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UBC Theses and Dissertations

Gravitating effect of gravitation. Lam, Dominic Man-Kit 1967

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THE GRAVITATING EFFECT OF GRAVITATION BY DOMINIC MAN-KIT LAM B.Sc, l a k e h e a d U n i v e r s i t y , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF PHYSIOS We a c c e p t t h i s t h e s i s as c o n f o r m i n g to t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER,1967 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 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 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 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 n o t 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 . Depa r t m e n t 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, C a n a d a A B S T R A C T The work r e p o r t e d i n t h i s t h e s i s i s "based on the v i e r b e i n f i e l d f o r m u l a t i o n o f g r a v i t a t i o n a l t h e o r y , used i n c o n j u n c t i o n w i t h the method o f the compensating f i e l d . I t i s shown t h a t the most g e n e r a l l i n e a r e q u a t i o n s of second o r d e r f o r a t e n s o r f i e l d , w h i c h a r e i n v a r i a n t u nder o r i e n t a t i o n s o f t h e l o c a l i n e r t i a l frame and under gauge t r a n s f o r m a t i o n s o f the v i e r b e i n f i e l d components' a r e i d e n t i c a l w i t h E i n s t e i n ' s f i e l d e q u a t i o n w r i t t e n down i n t h e weak f i e l d a p p r o x i m a t i o n . An a t t e m p t i s made to t a k e i n t o a c c o u n t any p o s s i b l y e x i s t i n g g r a v i t a t i n g e f f e c t o f g r a v i t a t i o n by a p p l y i n g the method o f the com-p e n s a t i n g f i e l d t o t h e weak f i e l d L a g r a n g i a n , r e s u l t i n g i n a s e t of n o n l i n e a r f i e l d e q u a t i o n s . The i n v a r i a n c e p r o p e r t i e s o f the m o d i f i e d f i e l d e q u a t i o n s a r e examined, and some s p e c i a l s o l u t i o n s a r e e x h i b i t e d . - i i -i i i CONTENTS - ,' •. ABSTRACT . . . i i ACKNOWLEDGMENTS i v 1. The Vierbein description of the g r a v i t a t i o n a l f i e l d . . 1 2. The method of the compensating f i e l d ....9 3. Construction of f i e l d equations.... .......13 4. Special solutions to the f i e l d equations 19 BIBLIOGRAPHY. 25 Appendix- Derivation of some formulae i n chapter 2 26 Acknowledgements I am indebted to Dr. F. A. Kaempffer f o r sug-gesting the topic of th i s thesis and f o r his stimulating supervision and invaluable a i d . I wish to thank Miss Y. H. Wong f o r her most p r o f i c i e n t typing. I am also g r a t e f u l f o r a scholarship given me by the National Research Council. 1 1 . The V i e r b e i n d e s c r i p t i o n o f the g r a v i t a t i o n a l f i e l d . I n d e s c r i b i n g the g r a v i t a t i o n a l f i e l d , one u s u a l l y c h a r a c t e r i z e s i t b y a m e t r i c f i e l d , , s u c h t h a t t he t r a j e c t o r i e s o f t e s t p a r t i c l e s a r e g e o d e s i e s i n t h a t f i e l d . As i s w e l l known ( s e e , f o r example , Landau and L i f s h i t z , 1959) E i n s t e i n p r o p o s e d i n 1916 as f i e l d e q u a t i o n s f o r the f r e e g r a v i t a t i o n a l f i e l d R^x>=° ( 1 . 