asymmetry i n n e u t r i n o emission ( i e the d i f f e r e n c e d i v i d e d by the sum, of n e u t r i n o and a n t i n e u t r i n o number l o s s rates) vs. p o l a r angle & f o r r o t a t i o n parameters «=. 1, . 3 and . 7. F i g . 13., Schematic i l l u s t r a t i o n of the b l a c k hole e v a p o r a t i o n process f o r n e u t r i n o s , with the F e r m i - D i r a c f a c t o r l i s t e d at top. F i g . 14. Large s c a l e baryon and ant i b a r y o n c u r r e n t s caused by n e u t r i n o s from r o t a t i n g black holes. The long arrows r e p r e s e n t the n e u t r i n o , a n t i n e u t r i n o , baryon, and ant i b a r y o n c u r r e n t s , whereas t h e s h o r t arrows r e p r e s e n t the r o t a t i o n axes of the evap o r a t i n g b l a c k holes. F i g . 15a. Time dependence o f 8, T, n, fit-, ^hk , and p** f o r f o r m a t i o n of b l a c k h o l e s a t time t d=M with number d e n s i t y n k K=^/M 3. (Logarithmic s c a l e s ) . Case p <y1/2/M. F i g . 15b. Time dependence of B, T, n, / r - , /Ohk , and />cr* f o r formation of black holes at time t e=M with number d e n s i t y n k K=yfi/M 3. (Logarithmic s c a l e s ) . Case p>yVz/ft. F i g . 16. Boundaries (u,v=+L) on which the s c a l a r f i e l d i s r e q u i r e d to be p e r i o d i c , i n order t o r e g u l a t e the i n f r a r e d d i v e r g e n c e s i n the • p a r t i c l e number c a l c u l a t i o n . The 273 i n t e r a c t i o n r e g i o n i s shaded. 274: SINGULARITY (r=0) + FUTURE NULL INFINITY r=0 FIG.la. RADIALLY INGOING SPHERICAL WAVES (u=const.) FIGURE lb, to cn 276 I I I I 0 .5 1.0 1.5 M w F I G U R E 2 a . 277 FIGURE 2 b. 279 FIGURE 4a. dcodt * S 9 9 9 ( I ,m) = (1/2,1/2) FIGURE oc=.999 ( l,m) = (5/2,5/2) O.D 0.4 0.8 281 FIGURE 4f. dcodt OC =.999 ( I ,m)= (11/2,11/2) - r -0.4 283 284 285 2 8 6 j-2 8 7 288 289 290 291 292 293 294 295 296 297 FERM1-DIRAC FACTOR: {EXP[(E - Y - I t > / k T > l ] FIGURE 13. 299 FIG.,14 300 301 302 303 APPENDIX B. DETAILS_OF_CALCULATIONS_F 1., Angular F u n c t i o n s The method f o r s o l v i n g f o r the angular e i g e n f u n c t i o n s i s d e s c r i b e d here. Eguations (6.3) of chapter 2 were i n t e g r a t e d from #=0 t o <5>=77/2. The matching of S / and at S=T?/2, by egn (2.14c) , gave a check on the c o n t i n u i t y of S ; and S t and thus on the accuracy of the eig e n v a l u e s used. The s t a r t i n g v a l u e s f o r the i n t e g r a t i o n were obtained by a power s e r i e s s o l u t i o n t o egns (6.3) f o r S ( and S^. One f i n d s , f o r m>0: Sx=Qr + l (a v+cJ>2 + ...) (B1) with the c o e f f i c i e n t s given by: aa_=-k/(2m*1) c ,= (ka^-z-m/3)/2 c, = (-kc^+ (z*m/3)a a)/(2m+3) (B2) For mO one o b t a i n s : S l=&- r n(1 + c i e 2 * (B3) with c o e f f i c i e n t s : 304 a(=-k/(2m-1) c f t=(ka,-z-m/3)/2 c , = (-k=a • (z*m/3) a, ) / (2m-3) ( B 4 ) Sample r e s u l t s of the c a l c u l a t i o n s of S, and S^ are p l o t t e d i n f i g u r e 6 f o r 1=3/2 and 1=5/2. 