Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Gravitational energy and conserved currents in general relativity Keefer, Bowie Gordon 1971

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1972_A1 K44.pdf [ 6.26MB ]
Metadata
JSON: 831-1.0084878.json
JSON-LD: 831-1.0084878-ld.json
RDF/XML (Pretty): 831-1.0084878-rdf.xml
RDF/JSON: 831-1.0084878-rdf.json
Turtle: 831-1.0084878-turtle.txt
N-Triples: 831-1.0084878-rdf-ntriples.txt
Original Record: 831-1.0084878-source.json
Full Text
831-1.0084878-fulltext.txt
Citation
831-1.0084878.ris

Full Text

I I 2 Z S GRAVITATIONAL ENERGY AND CONSERVED CURRENTS IN GENERAL R E L A T I V I T Y by BOWIE GORDON KEEFER B.Eng., R o y a l M i l i t a r y C o l l e g e o f C a n a d a , 1965 A T H E S I S SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f P h y s i c s Ule a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA December, 1971 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 the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 reference and study . I f u r t h e r agree t h a t permiss ion f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or 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 ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . P h y s i c s Department of •  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8. Canada Date 15 M a r c h 1972 ABSTRACT The p r o b l e m of t h e d e f i n i t i o n of g r a v i t a t i o n a l energy i s r e c o n s i d e r e d . In t h e E i n s t e i n t h e o r y a l l m a t t e r and f i e l d s e x c e p t g r a v i t y must have a w e l l d e f i n e d l o c a l d i s t r i b u t i o n of energy t h a t i s d e s c r i b e d by t h e energy-momentum t e n s o r . A g r a v i t a t i o n a l "energy-momentum c o m p l e x " may be d e f i n e d i n a n a l o g y t o an energy-momentum t e n s o r . However t h e r e i s an i n f i n i t e number of e x p r e s s i o n s f o r t h e g r a v i t a t i o n a l c o m p l e x , and each e x p r e s s i o n must depend e x p l i c i t l y upon t h e c h o i c e of r e f e r e n c e s y s t e m . F o l l o w i n g a r e v i e w of e a r l i e r w o r k s , a s t u d y i s made of p h y s i c a l and g e o m e t r i c c o n s i d e r a t i o n s w h i c h might s e l e c t u s e f u l l y d i s t i n g u i s h e d g r a v i t a t i o n a l c o m p l e x e s and r e f e r e n c e f r a m e s . T h i s i n v e s t i g a t i o n i s c o n d u c t e d w i t h i n t h e v i e r b e i n f o r m u l a t i o n of g e n e r a l r e l a t i v i t y . C o n s e r v e d c u r r e n t s c o r r e s p o n d i n g t o g e n e r a l i z e d e n e r g y , momentum and s p i n a r e d e r i v e d f rom a c t i o n p r i n c i p l e s . These c u r r e n t s t r a n s f o r m as v e c t o r d e n s i t i e s under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s , but depend on t h e v i e r b e i n f rame c h o s e n . The e x p r e s s i o n s f o r t h e energy and momentum c u r r e n t s a r e not u n i q u e , as t h e i r g e n e r a l e x p r e s s i o n c o n t a i n s t h r e e a r b i t r a r y c o n s t a n t s . P h y s i c a l examples a r e i s e d t o t e s t p o s s i b l e c h o i c e s of t h e s e c o n s t a n t s and p o s s i b l e v i e r b e i n f r a m e s . The g e n e r a l i z e d v i e r b e i n energy and momentum c u r r e n t s a r e c a l c u l a t e d f o r a s y m p t o t i c a l l y f l a t , r a d i a t i v e s p a c e - t i m e s . The p h y s i c a l r e q u i r e m e n t s t h a t t h e energy of an i s o l a t e d s y s t e m i i i c a n n o t i n c r e a s e when t h e r e i s no i n c o m i n g r a d i a t i o n , and t h a t t h e r e , be i n v a r i a n c e under v i e r b e i n t r a n s f o r m a t i o n s r e s p e c t i n g boundary c o n d i t i o n s a p p r o p r i a t e t o t h e a s y m p t o t i c symmetry g r o u p , a r e imposed on t h e g e n e r a l i z e d energy i n t e g r a l . These r e q u i r e m e n t s d e t e r m i n e a u n i q u e e x p r e s s i o n f o r t h e energy c u r r e n t w h i c h c o n t a i n s no s e c o n d o r d e r f i e l d d e r i v a t i v e s . S i n c e t h e boundary c o n d i t i o n s do not s p e c i f y t h e v i e r b e i n f rame e v e r y w h e r e , t h e d i s t r i b u t i o n of g r a v i t a t i o n a l energy i s not w e l l d e f i n e d even when t h e ' , c o n c e p t of t o t a l energy i s made l e g i t i m a t e by a s y m p t o t i c s p a c e - t i m e symmet ry . I t has been c o n j e c t u r e d r e p e a t e d l y t h a t a l o c a l d e n s i t y of g r a v i t a t i o n a l energy c o u l d be d e f i n e d even i n t h e a b s e n c e of s p a c e - t i m e s y m m e t r i e s t h r o u g h a s u i t a b l e c h o i c e of g r a v i t a t i o n a l complex and of r e f e r e n c e f r a m e . T h i s i s c e r t a i n l y a t t a i n a b l e i n a f o r m a l s e n s e , as i n v a r i a n t v i e r b e i n f rames a r e d e f i n e d by t h e p r i n c i p a l d i r e c t i o n s of t h e c u r v a t u r e t e n s o r and of t h e energy-momentum t e n s o r - o f m a t t e r . I t i s shown by t h e c o n s i d e r a t i o n of g r a v i t a t i o n a l r a d i a t i o n f i e l d s t h a t s u c h d e f i n i t i o n s w i l l not s u f f i c e t o l o c a l i z e g r a v i t a t i o n a l e n e r g y . iw CONTENTS ABSTRACT l i NOTATION v i ACKNOWLEDGEMENTS v i i 1. I n t r o d u c t i o n a . T h e P r o b l e m o f G r a v i t a t i o n a l E n e r g y 1 b. H i s t o r i c a l R e v i e w 13 2. G e n e r a l S y m m e t r i e s a n d C o n s e r v a t i o n Laws a . I n v a r i a n c e s a n d I d e n t i t i e s 3 4 b . S p a c e - t i m e T r a n s f o r m a t i o n s 3 9 c . Gauge F i e l d s a n d C u r r e n t s 4 5 3. V i e r b e i n F o r m a l i s m a . F u n d a m e n t a l s 50 b . V i e r b e i n C o v a r i a n t D e r i v a t i v e s 5 5 c . Dynamic V i e r b e i n C o n n e c t i o n 59 d. C u r v a t u r e T e n s o r 60 e. V i e r b e i n G r a v i t a t i o n a l L a g r a n g i a n s 62 4. V i e r b e i n E n e r g y , Momentum a n d S p i n C u r r e n t s a . G e n e r a l C o n s i d e r a t i o n s 64 b. V i e r b e i n S p i n C u r r e n t s 67 c . V i e r b e i n E n e r g y a n d Momentum C u r r e n t s 76 d. G e n e r a l E x p r e s s i o n s f o r C o n s e r v e d V i e r b e i n C u r r e n t s 85 5. V i e r b e i n R o t a t i o n Gauge 90 V , 6 . P h y s i c a l Examples a . S c h u u a r z s c h i l d m e t r i c 104 b . A s y m p t o t i c a l l y F l a t R a d i a t i v e M e t r i c s 109 1) A s y m p t o t i c S o l u t i o n of B o n d i and Sachs . . . 