1 ) T h i s a p p r o a c h has the o b v i o u s d i s a d v a n t a g e t h a t t h e r e i s no s i m p l e c r i t e r i o n f o r t e l l i n g w h e t h e r a g i v e n m e t r i c t e n s o r d e s c r i b e s a t r u e g r a v i t a t i o n a l f i e l d , a p s e u d o - g r a v i t a t i o n a l f i e l d , o r a m i x t u r e o f b o t h , w h e r e , b y d e f i n i t i o n , p s e u d o - g r a v i t a t i o n a l f i e l d s c a n be t r a n s -fo rmed away g l o b a l l y , and t r u e g r a v i t a t i o n a l f i e l d s can o n l y be t r a n s f o r m e d away l o c a l l y . A f a m i l i a r example o f p s e u d o - g r a v i t a t i o n a l f i e l d s a r e the c e n t r i f u g a l and C o -r i o l i s f i e l d s e n c o u n t e r e d i n c o o r d i n a t e sys t ems r o t a t i n g w i t h r e s p e c t t o a N e w t o n i a n frame o f r e f e r e n c e . I n the p r e s e n t v/ork, a new a p p r o a c h b a s e d on an i d e a by E i n s t e i n (1928) w i l l be a t t e m p t e d i n o r d e r t o a v o i d t h e above d r a w -back and o b t a i n , i f p o s s i b l e , s i m p l e r f i e l d e q u a t i o n s . F i r s t l y , one n o t e s t h a t i n the t h e o r y o f g r a v i -t a t i o n , one can a l w a y s f i n d a t l e a s t one l o c a l i n e r t i a l 2 frame by g o i n g i n t o f r e e f a l l . T h i s means t h a t one can always l o c a l l y t r a n s f o r m away a l l b u t the d i a g o n a l terms o f t h e m e t r i c t e n s o r . D e n o t i n g by c o o r d i n a t e s o f the c u r v i l i n e a r continuum and by c o o r d i n a t e s o f the l o c a l i n e r t i a l frame (.<j4=Jit) one can i n t r o d u c e now the t r a n s f o r m a t i o n f u n c t i o n s c o n n e c t i n g the d i s p l a c e m e n t s <Lx* ( l e 2 ) The main purpose o f t h i s f i r s t c h a p t e r i s t o deve-l o p a f o r m a l i s m e n a b l i n g one t o d e s c r i b e a l l the m e t r i c p r o p e r t i e s i n terms o f t h e s e f u n c t i o n s , w h i c h were g i v e n the name " v i e r b e i n " f i e l d by E i n s t e i n i n h i s o r i -g i n a l paper on t h i s s u b j e c t . By assuming t h a t the d e t e r m i n a n t J |^, does n o t v a n i s h , one ensures the e x i s t e n c e o f the i n v e r s e f u n c t i o n s f d e f i n e d by <tx* = - f * ^ dy* ( i . 3 ) w h i c h a r e c o n n e c t e d , because o f (1.2) , w i t h the -ft* by rf B < ( «* }k-h = <SS (1.4) Hence where (u«J i s the c o f a c t o r o f . 3 Once the v i e r b e i n f i e l d i s g i v e n , one can e a s i l y d e t e r m i n e whether i t d e s c r i b e s t r u e g r a v i t a t i o n a l f i e l d s o r n o t , s i n c e i f the a r e i n t e g r a b l e , t h a t i s i f where a v e r t i c a l b a r denotes p a r t i a l d i f f e r e n t i a t i o n , one can f i n d a g l o b a l i n e r t i a l frame ^ = | C^O a c ~ c o r d i n g l y l o o k upon the m e t r i c f i e l d d e s c r i b e d by the i n t h i s case as r e p r e s e n t i n g a p s e u d o - g r a v i t a t i o n a l f i e l d . Any t r u e g r a v i t a t i o n a l f i e l d i s t h e n c h a r a c t e r i z e d by the n o n i n t e g r a b i l i t y o f the TL <* . T h i s s u g g e s t s d e f i n i n g t h e f u n c t i o n s s 'ft "~ "ft p |oC (1.6) as t h e " t r u e g r a v i t a t i o n a l f i e l d s t r e n g t h s " , and the non-i n t e g r a b l e f u n c t i o n s -fl £ as " g r a v i t a t i o n a l p o t e n t i a l s " . T h i s i s done t o keep a c l o s e a n a l o g t o the w e l l known case o f e l e c t r o m a g n e t i c t h e o r y , where the " f i e l d s t r e n g t h s " Fp<^  = A*ip - Ap/* ( i.7) a r e d e r i v a b l e from the p o t e n t i a l s by d i f f e r e n t i a t i o n . The a r e a s e t o f s i x t e e n i ndependent f u n c t i o n s p e r s i s t i n g o f f o u r f o u r - v e c t o r s -ft* , fi* ft * ,-f\* whose components t r a n s f o r m under r e o r i e n t a t i o n o f the l o c a l i n -e r t i a l frame a c c o r d i n g t o — * v - R i * i (i .e) where i s an o r t h o g o n a l m a t r i x , t h a t i s 4 R; =5J <I.9) Because of the experimentally well established isotropy of i n e r t i a (Hughes et. a l . I960, Drever 1961) one must i n f a c t require invariance of the f i e l d equations under the transformation (1.8). In order