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A quantum mechanical treatment of the relativistic scattering of light by the sun Feser, Siegfried 1969

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A QUANTUM MECHANICAL TREATMENT OF THE RELATIVISTIC SCATTERING OF LIGHT BY THE SUN by SIEGFRIED FESER B . S c , U n i v e r s i t y of Manitoba, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the department OF PHYSICS We accept th i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, 1969 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date ^j^!yfoM • i i ABSTRACT This thes i s concerns i t s e l f with the a p p l i c a b i l i t y of quantum f i e l d methods, i n the f ixed f i e l d approximation, to problems invo lv ing a weak g r a v i t a t i o n a l f i e l d . Af ter in troducing general scat ter ing r e l a t i o n s , various c l a s s i c problems are reviewed to i l l u s t r a t e various approaches to so lv ing sca t ter ing problems. Newtonian and quantum mech-a n i c a l f i e l d methods are i l l u s t r a t e d using Coulomb scat -t e r i n g . C l a s s i c a l r e l a t i v i t y i s used to solve the bending of l i g h t rays by the sun. F i n a l l y , quantum f i e l d methods are used to solve the sca t ter ing of po lar ized photons by the sun. The a d d i t i o n a l problems of scat ter ing of l i g h t by a mass d i s t r i b u t i o n and by a r o t a t i n g mass are c a l c u l -able using t h i s method. i i i CONTENTS ABSTRACT . . i i LIST OF FIGURES i v ACKNOWLEDGEMENTS v INTRODUCTION 1 S I . BASIC RELATIONS BETWEEN SCATTERING CROSS,-SECTIONS AND DEFLECTION ANGLES 4 § 1 1 . CLASSICAL RUTHERFORD SCATTERING 1 1 S I I I . QUANTUM MECHANICAL RUTHERFORD SCATTERING 1 9 § I V . THE CLASSICAL EINSTEIN EFFECT - THE GRAVITATIONAL DEFLECTION OF LIGHT 30 §V. THE QUANTUM MECHANICAL EINSTEIN E F F E C T . . . 3 6 § V I . CONCLUSION 52 BIBLIOGRAPHY 54 APPENDIX A: THE SCATTERING MATRIX 56 APPENDIX B: THE SQUARE OF A DELTA-FUNCTION 6 l APPENDIX C: CONSERVED QUANTITIES 64 APPENDIX D: THE GEODESIC EQUATIONS AND INTEGRALS OF MOTION i v LIST OF FIGURES 1.1 IMPACT PARAMETERS AND SCATTERING ANGLES 4 1.2 THE DIFFERENTIAL CROSS-SECTION: 6 2.1 COULOMB SCATTERING 15 4.1 ORBIT OF A PHOTON IN A PLANE THROUGH THE ORIGIN OF A SCHWARZSCHILD OBJECT 33 5.1 GEOMETRIC IDENTIFICATION OF INCIDENT AND SCATTERED PHOTON 37 5;.2 DEVIATION FROM EINSTEIN'S RESULT-INCIDENT AND DETECTED POLARIZATIONS PARALLEL ' . . 4 8 5.3 DEVIATION FROM EINSTEIN'S RESULT-INCIDENT AND DETECTED POLARIZATIONS PERPENDICULAR..49 V AC KI\1 OWLED CEMENTS I am indebted to Dr. F . A . Kaempffer for suggest-ing the top ic of t h i s thes i s and f o r his generous g u i d -ance. I am g r a t e f u l to the Nat ional Research Counci l for the freedom t h e i r scholarship has given me. 1 INTRODUCTION In a l l observations of the def lec t ion of l i g h t by the g r a v i t a t i o n a l f i e l d of the sun one measures the angle of de f l ec t ion suffered by the l i g h t ray emitted from a distant s tar as i t passes near the sun's surface . Accordingly , the c l a s s i c a l theory of t h i s effect i s based on Fermat's P r i n -c i p l e which prescribes the geodesic as the ac tua l path . taken by l i g h t between source and observer. In most laboratory scat ter ing experiments, on the other hand, one measures an i n t e n s i t y as a funct ion of scat ter ing angle. Accordingly , both the c l a s s i c a l and the quantum mechanical theory of scat ter ing i n p a r t i c l e physics are aimed at y i e l d i n g the d i f f e r e n t i a l s ca t ter ing cross - sec t ion , i . e . the r a t i o between the scattered i n t e n s i t y i n t o a given s o l i d angle and the inc ident i n t e n s i t y per uni t area. One can, of course, describe p a r t i c l e scat ter ing also i n terms of the sca t ter ing angle i n the path of an incident p a r t i c l e . Even though i t has no immediate experimental relevance, because i t would requ ire , for example, the exper-imentor to f i r e an c £ - p a r t i c l e v/ith a given impact parameter past a nucleus; th i s a l t ernat ive treatment of p a r t i c l e scat-t e r i n g i s well-known and i t i s often-used as a bas is for the der ivat ion of s ca t ter ing cross - sec t ions . A l t e r n a t i v e l y , the g r a v i t a t i o n a l def lec t ion of l i g h t can be treated as a quantum mechanical scat ter ing problem, i n which the sun, for example, i s treated as the sca t ter ing 2 center which presents a sca t ter ing cross - sec t ion to an i n -cident beam of photons. This a l t ernat ive does not seem to be well-known, and i t i s the purpose of t h i s thes i s to rem-edy that def ic iency i n the l i t e r a t u r e on t h i s subject . To br ing out the equivalence of the two a l t e r n a t i v e treatments of scat ter ing i n general , the general re la t ions ex i s t ing between the concept of the sca t ter ing angle and the concept of the s ca t t er ing cross - sect ion are exhibited i n Sect ion I . The c l a s s i c a l example of a p a r t i c l e s ca t t er -ing i s the Rutherford sca t ter ing experiment, whose theory i s summarized i n Sect ion I I . The c l a s s i c a l theory i s not su i tab le when the scattered p a r t i c l e possesses i n t r i n s i c a t t r i b u t e s , such as sp in , and a f u l l quantum mechanical treatment of the scattered p a r t i c l e i s advisable even though the sca t t er ing center may be treated i n the so -ca l l ed f ixed f i e l d approximation, as i s explained i n Sect ion I I I , with the sca t ter ing of a Dirac e lectron i n a given Coulomb f i e l d serving as example. The c l a s s i c a l E i n s t e i n theory of grav-i t a t i o n a l l i g h t sca t ter ing i s reviewed i n Section IV, with emphasis on the analogy to c l a s s i c a l Rutherford s c a t t e r i n g . F i n a l l y , the quantum mechanical treatment of g r a v i t a t i o n a l photon scat ter ing i n the f i xed f i e l d approximation i s dev-eloped i n Section V, and as i n the case of the e lectron the i n t r i n s i c p o l a r i z a t i o n property of the photon can be hand-l e d most appropriate ly i n t h i s way. The present work should pave the way for the theore t i ca l treatment of more elaborate problems, such as the effect of the so lar mass d i s t r i b u t i o n and of the so lar ro ta t ion on the 3 e x t r i n s i c and i n t r i n s i c a t tr ibutes of inc ident photons, and the s ign i f i cance of such future work i s b r i e f l y d is cussed i n Sect ion V I . V FIGURE 1 . 1 IMPACT PARAMETERS AND SCATTERING ANGLES 4 i i . BASIC RELATIONS BETWEEN SCATTERING CROSS-SECTIONS AND DEFLECTION ANGLES Consider f i r s t the case i n which the concept of a 'path' of a p a r t i c l e can be def ined, and suppose one has a f i xed sca t ter ing center (taken to be the o r i g i n of the coordinate system). I f one i s in teres ted i n the s c a t t e r -ing of a uniform, i n i t i a l l y p a r a l l e l beam of i d e n t i c a l , noninteract ing p a r t i c l e s inc ident upon the sca t ter ing reg -ion with i n i t i a l v e l o c i t y V Q Q , then d i f f erent p a r t i c l e s i n the beam w i l l have d i f f erent impact parameters p (see F i g -ure 1.1) and hence w i l l be scattered a d i f f eren t amount ^CJ from t h e i r i n i t i a l d i r e c t i o n . The impact parameter i s defined to be the perpendicular distance from the o r i g i n to the asymptotic i n i t i a l path of the p a r t i c l e . I f dN p a r t i c l e s scat ter per uni t time through an angle between X> and %+d%, l e t (1.1) d ( r s i , n where n i s the number of p a r t i c l e s passing i n unit time through a un i t area of the beam cross sect ion (assumed uniform). 