UBC Theses and Dissertations

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UBC Theses and Dissertations

A calculation of gravitational radiation Nagata, Kenneth Wayne 1980

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A CALCULATION OF GRAVITATIONAL RADIATION B . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s t h e s i s as conforming by KENNETH WAYNE NAGATA to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ( a ) August 1980 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree at 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 , I a g r e e t ha t t he 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 t 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 the Head o f my Depar tment 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 . Depar tment o f Physi'g-s 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 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 14 AUJIAS*- 19 6 o ABSTRACT Approximate g r a v i t a t i o n a l f i e l d equations i n an a l t e r n a t i v e theory of g r a v i t y are solved f o r a c l a s s of boundary c o n d i t i o n s . The generation of g r a v i t a t i o n a l r a d i a t i o n from s p a t i a l l y bounded sources i s analyzed, and i t i s found that the theory p r e d i c t s the emission of d i p o l e g r a v i t a t i o n a l r a d i a t i o n . However, the d i p o l e r a d i a t i o n vanishes f o r slow-motion post-Newtonian sources. i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES ' i v LIST OF FIGURES v ACKNOWLEDGEMENTS v i 1. INTRODUCTION 1 2. THE GRAVITATIONAL FIELD EQUATIONS AND APPROXIMATE 5 EQUATIONS 3. SOLUTIONS OF APPROXIMATE EQUATIONS 11 4. RADIATION OF GRAVITATIONAL ENERGY 26 5. CONCLUSIONS 39 REFERENCES 40 BIBLIOGRAPHY . 41 APPENDIX : Expressions depending o n f n • 42 i v LIST OF TABLES Page Table I. L i m i t i n g Values of Speeds of Propagation v., v_, u 2 2 V LIST OF FIGURES Page Figure 1. Speeds of propagation v , v_, u 23 ACKNOWLEDGEMENTS I would l i k e to thank my adivsor Professor P. R a s t a l l f o r suggesting the t o p i c and f o r p r o v i d i n g a s s i s t a n c e throughout the research. I a l s o wish to thank Professor W. Unruh f o r h i s h e l p f u l comments. Acknowledgement i s a l s o made to the N a t u r a l Science and Engineering Research C o u n c i l Canada f o r awarding a Postgraduate Schola r s h i p f o r the period of research. INTRODUCTION Since E i n s t e i n . i n t r o d u c e d general r e l a t i v i t y as a theory of space, time and g r a v i t a t i o n , many a l t e r n a t i v e t h e o r i e s of r e l a t i v i s t i c g r a v i t a t i o n have been proposed. These t h e o r i e s a l l p r e d i c t r e s u l t s compatible w i t h the " c l a s s i c a l " experimental t e s t s of general r e l a t i v i t y - g r a v i t a t i o n a l r e d s h i f t of the frequencies of electromagnetic s i g n a l s , r e l a t i v i s t i c p e r i h e l i o n s h i f t s i n the o r b i t s of p l a n e t s , r e f r a c t i o n of electromagnetic waves and time delay of radar s i g n a l s due to the Sun's g r a v i t a t i o n a l f i e l d . Various other experiments may be performed to t e s t general r e l a t i v i t y and to p o s s i b l y e l i m i n a t e competing t h e o r i e s . For example, gravimeters have been used to search f o r anomalous Earth t i d e s which would i n d i c a t e "preferred-frame e f f e c t s " due to the Earth's v e l o c i t y r e l a t i v e to the r e s t frame of the Universe. Laser ranging experiments to corner r e f l e c t o r s l e f t on the Moon have been performed i n an attempt to detect d i f f e r e n t a c c e l e r a t i o n s towards the Sun f o r the Earth and the Moon, i n d i c a t i n g t h e i r departure from geodesic motion ( i . e . the "Nordtvelt e f f e c t " ) . So f a r n e i t h e r p r e f e r r e d -frame nor Nord t v e l t e f f e c t s have been observed. An experiment i s planned i n which the precession of a gyroscope o r b i t i n g the earth w i l l be measured to determine non Newtonian e f f e c t s [1], [2] ch. 40. The r e s u l t s of such " s o l a r system" experiments can be conveniently compared w i t h the p r e d i c t i o n s of metric t h e o r i e s of g r a v i t a t i o n w i t h i n the framework of the parametrized post-Newtonian 2 (PPN) formalism [ 1 ] , [2] ch. 39. This formalism summarizes the f i r s t (post-Newtonian) order c o r r e c t i o n s to Newtonian g r a v i t a t i o n i n a l a r g e c l a s s of metric t h e o r i e s of g r a v i t a t i o n i n terms of ten parameters. Solar system experiments then determine l i m i t s on the s i z e s of these post-Newtonian parameters. Thus to see whether a given metric theory of g r a v i t a t i o n agrees w i t h experiment, one computes i t s post-Newtonian parameters and compares them w i t h the experimental l i m i t s . On t h i s b a s i s , many t h e o r i e s can be r e j e c t e d . However, s e v e r a l t h e o r i e s - i n c l u d i n g the one used i n t h i s t h e s i s - have the same, or almost the same post-Newtonian parameters as general r e l a t i v i t