109 2) P a r t i c u l a r V i e r b e i n Gauges 114 3) A s y m p t o t i c V i e r b e i n Gauge 115 4) Weyl P r i n c i p a l V i e r b e i n 127 5) V i e r b e i n Gauge C o v a r i a n t T o t a l E n e r g y -Momentum 12B 6) S p i n I n t e g r a l s 134 c . P l a n e G r a v i t a t i o n a l Wave M e t r i c s 136 7 . C o n c l u d i n g Remarks 144 BIBLIOGRAPHY 146 v i NOTATION S p a c e - t i m e ( " w o r l d " ) c o o r d i n a t e i n d i c e s a r e Greek l a t t e r s w i t h range 0 , 1 , 2 , 3 . V i e r b e i n d i r e c t i o n i n d i c e s a r e l iafcin l e t t e r s w i t h range ( 0 ) , ( l ) , ( 2 ) , ( 3 ) . The p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o / x / i s d e n o t e d by . The c o v a r i a n t d e r i v a t i v e i s d e n o t e d by j ^ . D i f f e r e n t i a t i o n c o v a r i a n t w i t h r e s p e c t t o some gauge group s u c h as t h e v i e r b e i n r o t a t i o n group i s d e n o t e d . The s i g n a t u r e of t h e m e t r i c i s - 2 . The M i n k o w s k i m e t r i c t e n s o r i s w r i t t e n r^y^ . and t h e g e n e r a l m e t r i c t e n s o r ^ ° r y The u n i t d e n s i t y i s d e n o t e d by A= J^j , where J S detfj^. The E i n s t e i n g r a v i t a t i o n a l c o n s t a n t i s ACS £~n~(y , w i t h u n i t s c h o s e n so t h a t c = 1 . v i i A c k n o w l e d g e m e n t s I w i s h t o t h a n k D r . F . A. K a e m p f f e r f o r h i s g e n e r o u s e n c o u r a g e m e n t b e f o r e a n d d u r i n g my s t u d i e s 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 , a n d f o r many u s e f u l d i s c u s s i o n s . H i s d e r i v a t i o n o f t h e v i e r b e i n c o n s e r v a t i o n l a w s was t h e s t a r t i n g p o i n t o f t h i s p r o j e c t . T h e a d v i c e p r o v i d e d by D r . P. R a s t a l l , D r . M. H. L . P r y c e a n d D r . B. L. W h i t e has b e e n most h e l p f u l . I am g r a t e f u l t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a f o r a s c h o l a r s h i p a n d a b u r s a r y . T h e c o m p l e t i o n o f t h i s t h e s i s has b e e n e n c o u r a g e d i n t h e w o r k i n g a t m o s p h e r e p r o v i d e d by my a s s o c i a t e s a t L o c k h e e d P e t r o l e u m S e r v i c e s , L i m i t e d . 1 . I n t r o d u c t i o n a . The P r o b l e m of G r a v i t a t i o n a l Energy The E i n s t e i n t h e o r y of g r a v i t a t i o n r e q u i r e d a d r a s t i c r e v i s i o n of many b a s i c c o n c e p t s . The c o n c e p t of t o t a l energy as a c o n s e r v e d q u a n t i t y had been c e n t r a l i n Newton ian p h y s i c s , and r e m a i n e d most u s e f u l i n s p e c i a l r e l a t i v i t y . In s p e c i a l r e l a t i v i t y t h e d e n s i t i e s of e n e r g y , momentum and s t r e s s e s were combined i n t h e energy-momentum t e n s o r , w h i c h f o r example may be used t o c a l c u l a t e energy t r a n s f e r by e l e c t r o m a g n e t i c r a d i a t i o n . In g e n e r a l r e l a t i v i t y t h e e n e r g y -momentum t e n s o r a c q u i r e d f u r t h e r s i g n i f i c a n c e as t h e s o u r c e of t h e g r a v i t a t i o n a l f i e l d . But t h e r e i s no t r u e energy-momentum t e n s o r f o r t h e g r a v i t a i o n a l f i e l d i t s e l f , w h i c h means t h a t c a l c u l a t i o n s of energy t r a n s f e r by g r a v i t a t i o n a l r a d i a t i o n w i l l be f a r f r o m s i m p l e . T h e r e i s no good d e f i n i t i o n of a d e n s i t y o f g r a v i t a t i o n a l e n e r g y , and even t o t a l energy i s h a r d t o d e f i n e u n l e s s s p a c e - t i m e i s a s y m p t o t i c a l l y f l a t . -7-X The energy-momentum t e n s o r / ^ of s p e c i a l r e l a t i v i t y has t h e f o l l o w i n g p r o p e r t i e s ! ( 1 ) j s a t i s f i e s a c o n t i n u i t y e q u a t i o n or " l o c a l c o n s e r v a t i o n l a w " w h i c h i s e x p r e s s e d by t h e d i v e r g e n c e c o n d i t i o n (2) The energy-momentum t e n s o r o f f i e l d s whose d y n a m i c a l 2 e q u a t i o n s a r e summar ized i n a v a r i a t i o n a l p r i n c i p l e can be d e r i v Q d t h r o u g h t h e s t a n d a r d a p p l i c a t i o n of N o e t h e r J s (1918) Theorem, which w i l l be r e v i e w e d i n C h a p t e r 2 . T h i s method uses t h e i n v a r i a n c e of s p e c i a l r e l a t i v i s t i c f i e l d t h e o r i e s under r i g i d d i s p l a c e m e n t s of t h e c a r t e s i a n c o o r d i n a t e s y s t e m s . The d e f i n i t i o n of T/» t h u s o b t a i n e d i s o n l y u n i q u e up t o t h e a d d i t i o n of an a r b i t r a r y d i v e r g e n c e (Jy /v; where Uyt s i n c e ( l . l ) s t i l l h o l d s . T h i s f reedom may be used t o r e d e f i n e an energy-momentum t e n s o r t h a t i s s y m m e t r i c or t r a c e l e s s , and may be c a l l e d a " k i n e m a t i c " a m b i g u i t y ( S c i a m a , 1961) s i n c e i t has n o t h i n g t o do w i t h t h e d y n a m i c a l e q u a t i o n s . (3 ) The i n t e g r a l s of t h e energy-momentum t e n s o r o v e r a f l a t s p a c e l i k e h y p e r s u r f a c e r > J V ^ e - x = p - > « V ( 1 . 