'dies'' i s c a l l e d the e f fec t ive sca t ter ing cross -s ec t ion . Assume that p a r t i c l e s with impact parameters bet-ween p^iZs) and p^( j6)+dPi( %) scat ter through an angle between % and %+dX> • The subscript on p allows f o r the p o s s i b i l i t y of the r e l a t i o n s h i p between p and % not being 5 one to one. Then dN = ^27( Pi dpi n; (1.2) < r T d(T =^2tt Pi dpi . 1 I f , as usual, p and J6are one to one (which i s true f o r the usual case of JC a monotonic, decreasing function of p), (1.3) d<r = - 2 f f p ( ; t ) A £ d £ , 3X-where the negative sign r e s u l t s from the decreasing nature of )G{p). Or, since the s o l i d angle. dI2 between PC and X+dX i s 2 ?rsin£ d £ , ( 1 . 4 ) ri^_ - P ( ^ ) ^ P ( X ) rfr?. s i n X } ^ The case i n which the angle of scattering i s small i s of i n t e r e s t . Suppose that the deflection angle i s related to the impact parameter by where A i s some constant. (This i s a v a l i d r e l a t i o n f o r large p i n the case of Rutherford scattering as well as f o r E i n s t e i n scattering.) The p a r t i c l e s deflected between X and Pi+dX> are the ones that have impact parameters bet-FIGURE 1 . 2 THE DIFFERENTIAL SCATTERING CROSS r SECTION 6 ween p and pfdp (see Figure 1.2), where (1.6) p = £ ; | d P | = A 2 d ^ This area, the r i n g between p and p+dp, i s then, by defin-i t i o n , the d i f f e r e n t i a l cross -sect ion for s ca t t er ing into the angle between %j a n c * > that i s , into the s o l i d angle d-0.= 2?i: s in %> d% & 2 tt % d%. Thus 2 (1.7) d < r = | 2 ? r p d p | = , and (1.8) djr _ l27(p dp| _ A £ dj2 ~ 2n¥,&%-jK Next consider the case of a scattered p a r t i c l e des-cr ibed quantum mechanically as an inc ident plane wave, u U i ^ ) e _ i k i x , (1.9) kjx BCO^x0 - k^'X , x = (x 0 = t , x)> representing a state | k i , s ^ , where kj_ i s the four-momentum (4i = c = l ) and Sj[ includes other quantum numbers such as s p i n . This wave, when entering the sca t ter ing region has a p r o b a b i l i t y of being scattered in to another ( f ina l ) state ^ k f , 3 ^ 1 which again can be writ ten as a plane wave. Under these conditions the amplitude for s ca t t er ing from | ^ i » s j ^ to a d i f f erent state ^ f j S ^ J , i n the f i r s t Born approxi -mation has the form ( 1 . 1 0 ) < k f , S f | i JXintd4x|ki'?i^ = ^ f ' s f | M ) ^ i ' s i > SWf*>i in a f i x e d , time-independent external f i e l d , where oC^n^ i s the Lagrangian density governing the i n t e r a c t i o n , and U)±, COf are the energies of the scattered p a r t i c l e i n the i n i t i a l and f i n a l states (see Appendix A ) . The de l ta - func-t i o n re su l t s from the in tegrat ion over the time coordinate of the sca lar product of the i n i t i a l and f i n a l plane waves and ensures conservation of energy. However, i f one neg-l ec t s the r e c o i l of the source, one does not have conserva-t i o n of momentum. Using the r e l a t i o n , v a l i d f o r .any d e l t a -funct ion (see Appendix B) , (i.n, L§:<v^>]2=%^ j«. one has the t r a n s i t i o n p r o b a b i l i t y per uni t time for f ixed i n i t i a l and f i n a l s tates , (1.12) W f l - l ^ f ^ f l f c n t ^ l " ! - ^ ! 2 2 W 1 1 8 The incident intensity I i s defined as the number of incident particles per unit time per unit area of cross section of the. beam. Thus, I equals the particle density times the incident speed. If one normalizes the incident plane wave to yield one particle per volume V, ( 1 . 1 3 ) I = f , where v is the incident speed. The cross-section for scattering into a definite f i n a l state is (1.14) ^ = W F L | . The differential cross-section for scattering into the set of f i n a l states between kf and kg+dkg numbering (1.15) dNk f = — d k f i s (1116) do- = wfi | dN , U , 1 7 ) S { ^ V ) ^ 1 ) ^Kk^MMlki'si>|2 d k f -If one can assume that the interaction i s spherically sym-metric then the cross-section for scattering into the solid angle dH i s given by 9 (2 ?r )**- v where dk^ has been replaced by the expression integrated over d I 2 , i . e . , k^.2 d | k f J dQ and the rest of equation ( 1 . 1 7 ) has been l e f t unchanged due to the s p h e r i c a l symmetry. Since 2 _ „ 2 , v 2 ( 1 . 1 9 ) ^ f = m + k f and ( 1 . 2 0 ) cjfdu)f = ( k f | d | .k f | , equation ( 1 . 1 6 ) becomes, ( 1 . 2 1 ) ^ = ^ - ^ \ ^ f . ' f \ ^ , < ^ \ 2 Uf| « t W , I f the detector.responds only to p a r t i c l e s scattered i n t o the s o l i d angle dQ.{&,0), one must integrate over a l l dcJf keeping 8, 0 constant. Since the inc ident v e l o c i t y may be expressed r e l a t i v i s t i c a l l y as ( 1 . 2 2 ) v = £ , the in tegra t ion over dVg can be c a r r i e d out at once with the r e s u l t 10 The above quantum mechanical analys i s has been based on the stated assumption that the plai e waves are normal-i z e d to y i e l d one p a r t i c l e per volume V. This normaliza-t i o n i s not Lorentz invar iant since the volume V i s con-t r a c t e d along the d i r e c t i o n of motion r e l a t i v e to a frame of reference . I f one wishes to have an invar iant prob-a b i l i t y , i t i s convenient to normalize the wave functions to one e lectron per invar iant volume V.m/cJ . With t h i s normalizat ion the fo l lowing c r u c i a l equations must be r e -wri t ten as i n d i c a t e d . ( 1 . 2 4 ) was ( 1 . 1 3 ) I . m V ( 1 . 2 5 ) was ( 1 . 1 5 ) dN, = Iffi dkf - f *>(2?F)3 Hence, equation ( 1 . 2 3 ) becomes (1.26) f l = l V U f ( E f | M | k i ( S i f The i n i t i a l and f i n a l states may contain information about the p o l a r i z a t i o n propert ies of the scattered p a r t i c l e . I f they are not observed, the expression ( 1 . 2 3 ) has got to be subjected to appropriate sums and averages over the respect ive p o l a r i z a t i o n s ta tes . 11 §11. CLASSICAL RUTHERFORD SCATTERING The laws of motion f o r a mechanical system can be derived by app l i ca t ion of Hamilton's p r i n c i p l e (or the p r i n c i p l e of l east a c t i o n ) . This approach necessitates that every mechanical system be characterized by a func-t i o n , c a l l e d the Lagrangian, which, for a system with s degrees of freedom, one writes L ( q ] _ , q 2 , • . . ;q^ , $ 2 1 • • • >qs>t) , or equivalent ly L(q ;q ; t ) for a r b i t r a r y degrees of freedom, where the q's are the coordinates , the q's are the conjug-ate v e l o c i t i e s , and t i s the parameter representing t ime. The condi t ion that the a c t i o n , i s an extremum for the ac tua l motion, q ( t ) , of the system i . e . f i x i n g the endpoints of the t r a j e c t o r y . For one deg ree of freedom one gets , ( 2 . 1 ) dt = 0, S q dt = 0, 12 where the f i r s t term i s zero because the endpoints of the t r a j e c t o r y are f i x e d . Since the r e s u l t i s true for a r b i -t r a r y S q, For a system with s degrees of freedom, the s d i f f erent functions q^(t) must be var ied independently to y i e l d (2.4) A ( ^ k . ) . 1^ - =0, i = l , 2 , . . . , s . dt 2> o q^ These are the E u l e r , or Lagrange's equations of motion for the system. Some important ' in t egra l s of motion' can be derived from the homogeneity and i sotropy of space and time. The energy of the system, E , (2.5) E = g q i I L L - L , i o Qi the momentum of the system, P, (where r a i s the pos i t ion of p a r t i c l e ' a ' ) , and the ang-u l a r momentum, M, (2.7) M = 2 ^ a x £ a > a 13 are a l l conserved (see Appendix C ) . For a system of two p a r t i c l e s i n t e r a c t i n g through a centra l po ten t ia l U t j r ^ - r ^ j ) , (2.8) L = h^iLi2 + hm2r22 - V(\rx-r2\). Let r zr^-r^ and define the o r i g i n of the coordinate system such that ni]_r-j_ + m2.r2 = 0. Then (2.9) r , _ m 2X ; r_ - ~mlL -1- mj_+m2 ^ m l + r a 2 and (2.10) L =-|-mf2 - U(r) , m s m l m 2 . m l - m 2 Thus the problem of the motion of two p a r t i c l e s i n t e r a c t i n g through a centra l force i s mathematically equivalent (by transforming to center of mass coordinates) to the motion of one p a r t i c l e i n a given centra l f i e l d U ( r ) . In c y l i n d r i c a l coordinates r , 0, z , equation (2.10) becomes (2.11) L = Jm ( f 2 + r 2 0 2 + z 2 ) - U ( r ) . Since M ( = r x p) i s conserved and perpendicular to r , the motion i s p lanar . Choose the z-axis such that z = 0. Thus, (2.12) L r hair2 + r 20 2) - U ( r ) . Since M i s the momentum conjugate to 0, see equation (2.6), (2.13) M = -|4 = mr20. D 0 Since dt ^ '0 d 0 (2.15) M = mr 2$ i s constant. From the d e f i n i t i o n of E , see equation (2.5), E = f + 0 "TT - lm( f 2 + r202) - U(r) , or d0 (2.16) E= | m r 2 + & I?