to eliminate pseudo-gravitational e f f e c t s , one must further require invariance under the transforma-t i o n * * — < + d . i o ) Y/ith a r b i t r a r y four-vectors A • This i s the exact ana-l o g to the "gauge invariance" of electrodynamics that i s , under transformations . A * A* + A , * of the potentials which leave f ^ i n v a r i a n t . Once the Ty^ are known, one can compute the me-t r i c tensor. In the following, l o c a l i n e r t i a l frames w i l l be taken to be cartesian coordinate systems, and thus one may take as metric of the l o c a l i n e r t i a l frame | j , The invariant l i n e element ds = biA d y v d y (1.12) -can then be expressed i n terms of dx^dx by (1.2) as dS* = A i - Z l J c L x ^ x ' . ( 1 . 1 5 ) ' Since also ds2 = Jocp d x ^ o t c c 1 (1.14) one has, by comparison S i m i l a r l y , one obtains rf = f*f?* a.") Thus one sees that the components of are f a c -torise d by the functions " f t ^ . Moreover, i f the vierbein f i e l d i s given, one can unambiguously compute the metric tensor. The converse, however, i s obviously not true. The encourges one to take the view that the vier b e i n f i e l d i s more fundamental than the metric tensor when one wishes to describe g r a v i t a t i o n a l e f f e c t s . With a l l these tools i n hand, one can now rework the theory of general r e l a t i v i t y i n terms of the vierbein f i e l d . For instance, the a f f i n i t i e s 6 r e a d and hence the e q u a t i o n o f the g e o d e s i e s , f o r example, (1 .19 ) can be r e p r e s e n t e d i n terms o f the v i e r b e i n f i e l d by (1,18) F u r t h e r m o r e , l o o k i n g a t e q u a t i o n (1,18) i t i s i n t e r e s t i n g t o n o t e t h a t one can i n f a c t s e p a r a t e out t r u e g r a v i t a t i o n -a l f i e l d s from p s e u d o - g r a v i t a t i o n a l f i e l d s i n the e q u a t i o n of the g e o d e s i c (1,19). T h i s d i s c o u r a g e s the v i e w , t a k e n by some a u t h o r s , t o l o o k upon t h e as " f i e l d s t r e n g t h s " s i n c e t h e may a c t u a l l y c o n t a i n b o t h t r u e g r a v i t a t i o n -a l and p s e u d o - g r a v i t a t i o n a l f i e l d s . As an i l l u s t r a t i o n , c o n s i d e r the l o c a l t r a n s f o r m a -t i o n s between an i n e r t i a l (d>"X?) and a r o t a t i n g c o o r d i n a t e (ji- 1^ ) system g i v e n by I \ o o  o 0 0 0 I I (1.20) w i t h d e n o t i n g the rows and the. eolurorfcS. By d i r e c t d i f f e r e n t i a t i o n , one f i n d s t h a t # f •„ = fiti * (1.21) 7 which mean that the -ft^ are integrable and hence a global transformation between the i n e r t i a l and ro t a t i n g coordinate system should be possible. This i s indeed the case, and the connection between the two systems i s established by the well-known transformtaion equation j j a = - z ' M tl u>x*) + X 2 CJ*>(£ COX*) ( 1 . 22 ) From equation ( 1 . 1 5 ) , one can obtain the metric tensor: ()f.w*;«<„)-f ' 0 o 0 1 0 - i w z' 0 0 1 0 ( 1 . 23 ) and notes that i t does not indicate e x p l i c i t l y whether the f i e l d i s true g r a v i t a t i o n a l or pseudo-gravitational, whereas the mere f a c t that A^^Ap^ implies immediately that the f i e l d must be pseudo-gravitational. For most p r a c t i c a l applications, one i s interested only i n weak g r a v i t a t i o n a l f i e l d s , that i s , i n small de-8 viations from the Euclidean metric. This can be done by writing: *j +fit (1.24) with an expansion parameter <j-, introducing thus new v a r i -ables I* . By comparing this with = ^ + (1.25) where i s the deviation from the pseudoeuclidean me-t r i c , one has, by (1.15), the r e l a t i o n Using (1.4), one can also express the <f ^  i n terms of the Is Is important to r e a l i s e that the do not from a symmetric c o e f f i c i e n t scheme. However, as (1,26) shows, i n l i n e a r approximation, only the symmetric part of contributes to the metric. Effects from the skew part of the vierbein f i e l d can show up only v/hen terms quadratic i n ^ are taken into account. 