2 ) y i e l d t h e t o t a l energy and momentum, w h i c h f o r an i s o l a t e d s y s t e m a r e w e l l d e f i n e d and c o n s t a n t s of t h e m o t i o n , T h i s i s a " g l o b a l c o n s e r v a t i o n l a w " . T o g e t h e r t h e components t r a n s f o r m as a fjree f o u r - v e c t o r . S e p a r a t e l y t h e y may be r e g a r d e d as t h e g e n e r a t o r s of r i g i d c o o r d i n a t e d i s p l a c e m e n t s i n t h e i r r e s p e c t i v e d i r e c t i o n s i n t i m e and s p a c e . In t h e E i n s t e i n (1916) t h e o r y of g r a v i t y t h e s o u r c e of t h e g r a v i t a t i o n a l f i e l d i s t h e s y m m e t r i c energy-momentum t e n s o r of m a t t e r . T h i s " m a t t e r t e n s o r " i n c l u d e s c o n t r i b u t i o n s f rom a l l f i e l d s and p a r t i c l 8 S « p r e s e n t . The k i n e m a t i c a m b i g u i t y of s p e c i a l r e l a t i v i t y i s removed by t h e g r a v i t a t i o n a l f i e l d e q u a t i o n s w h i c h s e l e c t t h e s y m m e t r i c t e n s o r . The l o c a l 3 c o n s e r v a t i o n law . (1 .1 ) i s b r o k e n by t h e g r a v i t a t i o n a l i n t e r -a c t i o n , s i n c e i t i s now t h e c o v a r i a n t d i v e r g e n c e of t h e m a t t e r t e n s o r d e n s i t y w h i c h v a n i s h e s -fx Q u a n t i t i e s s u c h as / ^ w i l l h e r e a l w a y s be t a k e n as t e n s o r d e n s i t i e s of u n i t w e i g h t . I t i s a l w a y s f o r m a l l y p o s s i b l e t o o b t a i n a l o c a l c o n s e r v a t i o n law i n g e n e r a l r e l a t i v i t y by a d d i n g a g r a v i t a t i o n a l "energy-momentum c o m p l e x " o r " p s e u d o - t e n s o r " t o t h e m a t t e r t e n s o r so t h a t X C T ^ O A - o ( 1 . 4 ) These names r e f l e c t t h e n o n - t e n s o r i a l t r a n s f o r m a t i o n p r o p e r t i e s of e x p r e s s i o n s f o r X./»* w h i c h can be found f rom t h e v a r i a t i o n a l f o r m a l i s m of t h e g r a v i t a t i o n a l t h e o r y . Any a t t e m p t e d d e f i n i t i o n of O ^ , s ~T^u'4'^yt4 a s t h e t o t a l d e n s i t y of e n e r g y , momentum and s t r e s s of a l l f i e l d s i n c l u d i n g g r a v i t a t i o n w o u l d be o b s c u r e , not o n l y because o f t h e dependence on c o o r d i n a t e sys tems but a l s o b e c a u s e of a f u n d a m e n t a l l a c k of u n i q u e n e s s . There i s an i n f i n i t e number of p o s s i b l e e x p r e s s i o n s f o r t h e g r a v i t a t i o n a l complex w h i c h a l l y i e l d an e q u a t i o n of c o n t i n u i t y s i m i l a r t o ( 1 . 4 ) , and f o r each e x p r e s s i o n t h e r e i s an i n f i n i t e number of a l l o w a b l e c o o r d i n a t e f r a m e s . The d i v e r g e n c e of an a r b i t r a r y a n t i s y m m e t r i c q u a n t i t y may be added t o c3j* w h i l e p r e s e r v i n g ( 1 . 4 ) 0^**A ~ O • 4 Here t h e r e i s no d y n a m i c a l c r i t e r i o n t o remove t h e k i n e m a t i c a m b i g u i t y . A n o t h e r s o u r c e of a p p a r e n t a m b i g u i t y a r i s e s f rom t h e p o s s i b i l i t y o f f o r m u l a t i n g t h e l o c a l c o n s e r v a t i o n law s t a r t i n g f rom a n o t h e r t e n s o r fo rm of t h e m a t t e r t e n s o r . The mixed components i n (1.4) c o u l d be r e p l a c e d by t h e c o n t r a -v a r i a n t components ~J~^'t m u l t i p l i e d by an a r b i t r a r y we igh t f a c t o r ^ / ^ = r ^ ) 2 » or p r o j e c t e d onto an a r b i t r a r y v e c t o r f i e l d * y i e l d i n g i n e q u i v a l e n t forms of t h e g r a v i t a t i o n a l complex i n each c a s e . The n o n - t e n s o r i a l a s p e c t o f any g r a v i t a t i o n a l complex might be e x p e c t e d f rom E i n s t e i n ' s p r i n c i p l e of e q u i v a l e n c e , w h i c h c o n t a i n s t h e key i d e a t h a t a f rame of r e f e r e n c e can a l w a y s be found i n w h i c h " g r a v i t a t i o n a l f o r c e s " v a n i s h at any g i v e n . p o i n t . The components F^otf of t h e a f f i n e c o n n e c t i o n p r o v i d e a c e r t a i n measure of t h e g r a v i t a t i o n a l f o r c e a c t i n g on a t e s t p a r t i c l e when t h e c o o r d i n a t e s a r e p r o p e r l y c h o s e n ; and a r e not t r u e t e n s o r components, b e i n g l i n e a r i n t h e f i r s t o r d e r d e r i v a t i v e s of t h e m e t r i c t e n s o r . The f i r s t d e r i v a t i v e s o f t h e m e t r i c v a n i s h i n some c o o r d i n a t e s y s t e m s a t any g i v e n p o i n t . The g r a v i t a t i o n a l complex w i l l i n c l u d e t e r m s q u a d r a t i c i n t h e a f f i n e c o n n e c t i o n . T e n s o r i a l c o m b i n a t i o n s o f t h e / i n t h i s o r d e r must be l i n e a r i n t h e Riemann t e n s o r . S i n c e t h e g r a v i t a t i o n a l f i e l d e q u a t i o n s a r e a l r e a d y c o n s t r u c t e d f rom c o n t r a c t i o n s ' a f ' t h e r Riemann t e n s o r , t h e r e i s no manner i n w h i c h a n o n - t r i v i a l g r a v i t a t i o n a l energy-momentum t e n s o r o f s e c o n d rank c o u l d be c o n s t r u c t e d d i r e c t l y f rom t h e Riemann t e n s o r . 5 A study of conservation laws in general re lat iv i ty is necessarily intimately linked to a study of reference frames in that theory. It is well established that the non-covariance of a gravitational complex can always be removed or at least quarantined by introducing supplemental structures into the geometry which wi l l serve as reference systems. The invariant specif ication of a suitable reference system wi l l absorb the non-covariance of a given complex. It is then possible f o r m a l l y to define appropriate components as densities and fluxes of "energy" and "momentum" referred to that reference system. Such concepts can at best be only as good as the selected reference system. Possible geometric structures which may be used as reference systems include particular coordinate systems, vector f i e lds , a second metric, or two-point functions referred to some or ig in . Local and nonlocal specifications may be considered. ft physically interesting reference system wi l l have something to do with the int r ins ic geometry of the space-time. It can be based on the symmetries of the space-time in the degenerate cases that symmetries exist, and then the connection to integral or global conservation laws is immediate. Such a connection also exists clearly in the presence of asymptotic symmetry. Reference systems of interest can also be based on the asymmetry of the space-time as displayed in i ts curvature structure which determines certain eigen-directions. It is 6 a l s o somet imes u s e f u l t o use a "gauge c o n d i t i o n " w h i c h may not s e l e c t a u n i q u e r e f e r e n c e s y s t e m , but w i l l r e s t r i c t c o n s i d e r a t i o n t o a c e r t a i n f a m i l y of a l l o w e d