2«._TE§B§gi§gJo? C o e f f i c i e n t s l a t h i s s e c t i o n the method f o r o b t a i n i n g the t r a n s m i s s i o n c o e f f i c i e n t s , f o r the c a l c u l a t i o n s of chapter 2, i s o u t l i n e d . The r a d i a l equations (6.5) can be w r i t t e n , f o l l o w i n g Page(1977) (note the change i n n o t a t i o n ) : (d/driiK/^)R L(r)=k/ A»« fl. (r) ( B 5 ) with + f o r i=1,j=2; - f o r i=2,j=1; and with: K=-(w(r 2*a*) *ma) = (K^x 2 + K lx+K 0) ( r ^ - r . ) ( B 6 ) x=(r - r f ) / ( r + - r _ ) K^=-(r +-r_) w K, =-2r +w K e=- ( ( r + 2 * a 2 ) w + ma) / ( r + - r _ ) 305 D e f i n i n g new r a d i a l f u n c t i o n s G and F by: fi =G*iP B =S-iF (B7) I a , the r a d i a l e g u a t i o n s (B5) become r e a l : ( A ^ d / i r - k ) G-KA~1/2F=0 (B8a) (Avzd/dr«-k) F-KA-^2G=0 (88b) B i t h the a i d of the i d e n t i t y ( r + - r _ ) 2 x (x+ 1) , and d e f i n i n g the new r a d i a l v a r i a b l e y by y=ln (exp (x)- 1), the eguations (B8) become: dG/dy=(k ( 1 - e - * ) / ( x V 2 (1*x) **) G • (K 2x2*K,x+K o(1-e-M)/(x(1+x)) F (B9a) dF/dy=- (k ( 1-e-*)/U * * ( 1*x) »*) G - ( K ^ x Z + K^x + K, (1-e-*) )/(x (1 + x)) F (B9b) A pure i n g o i n g mode at the h o r i z o n r ^ . {x approaching 0, y neg a t i v e i n f i n i t y ) i s : Gee e x p ( i K 6 y) F=iG (BIO) 306 Hear the hocizon one expands H, and R x i n power s e r i e s : :,=exp{iK f ry) x»* (a ,+b, x+c, x 2*...) ( B l l a ) B^=exp(iK ey) (1+b xx*c xx 2*...) (B11b) From egn (B5) one f i n d s the val u e s of the c o e f f i c i e n t s : a , = k/(2iK 0+1/2) b 2_=ka,-i(3K 0/2-K ( ) b, =[ k (b^-1/2) * i (K 0/2-K, ) a, ]/ (2iK 0*3/2) c z=[k (b, - a l / 2 ) - i ( 3 K 0 / 2 - K , ) b . ^ i ( 1 1 K 0 / 1 2 - K , *K2) ]/2 c, = [ k (c^-b^/2+3/8) * i (K../2-K ,) b, - i ( 13K 0/12-K t+K,)a, ]/{2iK 0+5/2) (B12) The s o l u t i o n s (B11) can be used t o give a c curate s t a r t i n g v a l u e s (x near 0) f o r F and G: S = (1/2) (B, + R^) F=(i/2) (B,-R^) IB13) Eguations (B9) are then i n t e g r a t e d to l a r g e y. F and G can be w r i t t e n : 2G=Z^ axp tys*ij +Z^exp (yS-iJ-iq^) (Bl»»a) 307 2 F = i Z ^ exp (S*i6+ij5) - i Z ^ e x p t ^ - i e - i ^ ) ) (Bl4b) Z^ and Z„4 are the f l u x e s of the i n g o i n g and outgoing waves. To ordar 1/x 2 Page(1977) f i n d s the expansions f o r the f u n c t i o n s i n the exponents of eqns (B14): /=k(k-1)/4K 2x 2 *0(x- 3) S=k (k*1) / 4K 2x 2 +0(x- 3) y=k (k-1) /2K zx -[ (k-1) (Kjt+ck) - c k ] / 4 K 2 x 2 +0(x- 3) ^=k (k+1)/2K ix - (k+1) (K^k+ck) / 4 K 2 x 2 +0(x~ 3) (B15) In eqn (B15) c i s given by: c=2(K, -K x) The phase cf does not vanish as x approaches i n f i n i t y , and i s given by: 4>= j d x ( K l x 2 * K 1 x + K # ) / ( x ( 1 + x) ) =const.+kx+clnx*0 (x- 1) (B16) One d e f i n e s the q u a n t i t y E i n t e r n s of the i n g o i n g and outgoing f l u x e s by: 308 =[ |2Gj2exp(-2 / i) • J2FJ 2exp (-2S)-exp {-/*•- £) 2Be (2S*2F) s i n -£) ] / (2exp )Im (2G*2F) cos ) From t h i s the t r a n s m i s s i o n c o e f f i c i e n t I B _ J 2 i s found v i a : |B_!