9 2. The method of the compensating f i e l d . Prom the discussion of the preceeding chapter,it i s obvious that any action p r i n c i p l e ( i t f i t , p = B X T R £ M U M (2.1) from which the f i e l d equations follow as Euler-lagrange equations PoC _ f<><C ) _ A ^i~\mJ = 0- <2-2) cannot be demanded to be invariant under global Lorentz transformations of the coordinates , because l o c a l i n e r t i a l frames i n the presence of g r a v i t a t i o n are, i n general, accelerated with respect to each other. One should, however be able to i n s i s t on invariance under l o -c a l Lorentz transformations characterized by s i x coordinate dependent parameters ^^(x)''^^ This invariance requirement acts as a constraint on the possible coupling between the g r a v i t a t i o n a l f i e l d and a given source f i e l d , as was f i r s t shown by Utiyama (1956). Just as the so-called "minimal coupling" between the electromagnetic potentials A^ , and a source f i e l d can be derived (London, 1927) by demanding that the e f f e c t of any phase transformation with coordinate dependent phase Afr) on the f i e l d ^ be compensated by a gauge-trans-formation A e (-*^ +* ,A(0 among the potentials, requiring a l l derivatives of ^* to occpur i n the conduction <Vf r Tfj< + U i^P 10 with a coupling parameter & to he determined by experi-ment so the "minimal coupling" between the g r a v i t a t i o n a l potentials v l * and a source f i e l d ^ can be derived from demanding that the relevant action p r i n c i p l e be invariant under l o c a l Lorentz transformation, requiring a l l derivatives of y to occur i n the conjunction where A m n i s the appropriate operator representation of A_m > 1 (Freeman, 1967) acting on the components of ^ , and where the components of the "compensating f i e l d " #^n(x) - -B^i*) transform according to c - s r + o * n + A * 6 : * + A m v (2.4) one can show (Kaempffer, 1965) that the derivatives (2.3) are i d e n t i c a l with the components of the covariant d e r i -vatives of \js i n the l o c a l i n e r t i a l frame, and obtain the r e l a t i o n s O ^ - r j L - t S V - (2.5) connecting the f i e l d s f>^ ° with the quantities describing the metric f i e l d . Using the r e l a t i o n s (1.4), one can solve (2.5) f o r the f i e l d s 6 ^ n » and upon substitution of the expression (1.18) f o r the a f f i n i t i e s one obtains (Appendix A.a) 11 showing n i c e l y that only true g r a v i t a t i o n a l f i e l d s (1.6) contribute to the in t e r a c t i o n terra i n (2.3), In p a r t i c u l a r , one obtains upon substitution of the tensor representation (Freeman, 1967) into (2.3) f o r the case of a tensor f i e l d T^)f- the ex-pression (Appendix A.b) ' ' (2 . 8 ) B y s i m i l a r procedure, f o r the case of a vector f i e l d with the vector representation A I ™ • ^ = o' w p (2.7a) one obtain the r e l a t i o n a.vvHr,.fV^^V+ffV»+r? V ' H ( 2- 8 a ) The expression (2 . 8 ) should enable one to incorporate the g r a v i t a t i n g e f f e c t of g r a v i t a t i o n into the theory i n the following manner. Suppose one has a Lagrangian L = L ( } f | f ; > f ) (2.9) y i e l d i n g the correct l i n e a r f i e l d equation f o r weak gra-v i t a t i o n a l f i e l d s . I f now this f i e l d T . s T i s i t s e l f 12 treated as a source of gra v i t a t i o n , one must apply (2.8) to ^ and construct according to Utiyama's p r e s c r i p t i o n , the Lagrangian containing the i n t e r a c t i o n term "by writing X = fl L (2.10) where L? i s obtained from l _ by replacing everywhere ^dplr ^ v dr7]^ • By m e a ^ s of the expressions (l:.24) and , (1.27)j one should then be able, i n p r i n c i p l e , to construct the Lagrangian to any order i n the coupling parameter ^ • To t h i s end, one needs the expansion (Appendix A.c): i -1 + f 7 w * $ * l - < V 7 i f - 3 ( h ? - ( 2 . i i ) and (Appendix A.d) M w r ( V i w t F + > f7 M r ) ^ ( 2 , 1 2 ) where TJ ^ S - < J « H F . which are obtained by s u b s i t i t u t i o n of (1.24) and (1.27) into the expression f o r and ^ r 1 ] ^ > and c o l l e c t i o n of the appropriate terms, 1 3 3. C o n s t r u c t i o n o f f i e l d e q u a t i o n s . As i s well-known, M a x w e l l ' s vacuum f i e l d e q u a t i o n s a r e t h e o n l y r e l a t i v i s t i c a l l y i n v a r i a n t l i n e a r e q u a t i o n s o f second o r d e r f o r a v e c t o r f i e l d s a t i s f y i n g t he c o n d i t i o n o f gauge i n v a r i a n c e . R e s t r i c t i o n t o l i n e a r i t y and second o r d e r o f the f i e l d e q u a t i o n s i m p l i e s t h a t t h e l a g r a n g i a n must he b i l i n e a r i n the p o r t e n t i a l s A<* and t h e i r f i r s t d e r i v a t i v e s . Now w i t h a v e c t o r f i e l d one can form f o u r l i n e a r l y i n dependent i n v a r i a n t s o f t h i s t y p e , namely i.- A A ; i^AotifAif > r3s^i<A fip; ^ ^ i p V (5.i) S i n c e I 4 d i f f e r s from T3 o n l y by a d i v e r g e n c e : i t s c o n t r i b u t i o n t o an a c t i o n p r i n c i p l e w i l l be t h e same as t h a t o f I 3 , and one has thus as most g e n e r a l L a -g r a n g i a n L * c , I , + c a I a + C 3 l a w i t h a r b i t r a r y c o e f f i c i e n t s , y i e l d i n g l i n e a r f i e l d e q u a t i o n s o f second o r d e r s I m p o s i t i o n o f i n v a r i a n c e under gauge t r a n s f o r m a t i o n s ky^^ 14 with a r b i t r a r y scalar f i e l d A yiel d s the conditions c, =o and cz = 'C3 , This reduces the f i e l d equations (3.4) to Maxwell's equations: . A^ j^ p - Apj^ - =o (3.5) Reduction to wave equations A*ie0 = 0 (3.6) i n the potentials i s accompanied by imposition of the t r a n y e r s a l i t y condition . A f i ^ = o (3.7) which eliminates the unphysical l o n g i t u d i n a l and timelike p o l a r i z a t i o n modes of the electromagnetic f i e l d i n the usual fashion (Kallen, 1958) An e n t i r e l y analagous treatment of a general ten-sor f i e l d ^ c / ^ of rank two v / i l l now be carried out, with the intention to in t e r p r e t the r e s u l t i n g l i n e a r f i e l d e-quations of second order as the g r a v i t a t i o n a l f i e l d equations f o r weak f i e l d s , which are of lowest order i n the expansion parameter ^ introduced i n chapter 1. One can form fourteen l i n e a r l y independent i n -variants b i l i n e a r i n the f i e l d s and t h e i r f i r s t de-r i v a t i v e s , namely: 15 J 7 = 7^/r|c<-Mf J I B = |o<^ jr7rc<ip ; I « j = 7*^1 ff\< (3 .8) S i n c e I„ , Il2 , I„ , I,4 , d i f f e r from I, , I, , I*, , I„ r e s p e c t i v e -l y o n l y by d i v e r g e n c e s : ' r (3 .9 ) j»4=i,o + n ^ V f 'f' v i fnfi \f t h e y need n o t be t a k e n i n t o t h e a c t i o n p r i n c i p l e s e p a r a t e l y , and t h e most g e n e r a l l a g r a n g i a n l e a d i n g t o l i n e a r e q u a t i o n o f second o r d e r i s t h e r e f o r e : L = £ c t lL (3.10) w i t h a r b i t r a r y c o e f f i c i e n t s . I n o r d e r t o i n c o r p o r a t e t h e i s o t r o p y o f i n e r t i a i n t o t h e t h e o r y , one r e q u i r e s i n v a r i a n c e o f A\ under t h e t r a n s f o r m a t i o n ( 1 . 8 ) , t h a t i s : 16 I n l i n e a r a p p r o x i m a t i o n , A 4 « - i * < + i l * * (3.i2)' t h e c o e f f i c i e n t s o f R- { mus t , so as n o t t o v i o l a t e t h e l i n e a r a p p r o x i m a t i o n , a l s o "be expanded as S*X + (3.13) 8 , w h e r e , as i s w e l l - k n o w n , i i s skew. S u b s t i t u t i n g ( 3 . 