- + U(r) , mr ^ = ( 2 [ E - U ( r ) l 4"? )* • dt v m L 'J m 2 r 2 ' A l s o , from equation (2.13), (2.17) d0 = M i . 2 ' mr and using the t h i r d of equations (2.16), one gets (2.18) d0 = ^ ? ( f [E-U(r)l - ~ ) " 4 d r ' . mr^ m L J nrr Thus, FIGURE 2.1 COULOMB SCATTERING 1 5 (2.19) 0 = \ M d r / r 2 j - + constant. J (2m [E-U(r)J-M2 / r 2)2 In a centra l f i e l d the path of a p a r t i c l e i s symmetric about r m ^ n . For an inverse square law force f i e l d the path i s a conic sect ion with the center of force as a focus, i n which case the symmetry i s c l e a r . From Figure 2.1, ( 2 . 2 0 ) % = | ? T - 2 0 O | . From equation (2.19), ,2 ( 2 . 2 D 0O = C M d r / r n _ i • J ( 2 m [ E - U ( r ) J - M 2 ^ 2 ) 2 Since E and M remain constant, one uses t h e i r i n i t i a l v a l -ues , E = imv 2 , , ( 2 . 2 2 ) M =7 m p V o o } where p i s the impact parameter. For Rutherford scat ter ing the force f i e l d i s Coulombic, ( 2 . 2 3 ) U ( r ) = * / r , oC = Z e 2 . Combining equations ( 2 . 2 1 ) to ( 2 . 2 3 ) one gets, 1 6 o o ••) dr (2.24) 0O = \ . ° ^ . l-(p 2/r2)_-r . . - v P 2 / r 2 M 2 < V m r v 2 ) 2 ' l m i r \ (2.25) A - cos" 1 (*/ m vooP) [ l + l ^ / m v o o 2 ? ) ] * Or, solving equation (2.25) for p, (2.26) p 2 _ < 2 t a n 2 0 A * 4 which becomes, using equation (2.20), (2.27) -P'zr-* c o t . m V 0 D Hence, applying the result of equation (1.4), {2*26) dfr ( * -)2 ^ . 2IHV002 Bin^(ilX'l) Equation (2.28) gives the cross-section in the frame of reference i n which the center of mass i s at rest. The transformation to the laboratory frame i s accomplished by the formula, (2.29) tan ex - m 2 s i n * , 9 2 = i ( ? T - ^ ) , m1+m2cos % where 0j_ and 0 2 are the angles between the directions of motion after the collision and the direction of impact. The subscript '2' denotes the particle which was originally 17 at rest in the laboratory. Using the fixed f i e l d approx-imation, i.e., m-]_«m2, then masm-^ , and %V-Q^. Thus equation (2.28) becomes (2.30) 4-2: =r(|f^) 2 where E-^  i s the incident kinetic energy. For small angle scattering one can replace sin(X) by X and hence the dif -ferential scattering cross-section in the fixed f i e l d ap-proximation, in the limit of small angle scattering i s (2.31) (^)Rutherford ~ ( E J ~ ^ * That i s , the factor A of equation (1.8) is (2.32) A=it | s 2 • 1 For Einstein scattering (see Section IV), (2.33) A = 4QM , where M is the solar mass. The following 'rule of thumb' is apparent. To go from electrodynamics to gravidynamics one must replace e . by GUco (in units such that " f i s c a l ) , or in the cgs-system of units e 2/fic by GM«^/c^, where c*> i s some characteristic frequency of the system. Finally, i t should be noted that the scattering of particles with intrinsic properties such as spin i s not 16-handled by the methods of t h i s sec t ion . The next sect ion i l l u s t r a t e s a f u l l quantum mechanical treatment of a scat-t e r i n g problem invo lv ing spin quant i t i e s . 19 § 1 1 1 . QUANTUM MECHANICAL RUTHERFORD SCATTERING The c l a s s i c a l theory of s ca t t er ing , as developed i n Sect ion I I , i s not capable of handling the sca t ter ing of p a r t i c l e s v/ith i n t r i n s i c propert ies such as sp in . In t h i s sec t ion , the scat ter ing of a Dirac e lectron by a given Coulomb f i e l d i s t reated , using the f ixed f i e l d (or ext-ernal f i e l d ) approximation and the general scat ter ing r e l a -t ions developed i n Section I , to i l l u s t r a t e how these i n t r i n s i c a t t r ibutes of p a r t i c l e s are handled. When t r e a t i n g a scat ter ing problem, one general ly i s concerned v/ith the i n t e r a c t i o n of two f i e l d s . The coupling of the systems i s achieved by postulat ing the existance of a coupling term i n the f i e l d equations which depends on the f i e l d variables of both f i e l d s . In some problems i t i s adequate to represent one f i e l d by a quantized f i e l d , i n -vo lv ing creat ion and destruct ion (absorption) operators, while the other f i e l d i s treated as a given c l a s s i c a l func-t i o n of the space-time coordinates. For the c l a s s i c a l l y given f i e l d we need no equation of motion, since i t i s assumed to be the given space-time funct ion; t h i s i s a great s i m p l i f i c a t i o n . Under cer ta in circumstances the sca t ter ing of an e lec-tron by the f i e l d of a nucleus can be treated as the motion of a Dirac e lectron i n a g iven, f ixed Coulomb f i e l d . The condit ions necessary to allow the concept of a f ixed f i e l d to be v a l i d can most e a s i l y be seen when one r e c a l l s the results-of- c la s s i ca l - two-body e l a s t i c s c a t t e r i n g . As i n 20 Section II, (see equations (2.9), (2;29), and the para-graph following (2.29)), the i n i t i a l l y stationary mass .mg remains essentially stationary (i.e., there i s a negligible momentum transfer to m2) i f m 2 » m 2 . Equally, the condition that the reduced mass, m = m^ m2/(m-^ -m2), (center of mass fixed) is approximately ra-^ i s that m2>>m-^ . Hence, the condition for the validity of the fixed f i e l d approximation i s m 2 » m 2 , when the fixed center of mass i s nearly coin-cident with the; position of m2. In the r e l a t i v i s t i c case the condition of a relatively large target mass can often not be met at the same time as the r e l a t i v i s t i c limit cond-it i o n k^>^ m for the incident particle. In that case one must apply the condition of negligible momentum transfer to the target. This condition is met i f the angle of scattering i s small; and hence a small change of momentum occurs. In a l l the cases that shall be discussed, this limitation w i l l not be serious since in a l l cases the scattering is very strong in the forward direction , that i s , d c r/d_a^ A/ The approximation in which the f i e l d of the nucleus i s treated as a classically given Coulomb f i e l d function rather than a quantized one (involving creation and dest-ruction operators) can be obtained from a completely quan-tized theory by identifying the expectation values of the f i e l d and current operators with the classical quantities (see Jauch-and Rohrlich, chapter lft, 1955). The approxima-tion i s accurate when the fluctuations about the expectation 21 values are small compared to the expectation value i t s e l f . The term 'Coulomb sca t t er ing ' general ly refers to the scat-t e r i n g of an electron by a given, f ixed Coulomb f i e l d , to a l l orders of th i s f i e l d ( in t h i s Section only the f i r s t order i s ca l cu la ted) . The effect of the presence of photons ( r e a l or v i r t u a l ) i s general ly re ferred to as ' r a d i a t i v e correc t ion ' (corresponding to the effect of the s o - c a l l e d r a d i a t i o n f i e l d i n c l a s s i c a l electrodynamics) and i s here completely neglected. The e lectron i s described by a plane wave so lut ion to the Dirac equation (3.1) for a free e l ec t ron . (3.1) ( 7>/h^ - m) Y ( x ) =0, where ^ i s i m p l i e d . Th onent Lorentz spinor e wave funct ion 'Y'(x) i s a 4-comp the (£*.