*=1-|Z^12/|Z^ I 2 =2/(E+1) (B18) 309 APPENDIX C. NATUSAL_1PLANCKJ__UNITS In n a t u r a l u n i t s the fundamental c o n s t a n t s c,/ft,3 and k are taken to have u n i t v a l u e . The s i z e of these u n i t s i n terms of standard (such as c.g.s.) u n i t s i s found by c o n s t r u c t i n g the standard u n i t from powers of c^«,G and k. Th i s i s done i n Table 2 below. Conversion From Natural To Standard U n i t s Mass: (Hc/G) 2. 18x10-5gm Length: (*G/c3) «« = 1. 62x10" 3 3cm Time: tKG/cS) i/* = 5. 39x10-**sec Temp: (£cS/G) v«/k= 1. 42x1032R Energy: (•ttcVG) = 1. 96x10 **erg=1.22x1022Mev Power: C5/G= 3. 63x10S9erg/sec Charge; (he) *2 = 5. 62x10-»esu=11.7e Density: c S / h G 2 = 5. 13x10 9 3gm/=m 3 Magnetic F i e l d : (8Tic7//h) "Z/G= 4. 29x10 S 6Gauss EG: e l e c t r o n mass=4.19x10- 2 3, proton mass=7.66x10-2°, s o l a r mass = 9. 13x10 3 7 310 APPENDIX D. THE RATIO OF NEUTRINO BARYON TO NEUTRINO iNTIBAeYON_CRUSS_SECTIONS_AT The c r o s s s e c t i o n r a t i o CJ"v$/C7g i s c o n s i d e r e d here. In the v e c t o r boson theory the n e u t r i n o nucleon d i f f e r e n t i a l c r o s s s e c t i o n i s ( S c i u l l i , 1974) : The minus sig n ( - ) i s f o r n e u t r i n o s on nucleons, the p l u s sign(+) i s f o r n e u t r i n o s on a n t i n u c l e o n s . H i s the nucleon mass and i s the boson propagator mass. I have i n c l u d e d the boson propagator f a c t o r (1*Q2/M v z ) - 1 , important at high energy. Here Q 2 i s the f o u r momentum t r a n s f e r squared; x and y are g i v e n by: Kinematics r e s t r i c t s x and y to 0<x<1,0<y<1 . E i s the H boson propagator energy, and E, i s the n e u t r i n o energy. One d e f i n e s the constant A by do-/dxdy=(G2ME y/-^[ (1-y)F^ + 2xy 2F,/2 + xy (1-y/2) F^ ] / ( H Q 2 / M W 2 ) (D1) x=Q 2/23E y = B / E l / Q2/o\» 2= 2HEy xy/H y y 2 = Axy (D2) F, ,Fx and F 3 are nucleon form f a c t o r s . The C a l l e n - G r o s s 3IT r e l a t i o n F i=2xF ( holds f o r a nucleon of spin 1/2 c o n s t i t u e n t s . F o l l o w i n g S c i u l l i (1974), we d e f i n e : g(x) = (F^-xF,)/2 g(x) = (F : L*xF^)/2 the •guark' and 'antiguark* components of the nucleon. One has g (x) =xg (x) where g(x)dx i s the p r o b a b i l i t y f o r f i n d i n g guark c o n s t i t u e n t s of the nucleon with f r a c t i o n a l momentum between x and x+dx. g(x) i s r e l a t e d i n the same way to the an t i g u a r k component of the nucleon. I n t e g r a t i n g egn (D1) over y y i e l d s : d<rV£ /dx= (G2M£y/ft-) (g/(1 + Ax) + f (Ax) g) (D3a) dove/dx=d<Tye/dx=(G2HEy/ir) (g/(1*Ax)+f (Ax) q) (D3b) In egns (D3) , f(Ax) i s d e f i n e d by: f (Ax) =(2+Ax)/(Ax) 2-2(1 +Ax) In (1 + A x ) / ( A x ) 3 (D4) In egn (D3b) I have used the e g u a l i t y of n e u t r i n o a n t i b a r y o n and a n t i n e u t r i n o baryon c r o s s s e c t i o n s (egual by CP i n v a r i a n c e ) . He s h a l l next i n t e g r a t e over x, u s i n g : 312 • Jdx/(1+Ax) =ln (1 + A) /A (D5a) i ^dx f (Ax) = ( (H1/A) 21n (1 + A) - 1/A-3/2) /A (D5b) D We assume the quark component g(x) dominates f o r nucleon s c a t t e r i n g so t h a t one can take q>>g i n egns (D3). A nucleon i s viewed of as c o n s i s t i n g of 3 valence quarks together with a sea of v i r t u a l quark- a n