1 2 ) and ( 3 . 1 3 ) i n t o ( 3 . 1 1 ) , and d r o p p i n g a l l t e rms q u a d r a t i c i n s m a l l q u a n t i t i e s , one o b t a i n s as t h e t r a n s f o r m a t i o n l a w f o r t he q u a n t i t i e s o f u n d e r r e o r i e n t a t i o n o f t h e l o c a l i n e r t i a f r a m e . As i s p e r m i s s i b l e i f one u se s i n s t e a d o f I L O , h o s i ( J H , + i » ) * i l « « i + t i n l f + ^ } f ) ( 3 . 1 ? ) t h e n t h e t r a n s f o r m a t i o n ( 3 . 1 6 ) on t h e L a g r a n g i a n o b v i o u s l y l e a v e s Za , L ^ , I ) 0 , i n v a r i a n t . The r e m a i n d e r g i v e s • C | s C 4 i C4 = C$ '> *C7=C8-zCq ( 3 . 1 8 ) These have t h e e f f e c t o f l e t t i n g ^ a p p e a r i n L o n l y i n t h e s y m m e t r i c c o m b i n a t i o n T <? S 7 * ? 4 " 7 ^ ( 3 . 1 9 ) 17 and thus a l l o w one t o r e s t r i c t c o n s i d e r a t i o n t o f A i o C i f ^ ^ (3.20) where A^ a r e a r b i t r a r y c o n s t a n t s . T h i s l a g r a n g i a n i s i d e n t i c a l w i t h the one c o n s i d e r e d by Fyss (1965). Upon v a r i a t i o n o f (3.20) w i t h r e s p e c t t o 1~lc^  , the f i e l d e q u a t i o n s f o l l o w as " A ( 0 CT^\rr + ^ pKrO = ° (3.21) I m p o s i t i o n o f i n v a r i a n c e under gauge t r a n s f o r m a t i o n s t ^ — + A « , P + A j^oc (3.22) f o r t h e purpose o f e l i m i n a t i n g p s e u d o - g r a v i t a t i o n a l f i e l d s r e q u i r e s t h a t upon s u b s t i t u t i o n o f (3.22) i n t o (3.21), t h e c o e f f i c i e n t s o f A^/p , A T T ^ ^ , /Jf/ocpf v a n i s h s e p a r a t e l y , and t h i s g i v e s t h e c o n d i t i o n s A » - Aa - o j A 4 * A 7 = o (3.23) ^ A ^ + A , e = o ; A 7 + A , o = o 18 l e t t i n g A.—I > one has A 7 = -z ; ; A , e = z ( 3 . 2 4 ) Thus, t h e l a g r a n g i a n r e d u c e s t o U * \ ^ l t ( "I*p|f * 2 VlfO - l^jrC]^Jt- *"|f Hp) (3.25) The c o r r e s p o n d i n g r e d u c e d f i e l d e q u a t i o n s now do n o t con -t a i n any a r b i t r a r y parameters and a r e e x p r e s s i b l e e n t i r e l y i n terms o f the t r u e g r a v i t a t i o n a l f i e l d s t r e n g t h s : 7 * f M f - *l<tlfr+)«fl«+s$*,fiir+ZrM? = 0 (3.26) where one d e f i n e s 'J<^  as g r a v i t a t i o n a l p o t e n t i a l s f o r weak f i e l d s due t o t h e f a c t t h a t (3.27) The a n a l o g y t o the c o r r e s p o n d i n g M a x w e l l ' s e q u a t i o n s (3.5) i s t h u s b r o u g h t i n t o e v i d e n c e and f u r t h e r j u s t i f i e s t h e usage o f the terms " G r a v i t a t i o n a l p o t e n t i a l s " f o r and " g r a v i t a t i o n a l f i e l d s t r e n g t h s " f o r . One a l s o n o t e s t h a t c o n t r a c t i o n o f the f i e l d s t r e n g t h s w i t h r e s p e c t t o t h e i n d i c e s «< and f i m m e d i a t e l y l e a d s t o r / f = 0 (3.28) 19 4 . Special solutions to the f i e l d equations. The f i e l d equations (3.26) agree i n l i n e a r ap-proximation with those of E i n s t e i n . For example, i f one takes as g r a v i t a t i o n a l potentials * ' r ~ ' ' 2 ' 3 (4.1) where cj? and ^ are scalar p o t e n t i a l s , then s u b s t i t u t i o n of (4.1) into the lagrangian (3.25) and the f i e l d equations (3»26) respectively give $]fr^O (4.4) Specifying to spherical symmetry, the f a c t that the f i e l d equations obtained by the "vierbein" f i e l d approach agree, at l e a s t i n l i n e a r approximation, with Einstein's f i e l d equations enables one to apply Birkhoff's theorem, which states that a l l s p h e r i c a l l y symmetric g r a v i t a t i o n a l f i e l d s are s t a t i c (see, f o r example, Tolman 1958), to this case, that i s (4.5) In terms of spherical coordinates: 20 The f i e l d e q u a t i o n s ( 4 . 