= 0,1,2,3) are J+x4 matrices , operating on the , obeying spinor (3-3) 22 t h e r e s t mass o f t h e p a r t i c l e i s d e n o t e d b y m, a n d t h e s p i n i s r e p r e s e n t e d b y t h e o p e r a t o r s ( s e e A . S . D a v i d o v , 1 9 6 6 ) ( 3 . 4 ) ^ ' z = - i ^ x Y ' y ( c y c l i c ) . D e s c r i b i n g t h e e l e c t r o n a s a q u a n t i z e d f i e l d , o n e c a n w r i t e x ) a n d / ( x ) , a s t h e F o u r i e r s e r i e s ( s e e F . M a n d l , 1 9 5 9 ) , ' V U ) ^ ^ ( u ( k , r ) a ( k , r ) e + i k x 4 u * ( k , r ) b t ( k , r ) e " i k x ) , ( 3 . 5 ) V ( x ) = ±2 ( u 1 k , r ) a f ( k , r ) e - i k x + u ( k , r ) b ( k , r ) e + i k x ) , •jfv k , r v / i t h x ) n o r m a l i z e d t o o n e p a r t i c l e p e r v o l u m e V . T h e o p e r a t o r s a ( k , r ) a n d b ( k , r ) , a s s o c i a t e d w i t h t h e p o s i t i v e f r e q u e n c y p a r t , a l l o w i n t e r p r e t a t i o n a s a b s o r p t i o n ( d e s t -f t r u c t i o n ) o p e r a t o r s , w h i l e t h e o p e r a t o r s a ( k , r ) a n d b ( k , r ) , a s s o c i a t e d w i t h t h e n e g a t i v e f r e q u e n c y p a r t , become c r e a -t i o n o p e r a t o r s . The f o l l o w i n g l emmas c o n c e r n i n g t h e a l g e b r a o f t h e *f-m a t r i c e s a r e u s e f u l . Lemma 1 : T r a c e Proof: T r tf« f t f ^ r f p f f f , 1^^777' u s i n g t h e p r o p e r t y + ^ = 0 * 23 Lemma 2: Trace K*"^* Y ^ = 4 • Proof: Tr =Tr i f ° 0 ^ , Lemma 3: Trace n W r 4 ( 6 ^ ^ - g ^ g ° r ) . Proof: Using f° + 2 g<^, Trace ^ 1 ^ * * * * = 2 g ^ r T r - T r t r K t* = fig^Vf -«'g^grf + T r X^T^tf , = 8 g ^ g ^ f -3gC^g<Tf 4 g g f P g ^ -<£rWV X^-For purposes of t h i s problem, t h i s completes the quantum mechanical d e s c r i p t i o n of the f r e e e l e c t r o n . The f i e l d of the nucleus i s described as a c l a s s i c a l f i e l d , A ^ ( x ) . Use i s made of the F o u r i e r s e r i e s ( 3 . 6 ) A^(k.-k f) = ^ d 3 x A ^ ( x ) e l l ^ i - ^ f ) ^ . I n a f u l l quantum mechanical treatment, t h i s plane wave decomposition (or i t s inverse) would be w r i t t e n as the;, sum of p o s i t i v e and negative frequency parts ( i n c l o s e analogue to equations (3*5)) and the c o e f f e c i e n t s A^(k) and A^ (k) would become c r e a t i o n and d e s t r u c t i o n operators. I n the e x t e r n a l f i e l d approximation, however, the A^(k)'s are so-c a l l e d c-numbers, not operators. The Dirac equation f o r a f r e e p a r t i c l e (3.1) and i t s a d j o i n t equation can be derived from the Lagrangian d e n s i t y (3-7> ^ D l r a c = [t° <^ 7 ^ ^ " «] Tu>, 24 by application of the Euler (Lagrange) equations which are derived in analogue to equations (2.4), treating %{x) and ^ ( x ) , («=0,1,2,3) as independent variables. In classical electrodynamics one can derive the Lag-rangian for a point charge interacting with ,a fixed elec-tromagnetic f i e l d from the Lagrangian of the free point charge by substituting p^-eA^ for p^ in the free Lagrangian. The Lagrangian describing the Dirac electron in a fixed electromagnetic f i e l d A^ i s similarily obtained by sub-stituting i D/'i xf* - eA^ for i 2/9 x*4 in equation (3.7). The electromagnetic interaction term in the Lagrangian density i s then (3.9) Xint = l. ?(x) [t°ff *J t(x) . If one expands the Lagrangian density (3.9) according to equations (3*5) one gets two types of terms. Type one w i l l contain one each of a creation and an absorption op-erator, these terms shall temporarily be retained since 25 they describe events in which there is no change of part-i c l e number; type two w i l l contain two operators of the same kind (i.e., both creation or both destruction). Terms of type two describe processes in which pair crea-tion or pair annihilation take place; since these are processes not being considered in the present elastic elec-tron scattering problem, such terms shall be neglected. Of the two terms of type one, only the one term involving electron creation and electron absorption (a^ and a) i s retained; the term involving positron operators i s of no interest. Thus the interaction Lagrangian density for elastic electron scattering can be written (3.10) £ i n t = i e 1 S ^ V . - i k ^ j f o y ^ , ^ e * i k » x , # k » J r " Thus, using result (1.10), (see also Appendix A), the amplitude A ^ for scattering from jk.^  ,s.^=| to ^ k f > sf|=^ff| i s given by ( 3.11) A f l = < } f J j ^ i n t d 4 * | J i > in the f i r s t Born approximation. (3.12) A f i = - i p ( d x ^ f u 1 k « , r ' ) e - i k ' x A / A(x )u (k» , r » ) J j r „ ; r „ r e ^ i k " ^ 0 , a t ( k f , r f ) a l ( k S ^ M a ( k ^ , r M a ( k i , r i ) $ 0 > , •26 (3.13) A f i = ^ u U i , r i ) ) l 0 ) f c / 4 A ^ ( k i - k f ) u ( k f , r f ) 2 ? r £(0±-cJf) , A^(iSi-k f) r ^ d 3 x A^(x) e i ( ^ i ^ f } ^ . Using r e l a t i o n s (1.11) and (1.12), 2 (3.14) w f i= 2 * e ( U f Y ^ U i V ^ U f ) A ^ ^ - k f ) using the f a c t that the A^ are r e a l . Using equation (1 . 2 3 ) , the d i f f e r e n t i a l scattering cross-section for electrons scattered e l a s t i c a l l y by a f i x e d electromagnetic f i e l d i s (3.15) 4-~r - - ^ [B A ^ k i - k , ) A ^ - k , ) ] ^ d s l " Un)2 L T 1 f * 1 ~ f J ^ i - ' where B = ( X X ^ ^ ^ M ^ ^ ^ u f ) . I f the incident beam of electrons i s unpolarized, one must average over the two possible spin states s^=l,2. I f the detector does not distinquifih. between spin states, one must as well sum over the f i n a l spin states s^=l,2. Thus, B of equation (3«15) becomes B, B. averaged and sum-med, I (3.16) s^"f-i u ( k f , s f ) ^ u ( l c i , s i ) u ( k i , a i ) ^ u ( k f , s f ) 27 But using (3.17) ^ u^(k,r) y(k,r) = ^ ( Y ^ t , where - i - O ^ k ^ m)= A i s the so-called energy projec-tion operator, ( 3.18) B = 1 21 T f f t k i ( t. • m l V * ML • m l , (3-19) B = _ 1 _ Tr Specializing to the case of a fixed, static, Coulomb f i e l d given by (3.20) AQ.(x) = -2a. , A , = 0 , (k = l,2,3), (xl i n the laboratory frame, or (3.21) A ^ - k f ) = . f r * g e . 2 ? T Z e " o i l . j k j - k ^ k 2 ( 1 _ c o s e ) 2 2 2 2 (using k*k = k cosB, C*J = k +m ), equation (3.15) becomes, - .2/) 2 _ (3. 2 2). U L s £ * l L * (2 7 T ) 2 Z 2 e 2 d l l ( 2 7 r ) 2 k 2(l-cos6) 2 ^ i ~ Using lemmas 1-3 above, equation (3.19) becomes (3.23) B =• ( 2 ^ i 2 - kikf + m2) , 28 (3 .24) B = r - i - Q (m 2 + 4" (1+cose)), ( 3 . 2 5 ) B = - ^ 2 ( 1 + ^ = o s 2 | ) , using k^k^ = - k cos0. Thus, (3 . 2 6 ) A £ Z A J ( l + ^ c o s 2 | ) . d/2- V 2 k 2 s i n 2 | / m 2 ? I f equations (3*5) had been normalized to u>jm par t -i c l e s per volume V, i . e . , mul t ip ly (3*5) by an a d d i t i o n a l f a c t o r -fa/m', one would have had to use equation (1 . 2 6 ) ins tead of ( 1 . 2 3 ) , to y i e l d i n place of (3.15) the fol lowing ( 3 - 2 7 ) R ( 2 ^ r T 2 L B V A-3 " • Instead of equation (3*17) one would have 2 (3 .28) ^ u - ( k . r ) u , ( k , r ) = L m). r=l r 2m Equations (3.18), (3.19) , and ( 3 . 2 3 ) would contain the fac tor l / 2 m 2 rather than the factor l / 2 * J 2 . Beyond equation (3.25) the two normalizat ions lead to i d e n t i c a l expressions. 2 2 For the n o n - r e l a t i v i s t i c l i m i t , k « m , one can write 2 m for cd and mv for k and hence, using E ^ E k / 2 m , 29 the usual c l a s s i c a l scat ter ing cross-sect ion i n the f ixed f i e l d approximation; see also equation ( 2 . 2 6 ) . For small s ca t ter ing angle, the r e s u l t in* the n o n - r e l a t i v i s t i c l i m i t i s i d e n t i c a l to the r e s u l t ( 2 . 3 1 ) . The r e l a t i v i s t i c l i m i t may also be of i n t e r e s t since i n the case of photons being scattered t h i s i s the only 2 2 2 2 case. I f k >^ m , tO i s near ly equal to k and Due to the f i xed f i e l d approximation one must l i m i t the a p p l i c a b i l i t y of ( 3 * 3 0 ) . C l e a r l y , the condit ion that the mass (not rest-mass) of the inc ident p a r t i c l e be much smaller than the rest-mass of the target nucleus i s not l i k e l y to be compatible with the r e l a t i v i s t i c l i m i t condi-2 2 t i o n , k » m , for the inc ident p a r t i c l e . The condit ion that n e g l i g i b l e momentum be transferred to the target nucleus must then be appl i ed , to y i e l d the r e s u l t 6^<1, i . e . , small angle s ca t t er ing implies small change i n mom-entum. Equation ( 3 . 