t i g u a r k p a i r s . He then n e g l e c t the x dependence of g and g and use egns (D5) to o b t a i n the c r o s s s e c t i o n r a t i o from egns (D3) : CT;B/CT»i=ln (1+A) /((1 + 1/A) 2 In (1+A) - 1/A-3/2) (D6a) = 3 f o r A<<1 =1 f o r A » 1 (D6b) The t r a n s i t i o n of the c r o s s s e c t i o n r a t i o from 3 to 1 o c c u r s f o r A of order u n i t y . The behaviour of the asymmetry**, (ff?s -Ovs ) /<^i& , i s given by: <<=(1/A+3/2)/ln(1 + A) -(2/A+1/A2) (D7a) =2/3 f o r A « 1 =3/(21nA) f o r A » 1 (D7b) R e c a l l t h a t A i s given by A = 2HEy /M^2. 3 1 3 The assumptions used are known to hold o n l y approximately. E.g. S c i u l l i , 1 9 7 4 notes that t h e r e i s a s m a l l a n t i g u a r k component (of order 10 percent) f o r nucleon s c a t t e r i n g . The above formulas are not pretended to be r i g o r o u s but merely as guides needed f o r the c a l c u l a t i o n s i n Chapter 3. F i n a l l y ^ here we w i l l j u s t i f y the formula (1.9) of Chapter 3 (the same as egn (2.6) of Chapter 4): CTve =e*s/(s+H^2) z (D8) T h i s can be d e r i v e d from the low e n e r g y 1 7 c r o s s s e c t i o n which i s . normally quoted i n t e x t s (e.g. F r a u e n f e l d e r and Henley,1974; a l s o qiven by S c i u l l i , 1 9 7 4 ) : CVg =10-3«cmz (E y/lGev) (D9a) =G 2i Ey/TT (D9b) Hith the a i d of the approximate r e l a t i o n between the weak i n t e r a c t i o n constant G and the proton mass M: G//2=13-s/M2 s«oV 2, but for neutrino energies of order I3ev or greater. * 314 eqns (D9a) and (D9b) are seen to be the same. To o b t a i n eqn (D8) we i n c l u d e the a d d i t i o n a l f a c t o r 1/n+s/M^2)2 f o r the f i n i t e mass of the boson propagator. , L a s t l y we use the r e l a t i a n s (see G a i l l a r d , 1 9 7 9 ) : GM^2=g2/4 J l 1 g=e/sin£«/ sin 2(9^,= . 2 3 (D10) toge t h e r with the formula f o r the square of the fo u r momentum t r a n s f e r : s=2ME y. In eqn (D10) L% i s the Heinberq angle and g i s the d i m e n s i o n l e s s weak c o u p l i n g constant. ZX5 APPENDIX E. THERMALIZATIQN_0F_BLACK_H3 Here we show that the p a r t i c l e s emitted by ev a p o r a t i n g black h o l e s t h e r m a l i z e r a p i d l y with the surrounding medium i n the e a r l y u n i v e r s e . The t i a e s of i n t e r e s t are before the end of the hadron e r a , so p a r t i c l e masses w i l l be n e g l e c t e d . The t h e r m a l i z a t i o n time ^ f o r a s i n g l e emitted p a r t i c l e i s : ^ = 1/(n<rv) (E1) The v e l o c i t y v i s roughly the speed of l i g h t : v=1, f o r the high e n e r g i e s of emitted p a r t i c l e s c o n s i d e r e d here. c3" i s the t o t a l c r o s s s e c t i o n f o r c o l l i s i o n with ambient p a r t i c l e s and n i s the number d e n s i t y of the ambient p a r t i c l e s . Due to the power law dependences of n, the s c a l e f a c t o r R aud the temperature T (see egn (3.3) of chapter 3, or t a b l e 1 f o r ^>1/10M), the time s c a l e f o r changes i n the Friedmann expansion i s t , times a numarical f a c t o r of order u n i t y . Consider n e u t r i n o s with emitted energy E of roughly E=1/M, where fl i s the black, hole mass. The n e u t r i n o s are the part o f the emission by the black h o l e s which i s slowest to t h e r m a l i z e . These i n t e r a c t with baryons of thermal energy E E of E g=T and number d e n s i t y n. The c r o s s s e c t i o n i s given by (see egn (4. 