3 ) and ( 4 . 4 ) r educe t o a p a i r o f n o n - l i n e a r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s t h a t can he s o l v e d r e a d i l y , g i v i n g s o l u t i o n s : <f> = A + • £ • ( 4 . 8 ) • = 6 + § ( 4 . 9 ) where A , B , C a r e c o n s t a n t s . C l e a r l y , f a r away f rom the "body p r o d u c i n g the f i e l d , t h e g r a v i t a t i o n a l f i e l d must he N e w t o n i a n , t h a t i s (f) I ; Jx \ cu ti-*tx> ( 4 . 1 0 ) Thus , ( 4 . 8 ) and ( 4 . 9 ) become <f> = * + TT ( 4 . 1 1 ) JJL = 1 - ^ ( 4 . 1 2 ) . These s o l u t i o n s a r e c l e a r l y i d e n t i c a l i n N e w t o n i a n a p p r o -x i m a t i o n w i t h the s o l u t i o n s o b t a i n e d by S c h w a r s c h i l d (1916) u s i n g E i n s t e i n ' s f i e l d e q u a t i o n s u n d e r the same c o n d i t i o n s . Now, i n ' a c c o r d a n c e w i t h , t h e p r e s c r i p t i o n d e v e l o p e d i n c h a p t e r 2 , t he g r a v i t a t i n g e f f e c t o f g r a v i t a t i o n can he t a k e n i n t o a c c o u n t by r e p l a c i n g i n the L a g r a n g i a n ( 3 . 2 5 ) the p a r t i a l d e r i v a t i v e s ^o<p|f by oV^^ , and f o r m i n g 21 <£ = H L' (4.13) w i t h the r e s u l t , up t o o r d e r ^ , where the ^ ocpj-p i n L a r e symmetric. The c o r r e s p o n d i n g f i e l d e q u a t i o n s a r e the n , c « 7 - A 7 ) w f f / i ^ K l o - ] +^6a1_w( 7)wff£ + 2 > ) ^ w e - r ) w ^ | o - ) + 2 7 o r £ l u > 0 j w , f f e + Ifi^ tru) ' (4.15) The L a g r a n g i a n (4.14) i s n o t r e o r i e n t a t i o n i n v a r i a n t . Indeed, i f one per f o r m s the t r a n s f o r m a t i o n (3.16), t h e n (4.14) becomes 22 O b v i o u s l y , t he L a g r a n g i a n (4.14) c a n n o t be made r e -o r i e n t a t i o n i n v a r i a n t s i n c e no a r b i t r a r y c o n s t a n t a p -p e a r i n the e x p r e s s i o n ( 4 . 1 6 ) . However one c a n a l w a y s f i n d a gauge so as t o ensu re t h e r e o r i e n t a t i o n i n v a -r i a n c e o f t he f i e l d e q u a t i o n s ( 4 . 1 5 ) . I n o t h e r w o r d s , one has t o s a c r i f i c e t h e gauge i n v a r i a n c e so t h a t r e -o r i e n t a t i o n i n v a r i a n c e o f the f i e l d e q u a t i o n s ( 4 . 1 5 ) may be p r e s e r v e d . I n t h i s r e s p e c t , one n o t e s t h a t E i n -s t e i n ' s f i e l d e q u a t i o n s a r e a l s o gauge i n v a r i a n t o n l y i n l i n e a r a p p r o x i m a t i o n . As i s d i s c u s s e d i n c h a p t e r 2 , a p p l y i n g the i d e a o f t he c o m p e n s a t i n g f i e l d t o the weak f i e l d p o t e n t i a l s s h o u l d i n p r i n c i p l e e n a b l e one t o c o n s t r u c t t he l a g r a n g i a n t o any o r d e r o f a c c u r a c y i n the p a r a m e t e r , t hus p r o v i d i n g a method o f o b t a i n i n g t h e f i e l d e -q u a t i o n s b y s u c c e s s i v e a p p r o x i m a t i o n s . C o n s i d e r as an example the case when ^ocp i s d i a g o n a l , t h a t i s l - p - k P + ( 4 - 1 7 ) where i s a s c a l a r p o t e n t i a l . S u b s t i t u t i o n o f ( 4 . 1 6 ) i n t o ( 4 . 1 4 ) and ( 4 . 1 5 ) . r e s p e c t i v e l g i v e ^ - V ^ O + a ^ ( 4 ' 1 8 ) 23 t in- n^hr=° ( f-w) By s i m i l a r procedure as that f o r (4.1), one obtains, as solutio n f o r the scal a r p o t e n t i a l ( | ) , | ' ^ ^ ( 4 . 2 0 ) where A i s a constant. Considering next the ease when I * , f = 1,3,3 > P - { 7 (4-2i) where <£> and are scalar p o t e n t i a l s . One finds that the f i e l d equations (4.15) become ^<W + 4/W = f (~3 W ' % / V ' 4/V/]f) (4.22) *htt= ? e-2 W +>/v) (4-23) which c l e a r l y give (4.19) when <^-/-t' • Substitution of the solutions (4 .11) and (4.12) into the quadratic terms of (4.22) and (4.23) then y i e l d 4* = ' * 4 + ( 4 . 