3 0 ) becomes i n th i s approximation ( 3 . 3 D d_£_ _ / 2 Z e i \ 2 1_ , e^<l. djTL ~ [ oO J Q4 30 §IY. THE CLASSICAL EINSTEIN EFFECT - THE GRAVITATIONAL DEFLECTION OF LIGHT According to the general theory of r e l a t i v i t y , the t r a j e c t o r y of a p a r t i c l e i s governed by the equations for a geodesic (see Appendix C ) , = The metric i s determined by the f i e l d equations (4 .2) = -%KT g ^ , where the cosmological constant has been taken to be zero. A so lu t ion of equation (4 .2) for the metric (or equivalent-l y , the i n t e r v a l ) i n empty space ( i . e . , = 0) surround-ing a g r a v i t a t i n g point p a r t i c l e , the Schwarzschild metric , s h a l l be used to describe the g r a v i t a t i o n a l effect of the sun. This i n t e r v a l i s s t a t i c , s p h e r i c a l l y symmetric i n space, and invar iant under time r e v e r s a l , and i n spher i ca l polar coordinates and time, i t i s wri t ten (4-3) d s 2 = ( l - 2 m / r ) d t 2 - ( ^ m / r ) " r 2 d e 2 ~ r ^ 1 * 2 ® d02. To solve equations (4 .1) with the condit ion (4.3)> one needs the values of the 3-index symbols. A l l are zero except the fo l lowing: 31 f l 3 , 3 f = l / r , f M = l E 8 $ ! > ( 4 . 4 ) [ 2 2 , l | = - r ( l - 2 m / r ) , [23,3} = cote, [ 4 4 , l f = r - ( l - 2 m / r ) ( m / r 2 ) , [ 3 3 , i | = - r s i n 2 e (l - 2 m/r), ^ 3 3 , 2 } - - s i n e cose, - l">?>°( \ • It i s convenient to introduce new variables, ( 4 . 5 ) Rsr/2m, and T=st/2m, where ( 4 - 6 ) r s = 2 m i s the Schwarzschild radius. For the case of the sun r g (sun)^ 3 x 10^ meters (equivalent to about 4 x lCpO kgms.). The surface of the sun i s approximately at a distance p = 2 . 3 x 1C>5 units from the origin. The charact-e r i s t i c time (the time taken by light to go r s meters) i s approximately 10"^ seconds. The time taken by a photon to travel the distance equal to the diameter of the sun i s several seconds (cf. TftJlO"^ seconds). These two charact-er i s t i c parameters determine the magnitude of the scatter-ing effect of a photon passing the sun near the surface of the sun. With the new variables ( 4 . 5 ) , equation ( 4 . 3 ) becomes, (4 . 7 ) d r 2 = (l-l/R)dT 2 - _ R 2 (d6 2 + sin 26 d0 2). 32 The trajectory of a photon in any plane through the origin which, because of spherical symmetry may be taken i n i t i a l l y at 0 = w / 2 , i s the null geodesic. Equation ( 4 . 7 ) gives where d9 = 0 due to the 0-component of equations (4.1), i.e., giving de/dssO i f 9 = T/ 2 and d0/ds=iO i n i t i a l l y . The angular momentum integral results from the 0-component of equations (4.1), yielding (see Appendix D, equation (D.10)), (4.10) R 2 0 = J, where the "dot" represents derivation with respect to X . The energy integral i s a result of the t-component of equations (4.1), yielding (see Appendix D, equation (D.15)), (4.8) (1-l/R) dT 2 - dR2 - R 2 d02 = 0, 1-l/R ( 4 . 9 ) (4.11) T (1-l/R) = K. It should be noted that for a photon J and K are i n f i n i t e , however, their ratio i s f i n i t e and that i s a l l that i s required in this problem. FIGURE 4.1 ORBIT OF A PHOTON IN A PLANE THROUGH THE ORIGIN OF THE SCHWARZSCHILD OBJECT 33 S u b s t i t u t i o n of equations (4.10) and (4-H) i n t o equation (4.8) g i v e s , (4.12) R2 = K 2 - J 2 Ifizii . R3 Using a new v a r i a b l e , ^ = l/R, and d i v i d i n g by J 2 = (0/^ >2)2 equation (4.12) becomes where p=J/K. The cubic term i s c h a r a c t e r i s t i c of E i n -s t e i n ' s theory as compared t o the Newtonian theory.. I f t h i s term i s e n t i r e l y neglected, one obtains f o r the t r a j -e c t o r y which begins f o r J = 0 ( i . e . , R = co) at 0=0, (4.14) 0=^ ( l - P 2 ^ 2 ) " 4 P - a r c s i n ( p j ) , or (4.15) R s i n 0 = p. This i s the equation of a s t r a i g h t l i n e w i t h impact para-meter p (see Figure 4.1). The exact s o l u t i o n of equation (4.13) i s (4.16) 0 = J ( 1 - P 2 £ 2 (1- y ) ) ~ 2 p d j . '0 L e t t i n g 34 ( 4 . 1 7 ) u = P y d - y ) * , and since p » l for the surface of the sun, expanding in powers of p-1 gives (4.18) y p = u (1 + u/2p +...)., (4.19) p d j = (1 + u/p + ...) du , and hence equation (4.16) becomes S-u (1 + u/p •» ...) (1 - u 2)"* du , o (4.21) 0, = arcsin u - (l/p) ((1-u 2)^ - 1) + ... . For u = l , one has du/d0 = O and thus also dy/d0=O, and this corresponds to the point of closest approach (see Figure 4.1). (4.22) 0(u = l) = + l/p, corresponding to a deflection of the trajectory in the amount l/p up to this point. By symmetry, the same amount of deflection w i l l be engendered. al ong the second half of the trajectory as i t proceeds from 0 (u = l) to 0 — ft + 2/'p. Hence the total deflection i s \ (4.23) % = 2/p. In more usual units ( 4 . 2 4 ) X - 4MG/p , 35 where M i s the mass of the sun, G i s the g r a v i t a t i o n a l constant, and p i s the impact parameter. Numerical ly , for a photon t r a v e l l i n g just past the surface of the sun 1 .74 seconds of an a r c . The a t -tempts at experimental v e r i f i c a t i o n of the E i n s t e i n effect and the d i f f i c u l t i e s that the detection poses are reviewed by A. A. Mikhai lov , 1959-Comparison of equation (4.24,) with equations ( 1 . 5 ) and (1.6) y i e lds ( 4 . 2 5 ) ^ = i L G M i 2 # d_n See also equations ( 2 . 3 1 ) , ( 2 . 3 2 ) , and ( 2 . 3 3 ) for a comp-ar ison of the small angle scat ter ing i n the cases of C o u l -ombic and g r a v i t a t i o n a l s ca t t erers . The geometric method employing the notion of geodesic does not consider i n i t s formulation the concept of po lar -i z a t i o n of photons. The pred ic t ion made i s thus independ-ent of any p o l a r i z a t i o n of the photons. The quantum f i e l d method used i n the next sect ion again arr ives at the re su l t ( 4 . 2 5 ) f or the unpolarized case, however, the p o l a r i z a t i o n effects are calcuable as w e l l . 36 § V . THE QUANTUM MECHANICAL EINSTEIN EFFECT The c l a s s i c a l E i n s t e i n e f f e c t , as discussed i n Sect-i o n I V , does not concern i t s e l f w i t h the quantum mechanical n o t i o n of the s p i n of a photon. P o l a r i z a t i o n e f f e c t s are. thus not e a s i l y handled by t h i s method. In analogue to S e c t i o n I I I , t h i s s e c t i o n concerns i t s e l f w i t h the s c a t -t e r i n g of a photon by a given, weak g r a v i t a t i o n a l f i e l d . A s o l u t i o n i s obtained u s i n g the f i x e d f i e l d approximation, the bounds on the v a l i d i t y of which are as s p e c i f i e d i n Se c t i o n I I I . The photon i s described by a plane wave s o l u t i o n to Maxwell's vacuum f i e l d equations, which are assumed to hold i n every i n e r t i a l frame, (5.1) - ^ = 0, ^ k where (5.2) F i k = i l i - 2 ^ , ^ k 2 y ± and where A ^ ( x ) , (1=0,1,2,3) i s the vector p o t e n t i a l . The plane wave s o l u t i o n s A^(x) can be F o u r i e r analyzed (see Mandl, F., 1959, Chapter 9) i n the usual way to des-c r i b e the quantized electromagnetic f i e l d i n terms of photons. FIGUREj 5 . 1 i GEOMETRICAL IDENTIFICATION OF INCIDENT AND SCATTERED PHOTON 37 5.3) A ±(x) = 4 , 2 5 ? - ^ - L ( k , s ) b ( k | 8 ) e - l k x + e £ ( k , s ) b f ( k , s ) e + i k x ] , with the normalizat ion being one p a r t i c l e per volume V . The operators b(k,s) and b^"(k,s) are interpreted as ab-sorpt ion and creat ion operators of photons of momentum k and i n spin state s; and the p o l a r i z a t i o n vec tors - for the photons are denoted by e^(k,s) , or more simply by . By introducing the uni t propogation vectors n^, (5.4) n i e \l*> , and by combining equations (5.2) axid (5.3), 5.5) F..U) r r ^ g S l / f 7  1 J <flpk « ' 2 (e*n.