y . These are as yet compatible w i t h a l l s o l a r system experiments. To f u r t h e r r e s t r i c t the c l a s s of v i a b l e t h e o r i e s of g r a v i t a t i o n , one must consider e f f e c t s beyond the post-Newtonian. Advances i n technology during the next decade may a l l o w measurements of higher-order (post-post-Newtonian) d e v i a t i o n s from Newtonian g r a v i t a t i o n . Refined s o l a r system experiments would be expected to provide f u r t h e r c o n s t r a i n t s on the v i a b i l i t y of a theory of g r a v i t a t i o n . One should a l s o consider cosmological models p r e d i c t e d by each theory and compare them-with experiment. For example, the constancy of the Newtonian g r a v i t a t i o n a l "constant" G, and the cosmological a c c e l e r a t i o n parameter must f a l l w i t h i n current experimental l i m i t s . Further s e l e c t i o n amongst t h e o r i e s of g r a v i t a t i o n may be p o s s i b l e by means of g r a v i t a t i o n a l r a d i a t i o n experiments [ 3 ] . 3 Some c u r r e n t l y v i a b l e t h e o r i e s p r e d i c t speeds of propagation of g r a v i t a t i o n a l r a d i a t i o n d i f f e r e n t from that of l i g h t . A g r a v i t a t i o n a l d e t e c t o r may be able to detect g r a v i t a t i o n a l r a d i a t i o n b u r s t s from n e a r l y supernovae, ^  and", the a r r i v a l times of the bursts could then be compared w i t h those f o r the corresponding electromagnetic r a d i a t i o n . In a d d i t i o n , d i f f e r e n t m e t r i c t h e o r i e s of g r a v i t a t i o n p r e d i c t as many as s i x p o l a r i z a t i o n modes of g r a v i t a t i o n a l r a d i a t i o n - general r e l a t i v i t y p r e d i c t s only two. C u r r e n t l y f e a s i b l e detectors may be able to measure a l l s i x p o s s i b l e modes and thus e l i m i n a t e some th e o r i e s of g r a v i t y . The d i s c o v e r y of bin a r y p u l s a r PSR 1913+16 i n 1975 and i t s subsequent observation have caused much recent i n t e r e s t i n g r a v i t a t i o n a l r a d i a t i o n c a l c u l a t i o n s . The observed r a t e of decrease of the pu l s a r ' s o r b i t a l p eriod i s a t t r i b u t e d to the l o s s of energy from the system v i a g r a v i t a t i o n a l r a d i a t i o n . I t has been claimed that the period decrease i s q u a n t i t a t i v e l y accounted f o r by the E i n s t e i n "quadrupole formula" f o r g r a v i t a t i o n a l energy l o s s i n general r e l a t i v i t y [ 4 ] . However, the approximations made i n d e r i v i n g t h i s formula (weak f i e l d , slow-motion Newtonian source c o n s i s t i n g of two poin t masses i n K e p l e r i a n o r b i t s ) [5] may not be a p p l i c a b l e to the binary p u l s a r . For example, o p t i c a l observations i n d i c a t e that the companion s t a r to the binary p u l s a r may not be a compact object and so cannot r e l i a b l y be treated as a poin t mass [6 ] . I n any case the d e r i v a t i o n of the formula i s pure l y formal and the v a l i d i t y of p r e d i c t i o n s of the r a t e of period decrease based on 4 the quadrupole formula have been se r i o u s l y questioned [7], [8], Attempts are being made to f i n d more mathematically meaningful approximation methods for t r e a t i n g systems such as the binary pulsar [9]. One would l i k e to know whether other currently v i a b l e metric theories of g r a v i t a t i o n can su c c e s s f u l l y predict the rate of period decrease of the binary pulsar. Calculations f or several theories using weak-field, slow-motion post-Newtonian source approximations have been made, and the theories predict dipole g r a v i t a t i o n a l r a d i a t i o n from g r a v i t i o n a l l y bound sources as compared to the quadrupole r a d i a t i o n predicted i n general r e l a t i v i t y . However, the masses of the bodies and the nature of the companion star i n PSR 1913+16 are as yet not s u f f i c i e n t l y c e r t a i n to enable one to select conclusively amongst the competing theories [10]. In t h i s t h e s i s , the above-mentioned doubts about the v a l i d i t y of the usual formal manipulations are ignored, and weak-field approximations are used to l i n e a r i z e the f i e l d equations of an a l t e r n a t i v e theory of grav i t y . Solutions of the l i n e a r i z e d equations are found for c e r t a i n boundary conditions, and the solutions are used to. c a l c u l a t e the g r a v i t a t i o n a l energy radiated from a slowly-moving s p a t i a l l y bounded system of sources. I t i s found that the theory pr e d i c t s dipole g r a v i t a t i o n a l r a d i a t i o n , but that t h i s vanishes for slow-motion post-Newtonian sources. 5 2. The G r a v i t a t i o n a l F i e l d Equations and Approximate Equations The theory of g r a v i t a t i o n used i n t h i s t hesis i s described i n d e t a i l i n [11], and i s summarized here. The theory contains a pseudo-Riemannian space-time metric tensor f i e l d g, and a pseudo-Riemannian f l a t metric tensor f i e l d g ( i . e . , the Riemann tensor of g vanishes). In addition there are a covector f i e l d n and a r e a l scalar f i e l d ip. The f i e l d s are related by (cf .