2 4 ) where A, B, are constants. These solutions, however, d i f f e r i n order -^ with Schwars-ch i l d ' s solutions: 4>s ' = ' + 4 + (4.26) 24 A = ' "4 + ^ (4.27) T h i s s u g g e s t s t h a t t h e l a g r a n g i a n (4.14) he f u r t h e r mo-d i f i e d by r e p e a t i n g the method o f the compensating f i e l d . The m a t h e m a t i c a l c o m p l i c a t i o n s a r e , hov7ever, beyond t h e scope o f t h i s t h e s i s . 25 BJBTiTOGRAPHY D r e v e r , R. W. P., (1961), P h i l . Mag. 6, 683. E i n s t e i n , A., (1928), S. B. Akad. Wiss. B e r l i n , 217. Freeman, M., (1967), Ph. D. T h e s i s , U. B. C , C h a p t e r 3. Hughes, V. W.. R o b i n s o n , H. G., and B e t r o n - L o p e z , V., (I960), Phys. Rev. L e t t e r s 4_, 342. K a e m p f f e r , F. A., (1965), Concepts i n Quantum M e c h a n i c s , S e c t i o n 22, Academic P r e s s . K a l l e n , D. E., (1958), E n c y c l o p e d i a o f P h y s i c s , V o l . V, p a r t 1, 199. Landau, L. D., and L i f s h i t z , E, M., (1959), The C l a s s i c a l  Theory o f F i e l d s , 269, 299, A d d i s o n Wesley. London, I., (1927), £eits. F. P h y s i k 42,, 375. S c h w a r s c h i l d , K., (1916), B e r l . B e r . , 189. Tolman, R. C , (1958), R e l a t i v i t y , Thermodynamics and  Cosmology, 252, O x f o r d P r e s s . U t i y a m a , R., (1956), Phys. Rev. 101, 1597. ?/yss, W., (1965), H e l v e t i c a P h y s i c a A c t a , 38, 469. 26 A p p e n d i x - D e r i v a t i o n o f some f o r m u l a e i n c h a p t e r 2. a , By (2.5), one has " = f „ B j1 ( A . I ) S i n c e f * * * = T , ; *l {^ b\ hence, m u l t i p l y i n g ( A . l ) "by -A^ , one o b t a i n s M u l t i p l y i n g (A.2) by { ?^ , one has A l s o , S u b s t i t u t i n g (A.4) i n t o (A.3) A f t e r s l i g h t s i m p l i f i c a t i o n , one o b t a i n s w h i c h i s (2.6) b, l e t t i n g be T^p , one h a s , by (2.3) where and ^^•tffV^V^^?^"^^ <A-7) Multiplying the f i r s t terra of (A.S) by (A.7), and T^^ one has " ( t ^ t ^ - r ^ A ^ U n + t TV ^ (r^y 'ocn j + ( f + f fl" GrZs C T W ( J ) - a(f ^  ? Q ^ ) ( f ? + f ^  T^) (A.8) Prom the second and t h i r d terms of (A.6), one s i m i l a r l y obtains f ^ V ^ f W C ^ p , f w T ^ t + f ^ t - r ^ - f (A.9) and "**;f - G | ^ r u ^ + f $ ^ w f ^ p + f ^ / p i o ^ p (AilO) Since (A.9) and (A.10) are equal, thus one has, from (A.5) 28 By renaming th e r u n n i n g s c r i p t s and a f t e r s l i g h t r e -a rrangements, one o b t a i n s : w h i c h i s (2.8) c , S i n c e C o n s i d e r 71 = 3 , one has 4 ; fx 2 iiax -11 a) - nwx ~k)r^K (mi -fi\%\) B u t = +^71* hence expanding -/l up t o o r d e r o^2 , one o b t a i n s < - u M pun 'ftVU&il) <K >}f,)(H*i-p] One can g e n e r a l i s e t h i s e x p a n s i o n t o t h e c a s e f o r any i n t e g e r n » g i v i n g A = ' + f ( - ? ) * ( A " B j 2 ( A . l l ) 29 where Now, since • ^"J^A+iS (A.12b) thus solving f o r A and B i n (A.12a), (A.12b) and sub-s t i t u t i n g the values f o r A, B into (A.11), one obtains One notices that the running s c r i p t s of the nj's can a l l be made subscripts by m u l t i p l i c a t i o n with the appropriate & 's . Thus This completes the derivation f o r the expression (2.11). d, To obtain an expression f o r ^ j - 1 ] ^ up to order , one f i r s t expands i n terms of the ^ s . Prom (1.24) and (1.27), 30 one n o t e s t h a t Gr^ p-p- = (^) ^OL^V S i n c e and hence Now ( A . 1 5 ) ^r1*,p~ ^ ( S / f " J ^ r- A hnjocp ^ ^ ^ C '' ( A . 1 6 ) where A „ „ ; , p ^ =isa Y ^ - ^ p ^ ) + y & X + ^ A ^ ( A . I ? ) Thus 31 7 *>« |57 ^  * *+7^(3)71, * r + f 7 ^  ^  + 706 » 7 r> ^ + 1 ** 1 S i m i l a r l y 7*n 7^  ^ 7w'(3,i+y^^7w7'7u^^^>i/Ho +"7°t?,>/i,j''7/?-'w''0 7 7w^ 7n^w ^  1 nf 7wn 7*'w f) ( A . 1 9 ) Expressions (A.16), (A.18), and (A.19) immediately give the expansion (2.12) . •f 4 

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