f r a m e s . s y m m e t r i e s i s e v i d e n t i n s p e c i a l r e l a t i v i t y . There t h e e x i s t e n c e o f g l o b a l c o n s e r v a t i o n laws r e s u l t s f rom t h e homogene i t y and i s o t r o p y of M i n k o w s k i s p a c e - t i m e . The whole s p a c e - t i m e a d m i t s p r e f e r r e d c a r t e s i a n c o o r d i n a t e s y s t e m s w h i c h d e s c r i b e g l o b a l i n e r t i a l f r a m e s . The d i r e c t i o n s of r i g i d d i s p l a c e m e n t s and r i g i d L o r e n t z r o t a t i o n s a r e w e l l s p e c i f i e d once a s e t of o r t h o g o n a l axes i s s e l e c t e d a t some o r i g i n . In g e n e r a l r e l a t i v i t y w i t h o u t g l o b a l s y m m e t r i e s t h e c o o r d i n a t e d i s p l a c e m e n t s g e n e r a t e d by a n a l o g u e s t o energy and momentum a r e no l o n g e r r i g i d . Each c o o r d i n a t e t r a n s f o r m a t i o n may be d e s c r i b e d by a v e c t o r f i e l d T , and t h e i n f i n i t y of l o c a l c o n s e r v a t i o n laws i s a s s o c i a t e d w i t h t h e i n f i n i t y of v e c t o r f i e l d s e q u a l l y q u a l i f i e d t o s e r v e as d e s c r i p t o r s f o r t h e g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n group u n d e r l y i n g t h e t h e o r y . I f i n t h e v a r i a t i o n a l d e r i v a t i o n of g e n e r a l i z e d energy and momentum e x p r e s s i o n s t h e v e c t o r f i e l d j i s r e t a i n e d e x p l i c i t l y , i t i s p o s s i b l e t o o b t a i n f o r m a l c o v a r i a n c e t h r o u g h r e p l a c i n g t h e s e c o n d rank g r a v i t a t i o n a l complex by a v e c t o r c o m p l e t e n o n - c o v a r i a n c e o f t h e o l d complex can be a b s o r b e d i n t h e e x p l i c i t dependence of t h e new c u r r e n t on t h e a r b i t r a r y The g r e a t p h y s i c a l i m p o r t a n c e of s p a c e - t i m e g r a v i t a t i o n a l By s u i t a b l e c o n s t r u c t i o n t h e v e c t o r f i e l d The l o c a l c o n s e r v a t i o n law t a k e s t h e f o r m 7 transforms as a contravariant vector density. The vector f i e l d J" has spli t a "current" from the matter tensor, and has served to construct a gravitational current of generalized energy or momentum. A vierbein is a set of four orthogonal unit vectors. If the vector fields associated with the above currents are restricted to be vierbein fields, then an energy current and three momentum currents can be associated formally with each possible choice of vierbein and with some choice of expression for the currents. The use of unit vectors rather than completely arbitrary vectors as coordinate transformation descriptors means that the transformations generated by the conserved quantities are scaled to the units of time and space of a local observer. The orthogonality conditions for the vierbein fields have the consequence that the vierbein components can be used as gravitational potentials instead of the components of the metric tensor. Equivalent vierbeins are related by Lorentz transformations under which the theory is invariant and with which are associated conserved "spin currents". These features bring the theory of gravitation in vierbein language into very close contact with the general theory of gauge fields, of which the prototype is electromagnet ism. a The vierbein formalism is also a strong bridge between d i f ferent ia l geometry and physics. It seems to be well suited to an examination of the extent to which the expression for the gravitational currents can be made unique and then can be fixed in some sense by a good choice of geometrical reference system. Even though it seems clear that there is no salient reason to expect a local density of gravitational energy having physical significance to be definable, a considerable number of attempts have been made in this direct ion. Proposals have been made to select one expression or another, generally using some gauge condition to contain the n o n - c o v a r i a n c e » but these tentatives have been on the whole inconclusive. However i t is true that invariant local currents determined by the Riemann tensor can be defined in a manner which to some extent complements the use of symmetry directions to f ix the same currents, so the question of the local ization of gravitational energy has not been entirely settled negatively. The main objectives of the present investigation are (1) to set up a f lex ib le formalism for the conserved currents and possible reference systems in the vierbein formulation, and (2) to test this general formalism against physics using known stat ic and radiative metrics in order to eliminate as much arbitrariness and non-physics as possible. 9 The r e m a i n d e r of t h i s c h a p t e r i s a s u r v e y of r e l a t e d e a r l i e r i n v e s t i g a t i o n s i n t o t h e n a t u r e o f g r a v i t a t i o n a l e n e r g y . The s e c o n d c h a p t e r s e t s t h e b a c k -g round f o r a v a r i a t i o n a l t r e a t m e n t o f s p a c e - t i m e c o n s e r v a t i o n l a w s . M o e t h e r ' s theorem (1918) i n t h e form p r e s e n t e d by T rautman (1962) i s r e v i e w e d . Some a s p e c t s o f t h e g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n group and s p a c e - t i m e c o n s e r v a t i o n laws a r e nex t c o n s i d e r e d . Then t h e g e n e r a l t h e o r y o f gauge ( " c o m p e n s a t i n g " ) f i e l d s i s o u t l i n e d b e c a u s e of s t r o n g a n a l o g i e s i n t h e v i e r b e i n f o r m u l a t i o n o f g e n e r a l r e l a t i v i t y p a r t i c u l a r l y r e g a r d i n g c o n s e r v a t i o n l a w s . The v i e r b e i n f o r m a l i s m i t s e l f i s p r e s e n t e d i n some d e t a i l i n C h a p t e r 3 . Up t o t h i s p o i n t t h e r e i s n o t h i n g a t a l l new. The e x p r e s s i o n s f o r c o n s e r v e d " e n e r g y c u r r e n t s " and "momentum c u r r e n t s " c o r r e s p o n d i n g t o v i e r b e i n - d e s c r i b e d c o o r d i n a t e t r a n s f o r m a t i o n s and " s p i n c u r r e n t s " c o r r e s p o n d i n g t o v i e r b e i n L o r e n t z r o t a t i o n s a r e c o n s i d e r e d e x t e n s i v e l y i n C h a p t