[11] (4.3a), (4.4a), (4.8), (4.9), and (4.29)). (2.1) , +4 where T = (-det g.- ~f, V = (-det g ) 2 , lower-case Greek indices have the range {0,1,2,3} and obey the summation convention. Since g i s a f l a t metric, there e x i s t charts, c a l l e d g i n e r t i a l charts, i n which (2.2) 3" r , r -*4 6 where TI = -6 , n. = 6 , lower-case L a t i n i n d i c e s have the range y o ; ' . y o m n m n ° {1,2,3} and a l s o obey the summation convention. In a g- i n e r t i a l chart where the are small ( i . e . \-In™ I << |n n I) , eq. (2.1) i m p l i e s that e^-1 = - n n 2 + n n . Thus, 0 m m i n a re g i o n where if < 0 one can choose the n^ to be r e a l , and the space-time me t r i c w i l l d i f f e r l i t t l e from the "newtonian" metric g = -<5_ e 2^, g = 6 e-2^7 In a region where ty > 0, the spacetime met r i c w i l l be n e a r l y newtonian i f the n^ are imaginary. However, the n^ cannot be r e a l i n one space spacetime r e g i o n and imaginary i n another, so that i f ty takes both p o s i t i v e and negative values i n a re g i o n of space-time, then the space-time metric cannot be everywhere n e a r l y newtonian ( c f . [11] remarks f o l l o w i n g ( 2 . 2 ) ) . The Lagrangian d e n s i t y of the g r a v i t a t i o n a l f i e l d i s given by ( c f . [11] (4.14)) y 2^ =.2* where N = h n. ..= e - e , semicolons denote c o v a r i a n t d e r i v a t i v e s y w i t h respect to the spacetime me t r i c g, and the f u n c t i o n F i s ( c f . [11] (5.8)) (2.4) F(f0 « -N/K . *k(n-N»-) - e^O-e^Vlk-rtkO+e 8"*' ) , -4 where k = Gc , G i s the g r a v i t a t i o n a l constant, and c i s the speed of l i g h t . The f i e l d equations i n a g i n e r t i a l chart are ( c f . [ 1 1 ] ( 7 . 7 ) ) . 7 (2.5) 8£j$r\n - ($£jS^ir)>s. - e.-^d^T^v-T1"^, where commas denote p a r t i a l d e r i v a t i v e s w i t h respect to the space-time v a r i a b l e s and the f u n c t i o n a l d e r i v a t i v e 6 corresponds to the independent f i e l d s n , n , e w i t h y < v , g r y y,v' & y v ' °uv-,ir w i t h y < v , and the n o n g r a v i t a t i o n a l f i e l d s ( i . e . sources) whose stress-momentum tensor i s T. The components T^V of T s a t i s f y ( c f . [11] (4.15), (4.17) (2.6) To study g r a v i t a t i o n a l r a d i a t i o n i n the f i e l d equations (2.5) are replaced by approximate equations which are e a s i e r to s o l v e . Approximate f i e l d equations are presented i n d e t a i l i n [12]. Here the r e d u c t i o n of the f i e l d equation (2.5) to approximate equations i n the case when ty •> ty ^ 0 at s p a t i a l i n f i n i t y i n a g i n e r t i a l 0: chart i s o u t l i n e d . G r a v i t a t i o n a l r a d i a t i o n i n the case when ty -»- 0 i s trea t e d i n [12] . C a l c u l a t i o n s are s i m p l i f i e d i f new v a r i a b l e s q^ are introduced, defined by (2.7) . ^  -8 The transformation (2.7) i s non-singular provided that ty never vanishes. The are taken to be r e a l , w i t h q D < 0 , whether ty .is negative or p o s i t i v e ( c f . [12] remarks f o l l o w i n g ( 3 . 1 ) ) . From (2.7) and (2.1) i t f o l l o w s that (2.8) ^ U t v - 'I , which i n a g i n e r t i a l chart becomes q 0 2 = 1 + < L J 1 a 1 ' Thus ty and the q^ can be chosen as independent f i e l d s , and the Lagrangian d e n s i t y may be expressed i n terms of these v a r i a b l e s ([12] ( 3 . 3 ) ) . Approximations of f i e l d s a r e made on the b a s i s of "order of magnitude" assignments. The expression 0^n^ denotes terms which are of the order of magnitude V nc n of sm a l l e r , where V i s a t y p i c a l speed of the sources. The components of the stress-momentum of the , , moo „ , „mo „ , mmn _ sources are assumed to s a t x s f y kT = O ^ ) ' ^T = (3)' = (4) (e.g. an i d e a l f l u i d ) . One a l s o assumes that i n the neighborhood of the sources, time d e r i v a t i v e s are an order of magnitude smaller than space d e r i v a t i v e s . Thus kT°°,m = » kT°°,o = 0(3)> e t c . In the case when ty 1 -> ty^ at s p a t i a l i n f i n i t y i n a g i n e r t i a l chart x', one assumes that ty 1- tyQ= ^(2) a n c^ ^'m=^(3) ' Far from the sources, space and time d e r i v a t i v e s may be comparable i n s i z e . Thus i t i s assumed that ty1 • = 0.