10) of chapter 3 ) : 31G The r a t i o of the t h e r m a l i z a t i o n time t o the t i m e s c a l e t i s given by: ? v/t=2. 5x10»T/(nMt) (E2) =1.4x10s/M f o r /<1/10H = 1 0 s / g 2 ^ a - * / 3 t » / 3 f o r ^>1/10H t i<t<100M' For the case of s m a l l black hole d e n s i t y , n and T are g i v e n by egn (3.3) of chapter 3, whereas f o r the case t h a t the black h o l e s a l t e r the standard Friedmann expansion, t a b l e 1 has been used. T h i s r a t i o i s much l e s s than u n i t y f o r a l l masses of i n t e r e s t (H of o r i e r 2 X 1 0 1 2 ) . Only the time p e r i o d of t i<t<100n 3 i s considered e x p l i c i t l y here f o r ^ s>1/10H, s i n c e t h i s i s the p e r i o d when the energy d e n s i t y i n the black hole emission becomes important, %/t i s l a r g e s t than f o r t=100H* g i v i n g f y / t < 5 x I O S ^ Z ^ J J - ^ 3 fDr/3>1/10H. For masses of i n t e r e s t (M=2x13*2) one has ^/t<40^ 2 / 3. Thus except f o r very l a r g e i n t e n s i t i e s of black hole formation (^f>4x10-3) , the t h e r m a l i z a t i o n occurs r a p i d l y r e l a t i v e to changes i n the ge n e r a l c o n d i t i o n s of the u n i v e r s e (%, <<t). The dep a r t u r e from e g u i l i b r i u m f o r the medium surrounding the b l a c k hole i s measured by: 317 (black hole energy emission r a t e ) x (black hole number density) x ( t h a r m a l i z a t i o n t i m e t ) / (energy d e n s i t y of medium) = D=(y/M2) (^frt-** ( B / A ) - 3 ) r y ( ^ m (E3) = 1.4x1 O ^ M - ^ t 3 * f o r ^ <1/10H = 1 0 ^ 2/3 M-*/3 11/3 for^>1/10M t„<t<100M 3 'a. The b l a c k h o l e number d e n s i t y i s given by n ^ of egn(fl.7b) of chapter 3. Here the t h e r m a l i z a t i o n time f o r n e u t r i n o s Vv has been used r a t h e r than the mean t h e r m a l i z a t i o n time: (Z n-cr- v-) _ 1 which w i l l be s m a l l e r , by a f a c t o r depending on the e n e r g i e s and types of emitted p a r t i c l e s . The upper l i m i t t o D, f o r the case ^<1/10M, i s when ^f = 1/10K: D<1.4x10s/n (t/IOOM 3)*z. The l a t t e r f a c t o r i s j u s t t d i v i d e d by the bl a c k hole l i f e t i m e and i s always l e s s than u n i t y (while the black holes are s t i l l e v a p o r a t i n g ) . The remaining p a r t i s f a r l e s s than u n i t y f o r a l l masses of i n t e r e s t . For ^>1/10)i, s e t t i n g t=100M 3 g i v e s the upper l i m i t on D of 5x10 s /g 2 A 3M _ l / 3. The upper l i m i t s here on D are the same as the upper l i m i t s on Ty/t above, but the a c t u a l value of D w i l l be l e s s s i n c e one would use the mean t h e r m a l i z a t i o n time r a t h e r than that f o r n e u t r i n o s . 

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