-e*n ) b f ( k , s ) e + i k x J J Jj The inc ident and scattered photons s h a l l be i d e n t i f i e d as fol lows (see Figure 5.1): n i s the d i r e c t i o n of the inc ident photon, n' i s the d i r e c t i o n of the scattered photon, (5.6) 0 i s the sca t ter ing angle, ( n « n ' ) — c o s © , 0 i s the so lar longitude. Then, with the convention of Figure 5.1, 38 / i \ / cose (5.7) n = 0 ) ; n' = I sine cosjZu . \0f \sine Since the photons are transverse one can choose (see Jauch, J'.M., and Rohrlich, F., 1955) ( 5 . 8 ) e Q = 0 . Denote by /ei(n,s) (5-9) e(n,s) =: I e 2(n,s) \ e3(ii,s) the polarization vector of a photon of propogation direc-tion n and polarization state s. If s = 1 means 'right hand, circular polarized' and s =• 2 means 'left hand, cir-cular polarized', then one has the standard result (see, for example, Freeman, M..J., 1967) (5.10) e(n,l)= e (n ,2) r_, 1 f-ngxio-ini ^ 2 ( l - n 3 2 ) ' ^ \ln/ Alternatively, i f s = l , 2 means the following linear polar-izations, the notation w i l l be (5.11) €(n,l) = J?r<g(l) t e(2)| = .. 1 », | -non L J f l ^ I (5.12) 6( 39 According.: po equations (5.7), (5.13) § ( n , 2 ) that i s , with n p a r a l l e l to the x^-axis , £(1) means po lar -i z a t i o n p a r a l l e l to the x^-axis and €.(2) means p o l a r i z a t i o n p a r a l l e l to the X2-ax i s . This completes the descr ipt ion of the photon i n so f a r as th i s problem i s concerned. The g r a v i t a t i o n a l f i e l d i s described c l a s s i c a l l y by means of the metric tensor g*/* . In the weak f i e l d approx-imat ion, which w i l l be used throughout, (5.14) rtsW? , ^- , with (5.15) Y * < * = ) V * \f«(> | <«f I . I f one spec ia l i zes to the case of a s t a t i c g r a v i t a t i o n a l f i e l d and performs a Four ier decornpostion, one gets (5.16) f (x) : f l i l r ^ f l , ] e - ^ £ « , where G 1 ^ i s the p o l a r i z a t i o n tensor and f(q) i s a func-t i o n ( i . e . , a c-number) not an operator, s ince the external f i e l d approximation-is being used. 40 In order to be able to use the formalism developed in Section I, an interaction Lagrangian for the coupling of the photon f i e l d to the gravitational f i e l d i s needed. A possible derivation of the total Lagrangian results from the so-called compensating f i e l d method (see Kaempf-fer, F.A., 1965)• Maxwell's vacuum f i e l d equations (5.1) can be derived from the action principle, Using the method of the compensating f i e l d , the Lagrangian density in general space-time coordinates (not necessarily Minkowski) can be written with (5.IB) L r= - £ F F ik (5.19) (5 .20) £ s = - * h F*/*F, where (5.21) h s det ( h 1 ^ ), where 41 (5.22) " dy^ = h k ^ dx*" , where the y k are the l o c a l i n e r t i a l coordinates and the x * are the coordinates i n the underlying continuum. (For a more complete descr ipt ion of the ' v i e r b e i n ' formalism of general r e l a t i v i t y see Kaempffer, F . A . , 1968.) In the weak f i e l d approximation, see equations (5*14) and (5.15), (5.23) h = 1 - & ) f V . Thus equation (5.19) becomes. (5.24) Zs = -h F % l k + J / l k ( S r S P i r F k s - i S i k F r S F r s ) . Subtract ing the term (equation (5.18)) leading to the vacuum f i e l d equations i n a Minkowski space, the term due to the g r a v i t a t i n g source i s (5.25) / l n t = hf* ( $ » F l r F k s - 4 Sik F r S F r s ) , (5.26) ^ l n t = i S r S ( Y 1 J - t V - ; „ S t J ) F i r F j s . For ease of nota t ion , k s h a l l represent both var iab le s , the momentum k and the p o l a r i z a t i o n state s. I f one i s in teres ted i n the scat ter ing of a photon from an i n i t i a l state | k ^ into a d i f f erent ( f ina l ) state ^k T ( through the effect of V * * ^ ' then i n the i n t e r a c t i o n Lagrangian 4 2 density only terms which contain one each of creat ion and destruct ion operators need be reta ined; the terms deal ing v/ith the creat ion and destruct ion of photon pairs can be ignored. As i n equation ( 1 . 1 0 ) , the amplitude for t h i s sca t ter ing i s given by and hence, using the normalizat ion of one p a r t i c l e per volume V, equation ( 1 . 2 3 ) appl ies and f o r a f i x e d , time-independent f i e l d . Using equation ( 5 . 2 6 ) with the Four ier decompositions ( 5 . 1 6 ) and ( 5 . 5 ) , equation ( 5 . 2 ? ) becomes, fM f , as i n equation ( 1 . 1 0 ) , i s defined by ( 5 . 2 9 ) e i ( A ) ' - ^ ) t d q d 4 x f where the fac tor A contains a l l the relevant p o l a r i z a t i o n tensors , 43 (5.3D A =riS r s ( e 1 J - i e m ^ r m 5 1 J ) ( e i n r - e r n i ) ( e [ n ' - e f n ^ Integrating equation (5.30) over dSc yields the delta-functions, UTT )^ S"(k' -q-k) S(<j'-*p), and allows the im-mediate integration over dq, so that, (5.32) <k ' )M|k> = ^Z-fiJtT1 f(k'-ki) • Thus, equation (5.28) becomes The Schwarzschild metric, which (as in the last sec-tion) shall be used to represent the effect of the sun, (see equation (4 . 3 ) ) , can be written in isotropic coordinates, r = (l-nn/2?)2 r = (x 2+y 2+z 2)^, x = r sinG cos0, (5.34) y r r sine sin0, z = r cose, in the form (5.35) ds 2 = (l-m/2r) 2 d t 2 _ ( i + m / 2 r ) 4 (dx 2+dy 2+dz 2), (l+m/2r)2 or, in the weak f i e l d approximation, as (5.36) ds 2 = (l-2m/r) d t 2 - (l+2m/r) (dx 2+dy 2+dz 2), 44-with m S M G. In the weak f i e l d approximation M .-2m/r 0 0 0 i n i 0 -l -2m/r 0 0 (5.37) g^ = \ 0 0 -l -2m/r 0 0 0 0 -l -2m/r Hence, wr i t ing Y 1 ^ ( x ) as f ( r ) £ 1 J for a s ta t ic , , c e n t r a l , weak, Schwarzschild f i e l d (5.38) f (r) =: 2GM/r. Or, (5 .39) f (q) 8 7TGM if At resonance, 4 7 T G M (5.40) f ( k ' - k ) ) -*tO=cof • < d 2 ( l - n . n ' ) Combining equations ( 5 . 4 ) , ( 5 . 8 ) , and (5.13) (5.41) | A | 2 = | ( e . e ' * ) ( l + n . n « ) - (e .n')(e»*.n)j 2 Using equation ( 5 . 6 ) , 45 (5.42) | A | 2 = |(e.e'*)(l+cos6) - (e.n')(e'*.n)| 2 Thus, equation (5.33) becomes (5.43) G2 M 2 (i-cose) 2 (e.e'*)2(l+cose)-(e.n') ( e f * .n)| 2, This i s the d i f f e r e n t i a l c r o s s - s e c t i o n f o r s c a t t e r i n g of photons i n t o the s o l i d angle d£l at the s c a t t e r i n g angle 9 w i t h f i x e d i n i t i a l and f i n a l p o l a r i z a t i o n s e and e'. I f the i n c i d e n t beam of. photons i s u n p o l a r i z e d , one must average over the two p o s s i b l e p o l a r i z a t i o n s t a t e s s r l , 2 . I f the detector does not d i s t i n q u i s h between p o l a r i z a t i o n s t a t e s , one must as w e l l sum over the f i n a l s p i n s t a t e s s' = l , 2 . Thus, |A } 2 of equation (5.42) becomes JAJ 2, | A| 2 averaged and summed, (5.44) | A | 2 - ^ 0 ± | A | 2 , (5.45) | A | 2 = s€{ (e.e'*)(e*.e')(H-cos9) 2 - 2(e.e'*) (e.n») ( e'*.n) (l+cos6)+( e-n') (e'*.n) (e*'n') ( e T .n). Because of the s p h e r i c a l symmetry, of this>;interaction one can take 0=0 f o r t h i s c a l c u l a t i o n aaad one e a s i l y obtains, using equations (5*7) and (5.10), (5.46) -2? ^ > ( e . e 1 * ) ( e - e 1 ) r 1 + cos 20, s =1 s -1 — — — ~ ( 5 . 4 7 ) s=l s ' - l 2 2 ( 5 . 4 8 ) s-1 s '=l : £ ^ ( e -n t ) ( e '^ .n ) (e* -n t ) ( e ' -n ) - ( l - cos 2 e ) 2 . F i n a l l y , ( 5 . 4 9 ) | A | 2 = ( 1 + C O S 0 ) 2 , independent of 0, as must be because of spher i ca l symmetry. Thus, equation ( 5 * 4 3 ) becomes which i s the same as equation ( 4 . 2 5 ) ca lculated by the c l a s s i c a l method i n Section IV. Rather than considering an unpolarized beam and a detector which i s unable to d i s t i n q u i s h between the po lar -i z a t i o n s tates , suppose the i n i t i a l beam of photons i s l i n e a r l y po lar ized i n the equatoria l plane of the sun. Suppose also that only scattered photons po lar ized i n the same d i r e c t i o n are detected. Then G 2 M 2 cot A ( 9 / 2 ) . Thus, for small s ca t ter ing angles 0 ^ < 1 , 47 (5.53) e - €(n,2) = ( 1 j , and (5.54) e' =r €(n',2) =r L V l - n 3 » 2 I t follows that '-V 0 (5-55) e - e ' = (5.56) e - n ' - n 2 » - , (5.57) e ' - n = -n 2 « V l - n . ' 2 ' 3 Equation (5.42) then becomes (5.58) A = . 