„. and q' = 0,„. ,0 (2) 4 m,u (3) except at p o i n t s where T y V ^ 0, i n which case ty1 = 0,_. and > 0 (3) q' m Q = 0 ( 4 ) ' S i m i l a r assumptions are made f o r the higher p a r t i a l 9 d e r i v a t i v e s . Such assumptions must be checked l a t e r f o r consistency w i t h the s o l u t i o n s of the approximate f i e l d equations. The Lagrangian d e n s i t y ' i n the g i n e r t i a l chart x' i s given to seventh order by [12] (5.11). Define a new chart x by (2.9) * e"k x'm , x° - t ^ x ' 0 , so that ty ty and g -> rr.. at s p a t i a l i n f i n i t y i n x provided o yv • yv that the q^ are n e g l i g i b l e . Note that i n the g i n e r t i a l chart x', the g' -^r* n at s p a t i a l i n f i n i t y even i f the q':. are n e g l i g i b l e . uv - yv m I f one assumes the constant ty^ 4 0 and i s not s m a l l , a l l f u n c t i o n s of ty^ appearing i n expressions f o r the f i e l d s or f i e l d equations are assigned the order of magnitude 0 ^ . Thus ty - tyQ = ^ ( 2 ) ' %i = ^ ( 3 ) ' and so on, i n the chart x e x a c t l y as assigned i n the chart x', s i n c e the expressions f o r the f i e l d s i n x d i f f e r from t h e i r counterparts i n x' only by f a c t o r s of order zero. The f i e l d equations to t h i r d order i n the chart x are given by [12] (5.14) and (5.15): (2.10) ( W ^ e * * - ^ - U s \ m t 0 0 - ( l - ^ ) ^ l n , n m - z a + e * * - ) ^ -te,7rk ( l - fi^)"(n-e^) T M 0 + O C4) 10 (2.11) +Z(H-ta+'y,(-U6*t--£e*- )4,oo * i ( | f e ^ r ' ( r W ' - l - £ 4 4 * * d 8 | - ) e i 1 » , » 0 =» 4*kT°° + O f t) . Define a s c a l a r f u n c t i o n Y by x ,~ = q • Then d i f f e r e n t i a t i n g ,-o Tn,m . (2.10) w i t h respect to x m and summing over m, using . '. . + 0,,-v , and i n t e g r a t i n g the r e s u l t i n g equation w i t h , o. (. J / Tom = _ Too + Q :m respect to x° gives [12] (5.16) (2.12) e-^'t>im -e.U'l>0O ' ( i + e ^ ^ ) > w where "f i s a f u n c t i o n of the s p a t i a l coordinates only. Eqs. (2.12) and (2.11) w i t h q = X are a l i n e a r system of p a r t i a l d i f f e r e n t i a l Tii,mo »oo equations of x and ty. I t s s o l u t i o n i s described i n the next s e c t i o n . 11 "3.;. S o l u t i o n s of Approximate Equations The approximate g r a v i t a t i o n a l f i e l d equations (2.10) and (2.11) are solved i n t h i s s e c t i o n . F i r s t the system (2.12) and (2.11) w i t h q = X-> i s w r i t t e n i n matrix form as \,mo ,,oo (3.1) im ,5^ .00 - 4? - k T ° ° + 0^ M u l t i p l y i n g (3.1) on the l e f t by one obtains 12 (3.2) + 11 oo -4n ( e ^ - i y ' ( £ 8 K 3 ) e 3 4 k T 0 0 + p + o c ^ where p = - e 3 ^ 9 f / 4 i r . The system of equations (3.2) can be decoupled i f the matrix A « e - e - e + 0 can be d i a g o n a l i z e d . This, i s always p o s s i b l e over the complex numbers, but not n e c e s s a r i l y over the r e a l s . The c h a r a c t e r i s t i c polynomial of A i s d e t ( A - A I ) , where I i s the 2 x 2 i d e n t i t y m a t r i x . Thus the eigenvalues °f A are the s o l u t i o n s of the quadratic equation, 13 ( 3 . 3 ) 0 * ll - U e l t 4 b ^ ^ - 4 e 4 - . 0 ^ The s o l u t i o n s X + of ( 3 . 3 ) are given by (3-4) -xt - i [ - i ( e ' ^ + 8 e e 4 - - + E ^ ^ ) * A VM , where A = $.(<L+^- (fi^'^^-^e^-{4e**° - 3e+ 4° - °> ) . One notes that A = 0 i f tyQ = 0 or tyQ = tyQ*, where I/Jq* % 0.25, A < 0 i f 0 < < IJJ and A > 0 f o r a l l other ty0 (see Appendix). Thus the matrix A i s d i a g o n a l i z a b l e over the r e a l s provided tyQ £ 0 . or i j ; 0 > ^0&. In the case when ty0 = 0, the f i e l d equation (2.10) i s s i n g u l a r , and i n f a c t the approximations used to obta i n eqs. (2.10) and (2.11) do not apply i f tyQ i - s s m a l l . However, t h i s case i s t r e a t e d separately i n [12] . Here only the cases when A > 0 and IJJ0 i s not small are considered. The eigenvalues X • of A are then r e a l and negative (gee Appendix), and can be w r i t t e n as ( 3 . 5 ) \ ± = - CeM^ 0 - v ± " * , 14 where the v1 are p o s i t i v e r e a l constants depending oh ipQ, A matrix P which d i a g o n a l i z e s A i s found to be (3.6) .(3 a-j* where the a + are nonzero but otherwise a r b i t r a r y constants, and The matrix i n v e r s e of P i s given by (3.7) a.U- -a.ft 15 where d e f P = - a + a _ e ^ o ( e ^ o _ l ) 2 ( e 1 2 * o + D C e 1 6 ^ + e 1 2 " ^ + e ^ o + 1)A %. One checks that (3.8) M u l t i p l y i n g eq. (3 . 2 ) on the l e f t by P 1 and using PP 1 = I gives 4 \ (3.9) P"' + ( e ^ + irp>p-' ,00 k T ° ° + o&) D e f i n i n g l i n e a r combinations ty and co of the f i e l d s ty and x by - + (3.10) to t and using (3.7) and (3.5), one obtains from (3.9) (3:11) 4> Vi ,00 - a . ( r . k T ° ° + ( 5 r ) + Q ' C 4 ) a + C r +kT o o + /s ( 0) * J 16 where y± * ^ + ("e^-ir 5 ) e s 4 (5 I f the sources are s p a t i a l l y bounded, i . e . i = sup { |£| ; T^Cx",?) f O } i s f i n i t e , then the matrix wave equation (3.11) has a s o l u t i o n corresponding to outgoing disturbances: (3.12) = (Jfetpy1 where R = | x-x' | ( [T°°] r e t= T 0 0(x 0-v--. 1R, x') . I n v e r t i n g eq. (3.10) and imposing boundary c o n d i t i o n s ty ty0 and X 0 at s p a t i a l i n f i n i t y g i v e the s o l u t i o n s to the approximate f i e l d equations (2.11) and (2.12): (3.13) 4,Cf,t) - \a + k (e^HV'e^ A~'/x { y. jV'fT°°J^ JV (3.14) T ( ( K » ? ) = - 4 - r * P" I (-"'CT00];,,*. JlV } + J" R " > J V + . 17 The solutions (3.13) and (3.14) may be expressed as (3.i5) f x ) - ^ 0 = j > x ' R - ' L s + T ° v - | , n ^ . r v - t r ) ] «-o f t ) , (3.16) t „ # m M - J ^ V R l i J * . ^ - ^ ' 1 ^ - T ° % ( . * 0 4 ,3?')] v O < + ) , where £ t = ± k 0"' e"5^" fif*- , 6± = i k (e'^+.i)-' e i J / ° • To obtain solutions f o r the qg, one m u l t i p l i e s eq. (2.11) by 2 e 3 * G ( l - e 4 , ( j ° r 2 and eq. (2.10) by -\a+^°)~1 and gets (3.18) -j^f„ »^«o - X +7^i^Wo -177I^oe l^iin - j r ^ T * 0 ^ o c 4 ) . Define the Fourier transform ty of i p - i f ' hy (3.19) 4<60-J/„ - |a+k 4-Ck) e'^ and define q^, T ^ s i m i l a r l y . Then the Fourier transforms of eqs. (3.17) and (3.18) are ( 3 - 2 0 ) L-(Tr^^ k>^v.