e r 4 . C e r t a i n r e s t r i c t i o n s on t h e t r a n s f o r m a t i o n p r o p e r t i e s of t h e s e c u r r e n t s a r e f i r s t made and j u s t i f i e d by c o n s i d e r a t i o n s o f max imal c o v a r i a n c e and economy. Then t h e s p i n c u r r e n t s f o r m a t t e r f i e l d s w i t h o u t g r a v i t a t i o n , f o r g r a v i t a t i o n , and f o r a somewhat m o d i f i e d t h e o r y o f g r a v i t a t i o n ( K i b b l e , 1961« S c i a m a , 1961) a r e examined u s i n g v a r i o u s L a g r a n g i a n s . The energy and momentum c u r r e n t s a r e c o n s i d e r e d s i m i l a r l y . These v a r i a t i o n a l d e r i v a t i o n s 10 a r e s t a n d a r d o r n e a r l y s o . The g r a v i t a t i o n a l s p i n c u r r e n t s a r e i d e n t i c a l l y c o n s e r v e d i n d e p e n d e n t l y o f t h e m a t t e r s p i n c u r r e n t s } and t h e l a t t e r a r e not c o n s e r v e d a t a l l s i n c e t h e r e i s no p l a c e f o r o r b i t a l a n g u l a r momentum h e r e . The v i e r b e i n t r e a t m e n t c a s t s l i g h t on t h e w e l l known B e l i n f a n t e ( 1 9 3 9 ) - R o s e n f e l d (1940) s y m m e t r i z a t i o n method f o r t h e energy-momentum t e n s o r , w h i c h i n v o l v e s t h e m a t t e r s p i n c u r r e n t s . As e x p e c t e d , t h e r e i s no u n i q u e n e s s i n t h e fo rm of t h e g r a v i t a t i o n a l energy and momentum c u r r e n t s , as t h e g e n e r a l s u p e r p o t e n t i a l depends on t h r e e a r b i t r a r y c o n s t a n t s . A l t h o u g h t h e v i e r b e i n c u r r e n t s a r e c o o r d i n a t e c o v a r i a n t , t h e y depend s t r o n g l y on t h e v i e r b e i n r o t a t i o n g a u g e . P o s s i b l e ways o f f i x i n g t h i s gauge f reedom u s e f u l l y a r e c o n s i d e r e d i n C h a p t e r 5 . D i f f e r e n t p r e f e r r e d v i e r b e i n s may be d e t e r m i n e d by t h e c u r v a t u r e s t r u c t u r e , by i s o m e t r i c s y m m e t r i e s t h a t may be p r e s e n t , o r by a w ide r a n g e of somet imes l e s s a m b i t i o u s gauge c o n d i t i o n s . In C h a p t e r 6 t h e f o r m a l e x p r e s s i o n s and r e f e r e n c e s t r u c t u r e s o f t h e p r e v i o u s two c h a p t e r s a r e a p p l i e d t o g e t h e r t o c o n c r e t e p r o b l e m s . The p h y s i c a l examples c o n s i d e r e d a r e t h e S c h w a r z s c h i l d m e t r i c , t h e a s y m p t o t i c a l l y f l a t B o n d i -Sachs r a d i a t i v e m e t r i c s , and t h e e x a c t R o b i n s o n p l a n e wave m e t r i c s . The g l o b a l c o n s e r v a t i o n laws f o r energy a r e s t u d i e d u s i n g t h e most g e n e r a l v i e r b e i n s u p e r p o t e n t i a l i n c o n j u c t i o n w i t h an a s y m p t o t i c " v i e r b e i n r a d i a t i o n gauge" w h i c h i s p r o p o s e d . T h i s gauge i s c l o s e l y r e l a t e d t o p r e v i o u s 11 r a d i a t i o n c o n d i t i o n s and t o t h e a s y m p t o t i c symmetry g r o u p i and i t i s e x p e c t e d t h a t any more s p e c i f i c gauge t h a t might be based f o r example on t h e c u r v a t u r e s t r u c t u r e wou ld have t o be c o n t a i n e d w i t h i n i t . The p h y s i c a l r e q u i r e m e n t t h a t t h e t o t a l energy measured a t r e t a r d e d t i m e s m o n o t o n i c a l l y d e c l i n e when t h e r e i s o n l y o u t w a r d s r a d i a t i o n i s q u i t e s u f f i c i e n t t o s e l e c t a u n i q u e e x p r e s s i o n ( f r o m t h o s e a l l o w e d by t h e assumed t r a n s f o r m a t i o n p r o p e r t i e s ) f o r t h e energy c u r r e n t i f any v i e r b e i n w i t h i n t h e " v i e r b e i n r a d i a t i o n gauge" i s a l l o w e d . T h i s u n i q u e e x p r e s s i o n i s t h a t w i t h o u t s e c o n d o r d e r d e r i v a t i v e s , c o n s i d e r e d by P l e b a n s k i (1962) and o t h e r s , and i s v e r y c l o s e l y r e l a t e d t o E i n s t e i n ' s " p s e u d o - t e n s o r " . I t i s h e r e c a l l e d t h e " c a n o n i c a l v i e r b e i n energy c u r r e n t " because o f i t s i n c l u s i o n o f o n l y f i r s t o f d e r d e r i v a t i v e s . The c a n o n i c a l energy c u r r e n t may not be f i x e d by p r e f e r r e d d i r e c t i o n s based on c u r v a t u r e s t r u c t u r e , as i s d e m o n s t r a t e d by t h e c a s e o f t h e S c h w a r z s c h i l d m e t r i c I t i s shown by examples t h a t o t h e r c a n d i d a t e e x p r e s s i o n s f o r t h e energy and momentum c u r r e n t s may a l s o y i e l d t h e c o r r e c t mass f o r a r a d i a t i n g s y s t e m , i f more r e s t r i c t i v e v i e r b e i n gauges a r e u s e d . But t h e s e gauges a r e not c o n n e c t e d i n any p e r c e p t i b l e way t o t h e u n d e r l y i n g c u r v a t u r e s t r u c t u r e , and seem t o l a c k any c o n v i n c i n g p h y s i c a l j u s t i f i c a t i o n . 12 S i n c e t h e v i e r b B i n r a d i a t i o n g a u g e i s n o t e x a c t a n d d o e s n o t s e l e c t u n i q u e f r a m e s e x c e p t raear f u t u r e n u l l i n f i n i t y , g r a v i t a t i o n a l e n e r g y i s n o t l o c a l i z e d w i t h i n t h e f i n a l l y a c c e p t e d f r a m e w o r k . The e x a m p l e o f p l a n e w a v e s i s u s e d t o i l l u s t r a t e t h e i m p o s s i b i l i t y o f a f u l l and m e a n i n g f u l s p e c i f i c a t i o n o f t h e d i s t r i b u t i o n o f g r a v i t a t i o n a l e n e r g y . T h i s i s i n l i n e w i t h s i m i l a r a r g u m e n t s b y P e n r o s e (1966) a n d o t h e r s . 13 b. Historical Reviem As soon as Einstein had an action principle for generally covariant f ie ld equations (1915a) sat isf ied by components of the metric tensor ^<*tf » n e w a s able to propose an expression for the energy and momentum of the gravitational f i e l d . In this early version the f i e ld equations were somewhat truncated as the matter tensor had to be traceless. The f u l l gravitational f i e ld equations were discovered by Hilbert (1915) and Einstein (1915b), and the action principle was brought into a concise form free from restr ict ions by Einstein (1916b). The action integral for the gravitational f i e ld interacting with matter f ie lds is w = (1-6) where j?