1 . Tn-,'(1+cose) + n » 2 , V w 2 ^ l l 2 J (5-59) A - _ i — - (cose(l+cos9)+sin 2e cos20) , ( l - s i n 2 e s i n ^ 0 ) 2 (5.60) A = 1 — (cose + cos 20 + cos 26 sin 2 0 ) , ( l - s i n 2 e s i n 2 0 ) * using simple trigonometric i d e n t i t i e s , and hence (5.61) I A ) 2 - ( l t c o s e ) 2 ( l . s l n 2 0 + c o s e - s l n 2 0 ) 2 . 1 (l-sin 2esin ' I 0 ) This re s u l t d i f f e r s from,the usual E i n s t e i n result by the FIGURE 5 . 2 DEVIATION FROM EINSTEIN'S RESULT -INCIDENT AND DETECTED POLARIZATIONS PARALLEL 48 f a c t o r (5.62) D r [ l ~ s i n 2 0 (1-cosQ)] 2 ( l - s i n 2 8 s i n 2 0 ) which, f o r small s c a t t e r i n g angle, 6j becomes approximately l - ( l - | i | i ) s i n 2 0 2 (5.63) D £ 2 24  l - ( e _ ^ ) 2 s i n 2 0 Thus, 4 (5.64) D £ 1 - -|- s i n 2 0 cos 20, where a l l terms of order 0^ " are r e t a i n e d . This d e v i a t i o n i s p l o t t e d as Figure 5.2. I f the i n i t i a l beam i s p o l a r i z e d ( l i n e a r l y p a r a l l e l to the po l a r a x i s of the sun, and the detector i s e f f e c t i v e only f o r photons p o l a r i z e d i n the e q u a t o r i a l plane of the sun, then '0 (5.65) e r £(n,l ) z [ 0 ) , and (5.66) e' - j(n«.2) = , 1 „, f n f ' I t f o l l o w s that (5.67) (e-e') r 0 , FIGURE 5.3 1 D DEVIATION FROM EINSTEIN'S RESULT -INCIDENT AND DETECTED POLARIZATIONS PERPENDICULAR 49 ( 5 . 6 8 ) (e-n1) rr ny , (5.69) (e»-n) - -Equation (5*42) thus becomes I n . ' (5.70). A l3 and (5.71) |A|2 r (1+cose) 2 (1-cosQ) 2 s i n 2 0 cos 20 l - s i n 2 0 s i n 2 0 which, for small scattering angle, 0, becomes approximately (5.72) |A|2 = ( 1 4 - C O S & ) 04 . Irl 2 , £- s i n 0 cos 0 4 The deviation from the Einstein r e s u l t i s plotted as Fig-ure 5«3« I f one takes (5.73) e r £(n,l)rf 0 and (5.74) e' r G(n« ,1) = — i s\ l - n 3 1 "7 n l ' V r 3' •n2» n;' l - n ~ ! then for small angle scattering 50 (5.75) |A|2 ~ ( l 4cos9 ) 2 (1- i £ s i n 20 c o s 20). The deviat ion from E i n s t e i n ' s re su l t has the same graphic behavior as i n Figure 5.2. That i s , i f the i n i t i a l beam of photons i s l i n e a r l y po lar ized and only photons which are l i n e a r l y po lar ized i n the same d i r e c t i o n are detected, then the cross -sect ion i s independent of t h i s d i r e c t i o n . I f one has (5.76) e - e(n,2) and (5.78) |A| 2 ^ ( l f c o s e ) 2 ( S i n 2 0 c o s 20). This dev iat ion from E i n s t e i n ' s r e s u l t i s as p lot ted i n F i g -ure 5.3. That i s , i f the i n i t i a l p o l a r i z a t i o n and the detected p o l a r i z a t i o n are perpendicular , then the r e s u l t i s independent of the i n i t i a l d i r e c t i o n of p o l a r i z a t i o n . The p o l a r i z a t i o n ef fects ca lcu la ted above, simply express the geometric fac t that as the o r b i t of the po lar -i z e d photon i s bent through the influence of the sun, the d i r e c t i o n of p o l a r i z a t i o n ( in general) changes, s ince the (5.77) e' = € ( n ' , l ) - 1 , A l l -n'3 2 then for small angle sca t ter ing 51 photon must at a l l times remain transverse . The maximum deviat ion from the c l a s i c E i n s t e i n effect occurs for 0 - K/k, i n which case, Dcz l-eVl6. For 8 & 1.74 seconds of arc (as i n the case of a photon t r a v e l -i > -1$ l i n g just past the surface of the sun), Or/16 2 x 10~ I t i s c lear that these p o l a r i z a t i o n effects are very small when compared to the scat ter ing of uhpolarized photons, which i s i t s e l f present ly just wi th in the range of detec t ion . 52 VI- CONCLUSION That the p o l a r i z a t i o n effect ca lculated i n Section V i s not i n observable range i s not to say that i t i s use-f u l to invoke the method i l l u s t r a t e d only to obtain resu l t s of u l t r a - f i n e d e t a i l of the already small r e l a t i v i s t i c e f f e c t s . Without leaving the problem of photons scattered by the sun, i t i s poss ible to suggest several other per t -inent problems. The effect of the non-spherica l nature of the sun should be ca lcu lable using the out l ined method. In the weak f i e l d approximation (see Eddington, A . S . , 1924)> neglect ing a l l cross-coupl ing terms the metric due to a mass d i s t r i b u t i o n fol lows d i r e c t l y from the usual Newtonian p o t e n t i a l of the mass d i s t r i b u t i o n . In the l i g h t of Dicke's concern with the so lar oblateness (see Dicke, R.H. '^ , | 1 9 6 7 ) , th i s c a l c u l a t i o n should be carr i ed out with the very small p o l a r i z a t i o n effect again being ca l cu lab l e . The deviat ion from E i n s t e i n ' s re su l t w i l l be small since the suggested oblateness, A r / r , i s of the order 10 ^. The p o l a r i z a t i o n effects and the effect of the non-s p h e r i c a l d i s t r i b u t i o n are both small refinements of the c l a s s i c a l r e s u l t . I t i s poss ible that the effect of the r o t a t i o n of the sun on the photons sca t ter ing s h a l l be of an order such that i t w i l l be more nearly measurable. Solut ions of E i n s t e i n ' s f i e l d equations f o r a ro ta t ing mass d i s t r i b u t i o n have been found (see T h i r r i n g , V . H . , 53 1 9 1 8 ; Kerr, P., I 9 6 3; B r i l l , D.R., and Cohen, J.M., 1 9 6 6 ; Cohen, J . M . , 1 9 6 7 ) . Using t h i s solution f o r the metric, the i n t e r a c t i o n Lagrangian can be obtained and a solution f o r the scattering of polarized photons can be obtained following the methods of Section V. I t i s suggested that by increasing the accuracy of the deflection measurement one might be able to detect an asymmetry i n the scattering of photons due to the presence of a preferred d i r e c t i o n caused by the rotation axis of the sun. 54 BIBLIOGRAPHY Aharoni , J . , The Spec ia l Theory of R e l a t i v i t y . 1965, Clarendon Press, London. Akhiezer , A . I . and B e r e s t e t s k i i , V . B . , Quantum E l e c t r o - dynamics . 1965, Interscience Publ i shers , N . Y . Anderson, J . L . , P r i n c i p l e s of R e l a t i v i t y Physics . 1967, Academic Press, N . Y . Bjorken, J . D . and D r e l l , S . D . , R e l a t i v i s t i c Quantum  Mechanics. 1 9 6 4 , McGraw-Hill Book C o . , N . Y . Bogoliubov, N.N. and Shirkov, D . V . , Introduct ion to the  Theory of Quantized F i e l d s . 1959, Interscience Publ i shers , I n c . , N . Y . B r i l l , D.R. and Cohen, J . M . , 1966, Phys. Rev . , 1/^, 1011. Cohen, J . M . , 1967, J . Math. Phys . , 6, 1477. Davidov, A . S . , Quantum Mechanics. 1966, NEO Press , Ann Arbor, Michigan. D i r a c , P . A . M . , The P r i n c i p l e s of Quantum Mechanics. 1958, Oxford Univ. Press , Glasgow. Dicke, R . H . , J a n . , 1967, Physics Today, p.55. Eddington, A . S . , The Mathematical Theory of R e l a t i v i t y . 1924, Cambridge Univ . Press , London. Feynman, R . P . , Quantum Electrodynamics. 1962, W. A. Ben-jamin, I n c . , N .Y. Freeman, M . J . , 1967, Ph.D. Thes i s , U . B . C . , Department of Phys ics . Golds te in , H . , C l a s s i c a l Mechanics. 1950, Addison-Wesley Publ ishing Co., I n c . , Reading, Massachusetts. Jauch, J . M . and R o h r l i c h , F . , The Theory of Photons and  E l e c t r o n s . 1955, Addison-Wesley Publ ishing C o . , I n c . , Reading, Massachusetts. Kaempffer, F . A . , Concepts i n Quantum Mechanics. 1965, Academic Press, Nev; York. Kaempffer, F . A . , V ierbe in F i e l d Theory of G r a v i t a t i o n . 1966, Phys. Rev . . 165. 