-^o+e^) k° J * - i^ 5?.K.k«i«. = Tj ^ u ) ^ , (3 .21) - k.km 4 «• |_ — k nk n - ^  k.'J t m «• 2. T^iV, e i + » - <-oC4 Equations (3.20) and (3.21) may be w r i t t e n as a ma t r i x equation: (3.22) = ftitkT v o c o , where B = [ B ] i s a 4 x 4 symmetric matri x given by uv a „ f. J ^ L k k ^ ^ d - e ^ . e ^ e l j h k x 1 Poo L C I . ^ K « R « «• TT^UyT^Ti^O K o J ' (3.23) 4 t ^ B ^ and ¥ = [¥ ] , T = [T ] are four-dimensional column v e c t o r s given (3 .24) - { , ^ w - , - c " ^ y - f 0 0 , T w - ^  T> -r- I 'I, Let [k| ''"k , e ^ d e f i n e an orthonormal b a s i s f o r FR3 ' 1 m m m ' ' '2 where |k|= (kmk ) . Then one has the orthonormality r e l a t i o n s (3.25) | i T r X l c , = = e £ = I , k MeS? = k we« - ^ e £ > = o The 4x4 matrix Q = [ Q ^ l given by (3.26) - Q0/. - L 0 , Q m l = I k Y k * , Q^- , c ^ - e ^ 1 T - l T s a t i s f i e s Q = Q , where Q denotes the matrix transpose of Q. 19 De f i n i n g a new column ve c t o r ¥ by (3.27) t ' - q T f , one has (3.28) *'0.t, * , ' s 1 , ^ 1 ^ ' ^ , ^ , ^ ' ^ ^ and ¥ s a t i s f i e s the matrix equation (3.29) B'"*' =» S T i k T ' + o where (3.'30) B' "uS*> KK yz&l^KK - eS^»iJ o o 2. p fi i ve 8 * * O 0 O ,s<k> — k t - - S — - ~ Ir x i s found by using the orthonormality r e l a t i o n s (3.25), and T i s given by ( 3 . 3 1 ) T 0 ' - ^ x - T 0 0 , T% - h 3 ! ^ T » » T ' ^ e<0f«° -r'=-U e^f-Writing out eq. (3.29) e x p l i c i t l y , one gets U [ ( i - ^ - y K'KI» + 7^4^7JT^) K. J 4 - ^  k„kw?m = _ - ^ T . o o w ) 5 (3.33) - ^ ^ f L4 + T ^ M ^ k p - e ^ ] ^ ^ -ifgj.feT- - , (3.34) [ U - ^ - ^ k ^ ] ^ - . - ^ ^ ' f - . 0 < + ) C > M ) . M u l t i p l y i n g (3.32) by | e" 3 ^ 0 ( l - e ^ 0 ) 2 , (3.33) by - i | k| ( l + e 8 * 0 ) and (3.34) by -2e 3^°, one obtains (3.35) [ - M . * 2 ^ £ ^ - • « * f - . Q „ . (3.36) i 6 « ^ ) k , k , k . ^ - ( [ . - ^ I ^ k . ' ] ^ . . - 6 . k ^ l W f -mo , (3.37) [ - k m U * " ^ k . i ] # ' ^ - . u ^ ^ f - ( j - . , ^ . Equations (3.35) and (3.36) are the Fourier transforms of (3.38) ^ i t^ e. 4. ' 4.0^^ , | e i k ^ t ^ - ^ r ^ o t f ) ) (3.39) - C . ^ ) 4 , ^ * ^ - f f - e ^ ^ - e . k ; ^ T " 0 .O t 4 ) , where one uses T° m,m = - T°°,o + 0 ^ ([12](4.5)). Eqs. (3.38) and (3.39) have the solutions (3.15) and (3.16). The Fourier transforms of the solutions are (3.40) * 4-n[S+ (kwk* -^k.*)"' T°°(.k) 4-J.(k.k»-J_»k.O"T00^)] + o C 4 ) , (3.41) („0c) - 4,U4 - ( k , k » - i x k.O" b t c o ( k ) ^ -(Uc-^ koO" ^ ,f00(lc)] * o ( 0 . Using the F o u r i e r transform of [ 1 2 ] ( 4 . 5 ) , one obtains (3.42) £„oo = - 4 , [ ^ ( k ^ - ^ ^ r - ^ f "W * 6 .(Uk»-i*k0"^ t^ k ) ] * . Equation (3 .37) has the s o l u t i o n (3.43) ^ e i k J ^ - W Y e f f ^ - O a ) C J . . . O , where 0 - 4 * ^ , . 1 ± * £ > o. I n v e r t i n g the change of coordinates eq. ( 3 . 2 7 ) , one has S u b s t i t u t i n g the expressions (3.42) and (3.43) f o r qjj and q i n t o eq. (3.44) and using the i d e n t i t y e j j ^ = - $\lkwkp one gets (3.45) y u - - ^ [ ^ o ^ k , - ^ ) - 4 ^ . ( k , k . - ^ y + ecu*- ^ r l ^ f ^ k ) 0' J_ (3.46) » J > k {mCk)e^ *- O W ) . 22 where one imposes the boundary condintion that the -*• 0 at s p a t i a l i n f i n i t y . Equations (3.15) and (3.46) give solutions to the third-order f i e l d equations (2.10) and (2.11). They are consistent with the assumptions ty-tyQ = 0(2) anc* *V = ^(3) " Three speeds of propagation appear i n the solutions to the f i e l d equations. Two of the speeds, v+ and v_, are associated with ty and the l o n g i t u d i n a l component of q^, while the transverse components of q^ propagate with speed u. These speeds are d i s t i n c t from each other and from the speed of l i g h t (see F i g . 1), except i n c e r t a i n l i m i t i n g cases noted i n Table I (see also the Appendix). Table I. Li m i t i n g values of speeds of propagation v+, v_, u Limiting case li m v+ li m v- li m u CO aa o + 1 I 1 « Q.54 ^0.82 4o •* *° i. o « 0.71 23 cWpVis o4 syezds of propAcpiw AS functions of ike. value 4o °f 4>M A* 6pAtia.| infinity. For d e f i n i t i o n s , see e<jS- (i.s) Av,d (3.4-3). To c a l c u l a t e higher-order c o n t r i b u t i o n s to the f i e l d s , one keeps terms to seventh order i n the Lagrangian d e n s i t y and obtains approximate f i e l d equations [ 1 2 ] ( 5 . 1 2 ) and ( 5 . 1 3 ) : ( 3 . 4 7 ) (|f e*h e- j4>,pp - Ze s S * , ° o • 0 - e B * 0 e - » * W m - Ui*d*) ^>MO I f the t h i r d - o r d e r s o l u t i o n s ( 3 . 1 5 ) and ( 3 . 4 6 ) are s u b s t i t u t e d i n t o the quadratic terms of eqs. ( 3 . 4 7 ) and ( 3 . 4 8 ) , one has ( 3 . 4 9 ) (U«4)e*S-,«, -*«*S»,« Hi~t^)e^lP,pm - z C \ * ^ t m n ( 3 - 5 0 ) 4 ^ - 2 . ^ ^ 4>0 ^ l ^ t t ^ S ^ ' 4nk(r-.T-) • Cfc,. The system of equations ( 3 . 4 9 ) and ( 3 . 5 0 ) d i f f e r s from ( 2 . 1 0 ) and ( 2 . 