/* is a Lagrangian density for the matter f ie lds and applies to the gravitational f i e l d . For given f ie ld equations the Lagrangian density is only defined up to the addition of any total divergence. The f ie ld equations of general re lat iv i ty are is the Einstein tensor density and is the symmetric energy-momentum tensor of matter. 14 The gravitational Lagrangian may be chosen to be the curvature scalar density by a consideration of invariance requirements (see Kaempffer, 1968) Einstein avoided the complication of second order derivatives of the f i e ld variables Jo0> in his variational formalism by adding a divergence to the curvature scalar in order to obtain a new Lagrangian density (1916b) This is not a true scalar density and accordingly does not lead direct ly to s t r i c t l y tensorial conservation laws. Einstein drew his gravitational complex, the famous gravitational energy-momentum pseudo-tensor, from his Lagrangian (1.11) by the canonical prescription The local conservation law sat isf ied is (1.13) i i ns te in ' s canonical complex C transforms as a tensor only under linear coordinate transformations. It contains f i r s t order f i e ld derivatives only. 15 The c o n s e q u e n c e s of t h e n o n - t e n s o r i a l t r a n s f o r m a t i o n p r o p e r t i e s of E i n s t e i n ' s complex were e x p l o r e d by some of h i s c o n t e m p o r a r i e s . I t was shown t h a t c o o r d i n a t e s y s t e m s c o u l d be c h o s e n so t h a t t h e " e n e r g y " d e n s i t y would v a n i s h everywhere i n t h e f i e l d of a m a s s i v e e b o d y ( S c h r o e d i n g e r , 1 9 1 8 ) , o r so t h a t t h e " e n e r g y " i n t e g r a l wou ld d i v e r g e n e g a t i v e l y i n f l a t s p a c e ( B a u e r , 1 9 1 8 ) . These c r i t i c i s m s of any i d e n t i f i c a t i o n of ~i .. as a d e n s i t y of g r a v i t a t i o n a l energy began t h e l o n g c o n t r o v e r s y on t h i s t o p i c . E i n s t e i n h i m s e l f m a i n t a i n e d t h a t a l o c a l d e s c r i p t i o n of t h e d i s t r i b u t i o n of g r a v i t a t i o n a l energy was not t o be e x p e c t e d f rom t h e t h e o r y . These d i s c u s s i o n s showed t h a t l o c a l p h y s i c a l meaning can o n l y be a t t a c h e d t o t h e c a n o n i c a l complex i f p r i v i l e g e d s t a t u s i s a l s o g i v e n t o some c o o r d i n a t e s y s t e m , i n seeming d i s c o r d a n c e w i t h t h e r e q u i r e m e n t of g e n e r a l c o v a r i a n c e . But t h e i n t e g r a l s f(T^^)d^-f(Ty^)dV ( i . u ) may be i n t e r p r e t e d as r e p r e s e n t i n g t h e . c o m p o n e n t s of t o t a l energy and momentum under r a t h e r r e s t r i c t i v e c o n d i t i o n s e s t a b l i s h e d by E i n s t e i n (1918b) and K l e i n ( 1 9 1 8 ) . The c o n d i t i o n s a r e t h a t t h e s p a c e be a s y m p t o t i c a l l y f l a t and t h e c o o r d i n a t e s y s t e m a s y m p t o t i c a l l y c a r t e s i a n so t h a t where f i s a s u i t a b l y c h o s e n ou twards d i s t a n c e p a r a m e t e r and d e n o t e s a q u a n t i t y v a n i s h i n g as /" W . Such c o n d i t i o n s can o n l y be a p p l i e d t o i s o l a t e d n o n - r a d i a t i v e s y s t e m s . 16 Then ^ t r a n s f o r m s as a f r e e v e c t o r under c o o r d i n a t e t r a n s -formations which preserve the above asymptotic conditions. The second condition can be weakened through an asymptotic harmonicity condition to include the possibi l i ty of outwards radiation (Trautman, 1957, 1962). Lorentz (1916) proposed to avoid the d i f f i cu l t i es of interpretation associated with noncovariance by instead identifying the negative of the Einstein tensor density as the energy-momentum tensor density of the gravitational f i e l d . This expression is a true symmetric tensor density, and is equal to the negative of the matter tensor density. But these properties arise from an essentially t r i v i a l def in i t ion. The Lorentz tensor is scarcely useful for a description of energy because i t is incompatible with energy transport in the absence of matter by gravitational radiation. The energy of gravitation would cancel that of matter, and in vacuo would necessarily vanish. The real i ty of gravitational radiation as a means of energy transfer has now been raised well above the level of conjecture by the theoretical works of Pirani (1957), Bondi (1957, 1959, 1962), and Sachs (1962) among others; and by the experimental work of Weber (1969). In the absence of any better formulation, most investigations into gravitational energy and i ts radiative transfer were based on Einstein's canonical complex for a long time. 17 An important step towards a deeper understanding came with the discovery by won Freud (1939) that the canonical complex could be expressed as the divergence of an ant i -symmetric "superpotential" T*. + t\ - cJ.V (1-l6) Then Landau and L i fsh i tz (1951) announced a new gravitational complex which was also associated with a superpotential so that This complex shares the properties of being homogeneous quadratic in the f i r s t order derivatives of the metric tensor, and of being non-cov/ariant, with the Einstein complex. It has the advantage of being symmetric and the disadvantage of being a density of weight two. Goldberg (1956) found an in f in i te hierarchy of complexes of increasing weights given by a similar generalization of the superpotential. The important work of Bergmann and his associates (Bergmann, 1949t Bergmann and Schi l ler , 1953) elaborated and elucidated the variational formalism of conservation laws that had originated with Noether (1918). In particular the role of superpotentials in the generally covariant theory was c l a r i f i e d . 