1 4 2 C ~ 55 K e r r , P . , 1 9 6 3 , Phys. Rev. L e t t e r s , 11, 237. Landau, L . D . and L i f s h i t z , E . M . , Mechanics. I960, Addison-Wesley Publ i shing C o . , I n c . , Massachusetts. Lorentz , H . A . , E i n s t e i n , A . , Minkowski, H . , and Weyl, H . , The P r i n c i p l e of R e l a t i v i t y , 1923, Dover Pub-l i c a t i o n s , I n c . , N . Y . Mandl, F . , Introduct ion to Quantum F i e l d Theory. 1959, Interscience Publishers I n c . , N . Y . Morse, P .M. and Fleshbach, H;, Methods of Theore t i ca l  Physics , part I , 1953, McGraw-Hil l Book C o . , I n c . , N .Y . Schweber, S . S . , An Introduct ion to R e l a t i v i s t i c Quantum  F i e l d Theory, 1961. Harper and Row. Publ i shers , N .Y . S i l b e r s t e i n , L . , General R e l a t i v i t y and G r a v i t a t i o n . 1922, D. Van Nostrand C o . , N .Y . T h i r r i n g , V . H . , 1918, Phys. Z e i t s c h r i f t , 1£, 33. T h i r r i n g , V . H . , 1921, Phys. Z e i t s c h r i f t , 22, 29. Tolman, R . C . , R e l a t i v i t y . Thermodynamics and Cosmology. 1934, Oxford at the Clarendon Press , London. 56 APPENDIX A: THE SCATTERING MATRIX (THE S-MATRIX) The amplitude f o r sca t ter ing from a state | i ^ to a state ^ f | , as used i n equation (1.10), can most e a s i l y be derived as fo l lows . In the in t erac t ion p i c t u r e , one can write (see, for example, Mandl, F . , 1959) (A.I) i l i = H I ( t ) {(t) Z>t where (A.2) H I ( t ) = J f t j t t j x ) dx jtyij(x) i s the i n t e r a c t i o n Hamiltonian densi ty . I f one examines the case i n which the i n t e r a c t i o n i s a d i a b a t i c a l -l y switched off i n the remote past and i n the remote f u t -ure , one can write the f i n a l state ^ f( £ (]>(t—*oo) i n terms of the i n i t i a l state | i ^ H < £ ( t - > -co) as fo l lows: (A.3) <jj(oo ) = S {(-co ), i n which the operator S i s the so -ca l l ed sca t ter ing matrix. In order to solve for S and a r r i v e at a form su i table for c a l c u l a t i o n purposes i t i s usua l ly necessary to assume that the i n t e r a c t i o n i s smal l , al lowing one to use (non-r igorous ly) a power ser ies i n the i n t e r a c t i o n . Successive approximations to the so lut ion of (A.I) y i e lds upon making the i d e n t i f i c a t i o n ( A . 3 ) , (A-.4) S = 1 - i C^j(x) dSc * ( - i ) 2 j^-UJdSc Hz(t»)dt» For the case i n which the Lagrangian density, o£(x), i s not a function of the derivatives of the f i e l d variable one has (A.5) # ( x ) =r -<£(x) . To f i r s t order of approximation (Born approximation) equa-tio n (A.4) then becomes 57 •h oo (A.6) S = 1 • i ^ aC (x) d 4x . - oo Hence, f o r scattering from a state | i ^ to a d i f f e r e n t state ^ f j , the amplitude i s A.7) A f i = < ^ f | i ^ ( x ) d 4x|i/> . The p r o b a b i l i t y f o r scattering i s , i n t h i s approximation, (A.8) |A f i| 2 |</f | J ^ ( x ) d 4x | i > | 2 . The form of the S-matrix involving the Lagrangian density rather than the Hamiltonian density holds even when the r e s t r i c t i o n that *£(x) not be a function of the derivatives of the f i e l d i s removed (see Bogoliubov, N . N . , and Shirkov, D.V.,1959, Section 18). The form (A.6) can 5B be a r r i v e d at by using only the condit ions of covariance, u n i t a r i t y , and causa l i ty of S, together with the corres -pondence p r i n c i p l e . I t fol lows that the i n t e r a c t i o n Lagrangian must be l o c a l , Hermitean, and covariant . The usual sca lar combination of f i e l d var iables automatical ly ensures covariance and the condit ions of the Hermi t i c i ty and of the l o c a l nature of £ represent subs id iary condi-t ions l i m i t i n g the choice of a sca lar An a l ternate approach (see Bjorken, J . D . , and D r e l l , S . D . , 1964) for the case of r e l a t i v i s t i c quantum e l e c t r o -dynamics, using a Green's funct ion or Feynman propogator approach leads to s i m i l a r r e s u l t s . . 59 APPENDIX B . THE SQUARE OF A DELTA-FUNCTION The de l ta - func t ion has the well-known i n t e g r a l rep-resentat ion ( B . l ) § (Of-o)L) I T j~ exp[i(V f- * \ ) t j d t . Thus, (U)f- ^ ) J j S t ^ - o ^ ) ] can be evaluated by se t t ing Of-cJ-t i n one of the f a c t o r s , wri t ten i n the i n t e g r a l representat ion ( B . l ) , to obtain B.2) [ $ " ( ^ - 0 ^ ) ] 2 - L S ^ - ^ ) ^ d t . With the i d e n t i f i c a t i o n (B.3) the t o t a l t ime, one has Jdt = T , ( B.4) ^ (^ f - .^ i ) ] 2 = f ^ ^ f - ^ i A l t e r n a t e l y , i n any p h y s i c a l l y r e a l i z a b l e s i t u a t i o n the l i m i t s on t are never -co to +co . More r e a l i s t i c a l l y , assume the t r a n s i t i o n takes place i n the time i n t e r v a l (-T/2, +T/2). Rather than a d e l t a - f u n c t i o n , one then gets C M , T l x p = s i n fe r " i i j • - T / t 60 Thus, the de l ta - funct ion squared becomes (B.6) 4 s i n 2 [(T/2) # The area under t h i s curve i s T/2TC. Thus f o r large but f i n i t e T, one gets ( B . 7 ) 61 APPENDIX C. CONSERVED QUANTITIES The Lagrangian of a closed (isolated) system, because of the assumed homogeneity of time, cannot depend explicit l y on time. Thus, dt <a\ 2qi 2> V from which, using Euler's equations (2.4), dL _ _d, . JjTL . dt - d t ^ i a A . ) > ( C 2 ) Therefore, the energy of a closed system, ( C 3 ) E = £ q L - L, remains constant during the motion. The Lagrangian of a closed system, because of the homogeneity of space, must be invariant under arbitrary parallel displacement of the entire system in space. Let-ting r a be the position of particle 'a 1, translate the system by an arbitrary, infinitesimal amount S r to get 6 2 Since Sr i s a r b i t r a r y , (C.5) 2— = °> from which, using Lagrange's equations ( 2 . 4 ) , ( C 6 ) = 0 . Therefore, the momentum of the system, ( C 7 ) P = » a i s a constant of the motion. The Lagrangian of a closed system, because of s p a c i a l i so t ropy , must be invar iant under a r b i t r a r y rotat ions of the whole system i n space. Consider an a r b i t r a r y , i n f i n -i t e s i m a l ro ta t ion %0 of magnitude 0 about an axis ind ica ted by the d i r e c t i o n of and note _ | _ r a - S jTx r a , ( C d ) £ i a = $0 x r a , from which, using Lagrange's equations ( 2 . 4 ) and the f i r s t two of equations ( C . 6 ) , 6 3 I C 9 ) 11 * £ a x p a = 0 . d Since i s a r b i t r a r y , the angular momentum, ( C.10) M = 2 ? r x p a > a i s conserved. 64 APPENDIX D. THE GEODESIC EQUATIONS AN D INTEGRALS OF MOTION The equations of a geodesic are determined by the condi t ion , ds i s s ta t ionary . F i x i n g the endpoints of the t r a j e c t o r y i n general space-time, the path can be deformed by an / . in f in i tes imal amount dx**". Applying the s ta t ionary condi t ion , ( D . l ) \ M d s ) = 0, with (D.2) d s 2 = g ^ dx^ d x 0 , ( g ^ i s not the s p e c i a l r e l a t i v i s t i c metric of § 1 1 1 ) , r e s u l t s i n ds =0, Integrat ing by par t s , and se t t ing the integrated part equal to zero (since the endpoints are fixed) gives , n , l i \ J d x ^ d x ^ ^ f A V C r dx^ dx"^ C cL ' ( D * 4 ) 1 WdT d T fZFhx ~ di (Vd5" + g ^ d s ~ ^ x { d s = ° -Since equation (D.4) must be true for a r b i t r a r y S x*"" , the coe f f i c i en t s must be i d e n t i c a l l y zero. That i s , t D - 5 J 2 d s ds ' ix?' *>7*) " — c * d s* 65 Or, mul t ip ly ing by g^ "** to get r i d of g 6 5 - , that i s , (D 7) i J dx^ dx^  - o d s 2 " + ( M * J dT" dT " °» which are the equations determining a geodesic. The angular momentum i n t e g r a l r e s u l t s from the 0-comp-onent of equations (4.1) as fo l lows: (D.S) ^ | - o, r ds ds ' which has the immediate s o l u t i o n , (D.9) r2di=zhi ds * where h i s a constant. Using the coordinates of equation (4.5) one gets , (D.10) R 2 M = J . ds ( 2 m ) 2 The energy i n t e g r a l r e s u l t s from the t-component of equations (4.1) as fo l lows: 

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