1 1 ) only by the orders of magnitude of the neglected terms and 25 the a d d i t i o n a l 4 7 r k T i m u term i n the sources. The s o l u t i o n methods of t h i s s e c t i o n g ive the fourth-order s o l u t i o n (3.5D . ^ - ^ ^ ^ l ^ r v ^ , ? ) ^T - t V - a g ) , ? ) + s . r ^ - i H J -%)• , . T - ( » . . ^ ^ ] • o ( f ) , where = * k (n-e1^-)"' a"**- C"''1- [ «t± * e J ^ ( i ^ e*4") f ] . The expression (3.16) f o r m i s good to f o u r t h order, as are the s o l u t i o n s f o r the tr a n s v e r s e p a r t s of s i n c e eq. (3.50) i s unchanged from eq. (2.10). Thus one has (3.52) t . « . - l M ^ \ « . « * ^ \ e « ^ ] . * ' V * ' 26 4.'. R a d i a t i o n of G r a v i t a t i o n a l Energy Once s o l u t i o n s to the g r a v i t a t i o n a l f i e l d equations are known, i t i s p o s s i b l e to c a l c u l a t e the r a t e of g r a v i t a t i o n a l energy r a d i a t e d away from s p a t i a l l y bounded sources. The components of the c a n o n i c a l g r a v i t a t i o n a l stress-momentum H ' i n a g i n e r t i a l G chart x' are defined by [12] (2.7) to be (4.1) - - ( S £ « f S * + , r ) . I f the sources are s p a t i a l l y bounded i n ;:.x' , then the r a t e of g r a v i t a t i o n a l energy r a d i a t e d out of the s p h e r i c a l surface ( x ' m x ' m ) ^ = a, where a > I , i s given by [12] (3.27): (4.2) -t^ti) * -cf?A4'K<l8'Z<o*(*'',r)x'>r'*md' , where r'= (x'w x '"V'\ x' 1'r't*s9', x'1 - e'easot/, x's«r'sinB smol', r'- a. The r a t e of r a d i a t i o n of g r a v i t a t i o n a l energy i s defined by [12] (3.28) as (4.3) -£ - Mm . -3 As noted i n [12], terms i n H ' a which are 0 ( r ' ) as r* -»- ro do not c o n t r i b u t e to ~£: In the case of small q^, one uses [12](3.10) and (3.11) to get I < I The r e s u l t s of section 3 imply ty -tyQ = ° ( 2 ) ' ^ 0 = °(3)' \ s = °(3' q' = 0 / / N . Thus to eighth order, the energy f l u x i s given by m,o \H) ( 4 . 5 ) l U i ; . ' - " T T ^ l ^ - U e ^ - e ^ ^ l ^ o *-^-»-e^ + e^ H'*.«l'.,o - Z ^ - ^ U ' . . - 4,'o - ° < 9 , • The coordinate transformation (2.9) and tensor transformation r u l e s give 28 r\ f ^ T°°/ „ ?\ C-J-l r'oo/ ,a -•U.tt'-Ti -II f " ( l ; ) - e - " * » T " • • ( * ' ) , where k' i s given by ( 4 . 7 ) k'G » e ^ k 0 , k'^= C ^ k , The s o l u t i o n s (3.51) and (3.52) i n x transform to ( 4 . 8 ) + ' M - 4 . - [ V T ' - 6 , * - £ £ , T ' ) . e - ^ , » T ' - " ( » ' . - ^ , t O where v+ = e^°v+, u'= e^°u. At t h i s point i t i s convenient to drop a l l primes, s i n c e the chart x of the l a s t s e c t i o n i s no longer needed. A l l subsequent expressions are to be understood to be expressed i n a g i n e r t i a l chart x. Define the one-dimensional F o u r i e r transform ty of ty-tyQ by 29 (4.10) 4^4o = 4Ck0)3?) ej and define and T s i m i l a r l y . From (4.8) and (4.9), one obtains (4.11) f (lc.,*) - ^ [ ( ^ - ( t . ^ n - e ^ V ^ M ) ) ^ " ^ (4.12) ^(k 0,l?)= - e s 4 ° j " T | % J ' d J k k - ^ t 7 k ) [ 6 + e ' t o ^ ' + 6 . e 1 ^ . 6 ^ ' ] ^ At large distances from the sources, i . e . i f |x|= r > > |k | ^ £, one has |x-|| = r - x V V r + 0(r~+)'. The expression (4.11) may be written as (4.13) ^ao,t) = 7 { e U , , X ' I ^ [ ^ T 0 0 ( k 0 ^ ) * e < 4 > T ^ ( k 0 ^ ) ] e ^ where a p = - k Q x p / v + r , b p = - k Q x P / v _ r . I d e n t i f y i n g the s p a t i a l i n t e g r a l s as inverse Fourier transforms, one writes (4.14) ? ( k o | i ) . lM*{e<V" [ S,T00(k0,*) + e - ^ ^ f m m ( k 0 ^ ) ] 30 Similarly, one has (4.15) - -£**'*?[e?'*\.*& t"X*> • e^ 'V^T f ~(k.,*) where a = |k|/v+, b = |k |/v_, d = -k x P/ur, d = Ik l/u. 1 o1 • T 1 o p o 1 o1 If the sources are assumed to move slowly, i.e. i f |kQ| = 0 ^ and v+/Z ^  v_/£ > > |kQ|, one may expand the Fourier transforms: (4.16) t^ C k^lc) - (»V' Ul% f^o,i)elk^f where in (4.16) and in subsequent expressions, k is one of a , b , P P P dp. Keeping terms to fourth order, one has (4.17) j il<0f™(ko,£) = ^ T * J > t ikDfnic.^) :koT'M'M(k0,'k*) - cv 0 . * o cs i The F o u r i e r transform of the conservation law T°° = -T° n + C> N ,o ,n (5) (cf . [ 1 2 ] ( 4 . 5 ) ) i s (4.18) ik0T°° - - f°" + O I n t e g r a t i n g eq. (4.18) over a l l space gives (4.19) J>3 ; k 0 T D O ( k o , t ) - O, w h i l e m u l t i p l y i n g eq. (4.18) f i r s t by £ m and then i n t e g r a t i n g over a l l space gives (4.20) j f % ikoTom(k0,%) = - k 0 ^ ^ f - o o ( k 0 ^ ) r * O t t ) . I f one de f i n e s the F o u r i e r transform of the d i p o l e moment of the sources as (4.20) P " Y U = I ^ T ° ° ( k . , t h m , 32 then from eqs. (A.17) and (4.19)-(4.21) one gets (4.22) { lkoT00LkDlk) - ( z T i Y H o k p D p ( k 0 ) v Q From dqs. (4.14), (4.15), (4.17) and (4.22) one obtains (4.23) i k . # M ) . " " T U ^ ' Vt 7 6 " l k . ) - £ t>*(k0)] * Ocs, • O(r-0, (4.24) i ( k B , n - r T [ e ^ * > ( k 0 ) * e i l ' ^ f 5 " V U l * Q*, • Ofr-O, (4.25) ^ M<k.,x) - ^ { e">x%+ D-flc.) *• e ' b ' * % _ ^ D7k 0^ W " * ' 9 D*(k„) - Dm(ko)]] «- O t o J * O ( r - ) , (4.26) l.^t^.x) = -e. 7 7 {e^ 7, — D * e T ^ — D"UO 33 The F o u r i e r transform eq. (4.10) may be w r i t t e n (4.27) ^U)-^0 = ( ^ . [ ^ k J ) e ; U ' * ^ (M)e i V° X°] . I f the sources c o n s i s t of a s i n g l e F o u r i e r component, i e , i f (4.28) T^(») = f n k . ; ) ^ ' v T ^ k ^ ^ e ^ , where k Q<0, then so do the f i e l d s ty-tyQ and q^. For such cases one has (4.29) +.