18 Th8 invariance group of general re lat iv i ty is the group of general coordinate transformations. An inf initesimal coordinate transformation may be written ^ = ^ + £° faY (1.18) where the are suitably smooth vector f ie lds which generate the transformations and are called descriptors. The index a , b , c , . . . labels different l inearly independent vector f ie lds which may be considered at once and which wi l l be associated with different similar ly labelled conserved quantit ies, c- is the inf initesimal parameter of the transformation in the direction given by the vector —> descriptor fa . It was found that a "strong" and a "weak" conservation law exists for each f . Here "strong" means independent of the f i e ld equations'validity, and "weak" means dependent on that va l id i ty . The specif ic structure of the conserved quantities depends upon the i n i t i a l choice of f i e ld variables and Lagrangian density in the action pr inc ip le . The strong conservation law always has the consequence that a superpotential exists. Then the weak conservation law follows from the f ie ld equations. T h e s t r u c t u r e o f t h e ff^  , w h i c h a l r e a d y d e p e n d s u p o n t h e i n i t i a l s t a t e m e n t o f t h e a c t i o n p r i n c i p l e a n d u p o n t h e c h o i c e o f d e s c r i p t o r s fa , m a y b e f r e e l y a l t e r e d b y a d d i n g a n a r b i t r a r y s k e w q u a n t i t y t o t h e s u p e r p o t e n t i a l . T h i s f o r m a l i s m w h i c h w i l l b e r e v i e w e d i n m o r e d e t a i l i n S e c t i o n 2 s h o w s t h a t o n e h a s w i d e l a t i t u d e i n c o n c o c t i n g c o n s e r v e d q u a n t i t i e s a s s o c i a t e d w i t h g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s . B u t t h e p h y s i c a l i n t e r p r e t a t i o n o f a n y o f t h e s e k i n e m a t i c a l l y o b t a i n e d f o r m a l e n t i t i e s i s a n o t h e r m a t t e r , h i n g i n g e n t i r e l y o n w h e t h e r t h e r e i s a n y p h y s i c a l c o n t e n t i n t h e s e l e c t i o n o f t h e e x p r e s s i o n T J ^ a n d t h e d e s c r i p t o r s W h e n t h e f0 d e s c r i b e t r a n s i t i v e t r a n s f o r m a t i o n s ( d i s p l a c i n g a l l p o i n t s ) , t h e c o r r e s p o n d i n g m a y b e c a l l e d t h e g e n e r a l i z e d e n e r g y a n d m o m e n t u m c u r r e n t s a s s o c i a t e d w i t h r e s p e c t i v e l y t i m e l i k e a n d s p a c e l i k e d e s c r i p t o r s . B e r g m a n n (1958) a p p l i e d h i s i d e a o f a s s o c i a t i n g c o n s e r v a t i o n l a w s w i t h v e c t o r f i e l d s i n a s i m p l e g e n e r a l -i z a t i o n o f t h e v o n F r e u d s u p e r p o t e n t i a l T h e n i n a g i v e n c o o r d i n a t e s y s t e m o n e o b t a i n s t h e E i n s t e i n c o m p l e x b y t a k i n g t h e c o m p o n e n t s J" — * da c o n s t a n t , s o t h a t t h e d e s c r i p t o r s c o n s t i t u t e t h e n a t u r a l b a s i s i n t h a t c o o r d i n -a t e s y s t e m a n d t h e g e n e r a t e d c o o r d i n a t e t r a n s f o r m a t i o n s a r e r i g i d . S i m i l a r l y t h e L a n d a u - L i f s h i t z c o m p l e x i s o b t a i n e d 20 by t a k i n g t h e components A Yaoe constant, which seems somewhat a r t i f i c i a l . This shows that the non-covariance of the Einstein complex results both from its derivation from a non-covariant Lagrangian,and from i ts association with r ig id coordinate transformations which can only be r ig id in the i n i t i a l coordinate system where the descriptor vectors are the basis. complex to an extent suff ic ient for a valid local izat ion of gravitational energy was made by miller (1958). This effort was mainly directed at the elimination of the strong dependence of the energy component of Einstein's canonical complex on purely spatial directions, which causes the appearance of f i c t i t i ous energy terms when polar rather than (cartesian coordinates are used in f lat space-time, (Bauer, 1918). miller was able to derive a complex without this dependence but which included second order derivatives of the metric tensor. This complex was also derived by mitskevich (1958), and can be extracted from the covariant Lagrangian density R by considering r ig id coordinate transformations (Mil ler, 1959» Magnusson, 1960). It is An attempt to remedy the non-covariance of Einstein's (1.20) 21 The Mrfller-roitskevich complex is a tensor with respect to purely spatial coordinate transformations, as required to give polar coordinates equal status with cartesian coordinates. But the non-covariance with respect to transformations affecting the time coordinate is aggravated (Mi l ler , 1959a). Komar was able to formulate a completely covariant conservation law (1959) by effectively combining key ideas from the work of Bergmann and of Mi l ler , which were the incorporation of the vector descriptor of the coordinate transformation into the conserved object and the use of a covariant Lagrangian as a starting point. He introduced the superpotential giving r ise to generalized energy and momentum currents in close analogy with the inhomogeneous Maxwell equations with the vector descriptor as vector potential . Komar derived this expression by generalizing the complex (1.20), but i t can be drawn direct ly from the curvature scalar density as Lagrangian density. In a given coordinate system the fflffller-ftiitskevich complex is obtained by taking the contra-variant components constant in Komar's superpotential, and the Lorentz tensor by taking the covariant components constant so that the superpotential vanishes. 22 Thus Komar completed the work of Bergmann by bringing f u l l covariance to i t . A powerful approach was indicated, which is to modify the structure of the local objects associated with generalized energy, momentum, and angular momentum so that they are true vector currents rather than two or three index nontensorial complexes (see Rayski, 1961). The matter energy and momentum currents derived from the symmetric energy-momentum tensor density of matter If the vector descriptors satisfy then since T^>\ =r O > T~\S> = T\\X = & (1.24) Here has been required to be a K i l l ing vector (see Section 2) corresponding to a generator of a group of motions. This is a most immediate characterization of the directions determined by the preferred cartesian coordinate systems in special re la t i v i t y . So in special re lat iv i ty the preferred vectors X associated with translations are the basis vectors of Ja the global iner t ia l coordinates, in which the components of the four energy