* +.o - Z« e l? | 4(iko?)* «• ^ ( i k ^ ] , where Re{z) denotes the r e a l p a r t of a complex number z. S u b s t i t u t i n g the expressions (4.23) and (4.24) i n t o (4.29), one obtains (4 .30) | o = - 2 £ £ Re { $ P*(UD w(k 0) • £ ^ DTkJD^CWJ •[ . §fe e«VVx' ] ==£ ^ (L)D-(kJ * 4/^ 4 )e;i^° ] Taking the average over a l a r g e r e g i o n of spacetime compared w i t h the wavelengths and frequency, the o s c i l l a t i n g terms give no net c o n t r i b u -t i o n , and one has (4.31) <l>lo) = -Z*T' Re{(%; * 5"(k.)Dn*Ck0} * Ocr-O, 34 where < > denotes the average taken over a l a r g e region of spacetime. S i m i l a r l y , ( 4 - 3 2 ) <Wl*,o> " T ? M(£%v;-?) ^  D7UD**(ke) (4.33) < i s , „ W > - - Z e 4 ' ^ Re{(v> € ) ^  6"(k.3D"*tk.3} «• 0<„ «• 06r»), (4.34) <ls,o4,0> - - e e ^ ^ ^ R e U ^ ^ ) ^ D ^ D ^ k o ) 3 + - 0 ^ 0 6 - ) . g I f one c o n t r a c t s eqs. (4.31)-(4.34) w i t h the u n i t v e c t o r x / r the angular v a r i a b l e s one obtains (4.35) jVfl-f <4„4,o> - " ^ ¥ ( % » + S ^ ( k o ^ ' C O «- 0<,^ 0O"0, (4.36) j ^ I l f ^ ^ V e U ^(^^ l ^!Vo " (yra. ) * O w * 0 £ r - 0 , (4.37) f j i l =r !„,,!,> - " f e - ^ ^ ( ^ ^ ^ ) P ^ D ^ ( k O «-0<„ •o<r-0 , (4.38) Wf<ls,04,0> = - f e - ^ ^ ( ^ % ^ ) D ^ D ^ ( k e ) + 0,,^O fr-0 , f (2 I T c where da = da d6 s i n 0 ( c f . ( 4 . 2 ) ) , (dfl x m x n / r 2 = (4ir/3) 6 mn S u b s t i t u t i n g eqs. ( 4 .35 ) - (4 .38 ) i n t o eq. ( 4 . 5 ) , one obtains (4.39) k ( r M i l ^ < Z , 0 J > - - kB* D"L\o) D M * 0 O { |( |* .+ £ ) - * e-M 4 '(i-e^-) x By (4.2) and ( 4 . 3 ) , the power l o s t from sources c o n s i s t i n g of a s i n g l e F o u r i e r component i s (4.40) £ - fk ^ D * ( U D " " ( U { 2.(§>* £0 - e-w,k<i-e*Mx r i /fe^.1 \ - j . i t ^ / b i t ° - < - M l <9) For small values of ty . < 0 , one has o ' (4.41) { 6 = ^ ( l - e ^ Y 1 »- o£/) and hence 36 (4.42) -g = I2^k 6 +D M (k 0)D M * (ko) ( l-e^T ' [ l fO( ( i-e^^] * O C „ as tyQ 0_. Thus the power l o s s has a p h y s i c a l l y reasonable s i g n as 0_, although i t becomes i n f i n i t e l y l a r g e . One r e c a l l s that the approximations used to ob t a i n the f i e l d equations (2.10) and (2.11) assumed that ty i s not small ( c f . [12] remarks f o l l o w i n g (4 .21) , remarks f o l l o w i n g (5.2) and (5.10) concerning analogous post-Newtonian equations) and that the approximations are not v a l i d f o r small ty . In p a r t i c u l a r , one cannot assume that q^ = • However, eq. (4.42) gives some hope that energy i s l o s t r a t h e r than gained by g r a v i t a t i o n a l r a d i a t i o n f o r at l e a s t some tyQ. Further work ( f o r example, numerical c a l c u l a t i o n s ) would be needed to v e r i f y that the energy l o s s ~£ remains p o s i t i v e f o r a l l a d m i s s i b l e values of ty^. I n v e r t i n g the F o u r i e r transforms (4.23)-(4.26) g i v e (4.43) 4,oW - i [^D m(x»-i) . B"iW"--5.)I + °*> * • (4.44) 4„to > - t[ ^ T D V i ) - fe^D-fc'-Ul * O^fOCr-L), (4.45) lmi,M - - ^ ° U ~^ D"(x--^ * 5«U'-te + 0^'-i) 3 7 (4.46) * ^ ^ ^ i > ^ ' % ) t | ¥ D ^ 5 > 5 ^ l J " H ) where the dot denote the d e r i v a t i v e . One may use ( 4 . 4 3 ) - ( 4 . 4 6 ) , ( 4 . 5 ) , ( 4 . 2 ) and ( 4 . 3 ) to obtain an expression f o r the energy loss 2f from sources c o n s i s t i n g of a smooth d i s t r i b u t i o n of frequencies. One notes **m o "in o that cross-terms such as D (x -r / v + ) D (x -r/v_) survive i n general, while f o r periodic sources the cross-terms correspond to best periods i n the expression for ~E which average zero over a long time i n t e r v a l . Integrating the equation ( 4 . 4 7 ) T P ° , D - - T P % " T~4v * O i b ) ( c f . [ 1 2 ] ( 4 . 5 ) ) over a l l space, one has ( 4 . 4 8 ) = - I A T ° ° ( , ^ + Ocs) Substituting the so l u t i o n ( 3 . 1 5 ) for ty-tyQ into eq. ( 4 . 4 8 ) , one gets ( 4 . 4 9 ) D'fcO - a^)^d\dh'T°0U,nf^,T00U°,r) 4 . 0 38 for slowly moving sources. .Under the interchange of x and x , the integrand i n (4.49) i s odd, hence the i n t e g r a l vanishes and one has (4.50) 0 ( s ) . Thus to fourth order, the D vanish and (4.51) £ = 0Ca>) . No energy i s l o s t by g r a v i t a t i o n a l r a d i a t i o n i f terms of ninth order are n e g l i g i b l e . 39 5. Conclusions It has been shown that the weak-field approximations to the f i e l d equations have wavelike solutions that correspond to three speeds of propagation. These solutions s a t i s f y ty -> tyQ and q^ 0 at s p a t i a l i n f i n i t y , provided that tyQ i s not too small, and that A > 0 (see remarks following eq. (3.4)). In the slow-motion approximation, the leading term i n the g r a v i t a t i o n a l energy loss i s due to dipole r a d i a t i o n . However, t h i s term vanishes for post-Newtonian sources. The r e s u l t s of t h i s t hesis are not incompatible with the E i n s t e i n quadrupole formula or the measured rate of decrease of the period of the binary pulsar. Higher-order contributions to the energy l o s s need to be computed to give more conclusive r e s u l t s . References [1] CM. W i l l , i n Experimental G r a v i t a t i o n , ed. B. B e r t o t t i (Academic Press, New York, N.Y., 1974). [2] C.W. Misner, K.S. Thorne and J.A. Wheeler, G r a v i t a t i o n (W.H. Freeman, San F r a n c i s c o , C a l i f , , 1973). [3] D.M. Eardley, D.L. Lee, A.P. Lightman, R.V. Wagoner and CM. W i l l , Phys. Rev. L e t t , \30, 884 (1973) . [4] J.H. Tayl o r , L.A. Fowler and P.M.". McCulloch, Nature 277 , 437 (1979). [5] P.C Peters and J . Mathews, Phys. Rev. 131, 435 (1963). [6] P. Crane, J.E. Nelson and J.A. Tyson, Nature 280^ 367 (197 [7] J . E h l e r s , A. Rosenblum, J.N. Goldberg and P. Havas, Astrophys. J . L e t t . 208, L77 (1976). [8] M. MacCullum, Nature 280, 449 (1979) . [9] J . E h l e r s , ed., I s o l a t e d G r a v i t a t i n g Systems i n General  R e l a t i v i t y (North-Holland, Amsterdam, 1979). [10] CM. W i l l , Astrophys. J . 214, 826 (1977). [11] P. R a s t a l l , Can. J . Phys. 57_, 944 (1979). [12] P. R a s t a l l , "Approximate g r a v i t a t i o n a l f i e l d equations," to be published i n Can. J . Phys. (1980). 41 BIBLOGRAPHY Crane, P., J.E. Nelson and J.A. Tyson. Nature 280, 367 (1979). Eardley, D.M., D.L. Lee, A.P. Lightman, R.V. Wagoner and CM. W i l l . Phys. Rev. L e t t . 30, 884 (1973). E h l e r s , J . , ed. I s o l a t e d G r a v i t a t i n g Systems i n General R e l a t i v i t y . North-Holland, Amsterdam, 1979. E h l e r s , J . , A. Rosenblum, J.N. Goldberg and P. Havas. Astrophys. J . L e t t . 208, L77 (1976). MacCullum, M. Nature 280, 449 (1979). Misner, C.W., K.S. Thorne and J.A. Wheeler, G r a v i t a t i o n . W.H. Freeman, San F r a n s i c o , C a l i f . , 1973. P e t e r s , P.C and J . Mathews. Phys. Rev. 131, 435 (1963). R a s t a l l , P. Can. J . Phys. 5_7, 944 (1979). R a s t a l l , P. "Approximate g r a v i t a t i o n a l f i e l d aquations ."To be publ i s h e d . T a y l o r , J.H., L.A. Fowler and P.M. McCulloch. Nature 277, 437 (1979). W i l l , CM. In Experimental G r a v i t a t i o n . Ed. B. B e r t o t t i . Academic Press, New York, N.Y., 1974. W i l l , CM. Astrophys. J . 214, 826 (1977). Appendix: Expressions depending on ty In sections 3". and 5 , several expressions appear which depend on the boundary value ty^. Here the values of tyQ f o r which the expressions are v a l i d are determined. To sim p l i f y notation, define (A.l) r m t«+. so that Z > 0 for any r e a l ty The discriminant A appearing i n eq. ( 3 . 4 ) i s given by (A.2) A - 4^-0 39(?) where 9ft) =• $s * 3 j+ - / 0;r 5 - 14 - 3 T - ° ) . Several cases are considered: 3 ( i ) If 0 < C < 1 , then ( c - l ) < 0 , hence A > 0 (to see that 6 ( 5 ) -•< 0 , one notes that £ 5 < ? 3 and 3c,k < 3 £ 3 together imply that e(r) < - 6 5 s - I 4 j l - 3 c - } < o ) . ( i i X v l f v ? = 1 , then A = 0 43 ( i i i ) For £ > 1, one notes that - 1) > 0. 0(1) = -32 and l i m 0(C) = 0 0. the continuous f u n c t i o n 0(c) must ' . •• have, a t : l e a s t : o n e p o s i t i v e zero. Resorting to numerical computation, one f i n d s the smallest p o s i t i v e zero CA = 2.79. I f 1 < z, < C*., ' then A < 0,. ( i v ) I f ? = ? A , then A = 0. (v) Numerical computation gives *0(?) > 0 f o r £^ < C < 3. For ? > 3, one checks that - |0]f * - I 4 j z - 3? - } > - I t ^ 3 , thus 0(5) > 5 s 1. 3 j * - = t 5 ^ 1 * 3$ - It, ) . One shows that ^ 35" - IL > O for J" > 3 , hence 9 ( j j ) > O . I f C > CA, then A > 0. From cases ( i ) - (v) and eq. ( A . l ) , one concludes that A > 0 i f , and only i f , *\>0 & D ^ {i>0 : < 0 o r 4o >4o* = U ** O.is] . The eigenvalues X + of the matrix A are given by (A.3) \± = i ( - A ± , where /Vr, ) - tCy* + S $-•• - 4X «• 5 ) ( c f (3.4)). One finds that (A.4) A1 -A = ViX?* - T * 1 ") > 0 44 for a l l C > 0, so i n p a r t i c u l a r A > A 2 and hence X + < 0 for ty - e- D. A has an absolute minimum at X, ~ 0.84, for which A - 4.17. Thus A > 0 and X_ < 0 for ty e D. One considers the boundaries of D: 6 ( i ) As 40 -co -» one has, by (AA), A1 - A. -» 0 , hence ; + -.O and" A_ -» -ACO) = . ( i i ) As 4» -» 0_ (r-* i_ one has A-^O, hence A t - 1 A(0 = - 2.. ( i i i ) As 4o ••'r'o* 5* + )» O M e has A - » 0 , h e r i c e A ± ^» (iv) As • 4* *o (t;-*«o), one has A- i • 6 S _ t +0(5-*)] , The propagation speeds v + are given by ( c f . (3.5)) (A.5) v ± . ( S * + o v V - 7 » t V V r . Since C 2 + 1 > 0 for a l l £ , and \ + < 0 for ^  & D, one can take the • v + r e a l and p o s i t i v e for Q^.--:6 D. The l i m i t i n g values of v + corresponding to the boundaries of D are: (i ) As ^ o - * - * 0 one. Was y 1 + l -» I , hence -vt-*JUm (-atYVl", i . e . , v+ -» » , v_ (%•)"''* « o . e i . ( i i ) As -v[/0 -* C>_ ( "5 -» I_ ), one has J' 1 + I -» Z , hence -vr^  -* I. ( i i i ) As -k-»4v*+ ( s -»3« + ) , v * -» *• 0 / A ( ? * ) ] ' / l « 0.56 . (iv) As -fy.-*- xr + = i *• OCj-') , v . - Z ^ s ' 1 -OCr*)-45 The constants a+ and B i n the P i n eq. (3.6), and the c o e f f i c i e n t s y + and <5+ appearing i n eqs. (3.11) and (5.3) are " a l l w e l l - d e f i n e d f o r ij; .& D. Note that s i n c e 0 ^ D, one has B 4 0 and det P 4 0 f o r ty:& D. 

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