and momentum currents form the familiar tensor / The problem posed by this approach of treating energy and momentum currents on the same basis as any other currents carrying some "charge" is whether the 's can be chosen to have intr ins ic geometric and physical s i g n i f i -cance in general. K i l l ing vectors are a natural choice. But groups of motions only exist in very special space-times, although this class of highly symmetric and . mathematically simple worlds includes nearly a l l known exact solutions of Einstein's equations. In more rea l i s t i c space-times of lesser symmetry other c r i te r ia may be possible. Komar (1962b, 1963) tr ied to weaken the K i l l ing condition in some way of suff icient generality to local ize gravitational energy, without much success. Various authors (Dirac, 1959; Arnowitt, Oeser and fflisner, 1961; Rayski, 1961, 1962; Komar, 1962b, 1963) have distinguished vectors orthogonal to minimal hypersurfaces upon which energy would be defined. Komar (1962b) also considered the invariant eigenvectors of the Uleyl conformal tensor but did not develop any published conclusions. Pirani (1959) derived an energy conservation law closely related to that of Komar by an approach much more intuit ive than the variational formalism. He considered a set of f i c t i t i ous privileged observers whose (in general non-geodetic)trajectories form a normal congruence The world-lines of these observers are orthogonal to a family of space-l ike reference hypersurfaces which define a common time coordinate* T.he curvature of the world-lines is the "gravitational acceleration" experienced by the observers • ^ 2 4 Here U is the timelike unit vector tangent to the world-l ines or four-velocity , with U / . Pirani defined the "gravitational mass" relative to this reference system and within a region of one of the selected hypersurfacessby a flux integral in analogy with Gauss' law in Newtonian gravitation theory. Then " s f ^ y s - &, f U^t C/**r ( 1 ' 2 5 ) In the above S is a closed two-surface with outward normal A? within the referenc  hyper rf , and the identity has been used. Now (1.25) may be written S" ( 1 . 2 6 ) is Pi rani 's superpotential di f fer ing from Komar's by a factor of two and in the restr ict ion to unit reference vectors. This factor of two is compatible in the Schwarz-schi ld metric with the use of the t ime-l ike K i l l ing vector^ paral le l but not equal to U in Komar's superpotential. The freedom in selection of superpotential may be exploited in other directions. For example, Cattaneo (1966) has considered amoung others the superpotential 25 An important modification of Komar's conservation law was devised and applied by Utinicour and Tamburino (1965,1966). Komar's formulation (1959) is adequate when a global K i l l i ng vector exists, but his attempts (1962b, 1963) to find well defined reference vectors when only asymptotic symmetries exist were inconclusive. Ulinicour and Tamburino were able to find geometric conditions firmly based on asymptotic symmetries and then modified Komar's superpotential to avoid ambiguities. This gave an elegant def init ion of total energy and momentum applicable to asymptotically f la t radiative space-times. The asymptotic symmetries of a space-time which is asymptotically f lat and which may be radiating outwards are described by the Bondi-ffletzner-Sachs group (Bondi, van der Burg,and fletzner, 1962; Sachs, 1962) which is defined at future null in f in i ty and which is somewhat larger than the Poincare group. Hence uniquely defined symmetry descrip-tors exist at future null in f in i t y , and these may be reached by the conformal method of Penrose (1963, 1964, 1965). Ulinicour and Tamburino established a well defined reference system by considering a closed two-surface bounding a region of a space-l ike hypersurface within which the total energy and momentum wi l l be evaluated. The two-surface <S specif ies outwards and inwards nul l directions at each of i ts points. The outwards nul l vector f i e ld U generates a nul l hypersurface which extends to future null i n f in i t y . 26 Reference vector f ie lds J are then defined uniquely in terms of the asymptotic BflflS descriptors throughout the null hypersurface by the propagation equation These conditions were chosen on the grounds of simplicity and because they are sat isf ied immediately by a global K i l l i ng vector* These reference vectors cannot be inserted direct ly into Komar's superpotential because they are only defined on the null hypersurface. Ulinicour and Tamburino found i t appropriate to modify the superpotential of Komar to where «7 is a nul l vector representing the incoming null direction and specified suff ic ient ly by / * % v ^ O , / r V ^ - / , ^ V > ; ^ ~ O- (1.30) These authors showed that the flux acroessthe null hypersur-face is well defined local ly and independent of the remaining freedom in ho t and further that this formulation gives a monotonically decreasing total energy of a radiating system in accordance with earl ier intuit ive conclusions by Bondi and Sachs. 27 The vectors considered so far have been quite extraneous to the dynamics of the gravitational f i e l d , except to the extent that they reflect i t s symmetries. of four orthonormal vectors, their components may be used as the gravitational f i e ld "variables in place of the components fo the metric tensor. The vierbein formulation of general re lat iv i ty is completely equivalent to the metric formulation. There is an additional gauge group corresponding to the six degrees of freedom of the homo-geneuous Lorentz group of vierbein rotations, since sixteen vierbein components are replacing ten metric components. This freedom leads to the particular advantage of the vierbein formulism which is the clear specif ication of directions in space-time unconfused by the coordinate label l ing of points. The vierbein vectors hfc provide a decomposition of the metric tensor 1 But when the vectors are restricted to be a vierbein (1.31) and sat isfy the orthogonality conditions (1.32) It has long been realized that the gravitational f i e l d equations may be written (Rosenfeld, 1930) 28 and T\ = T ^ 4 / ~ = S/^ ( 1-35) are the vierbein energy and momentum currents of matter. Of course vierbein vectors may be introduced direct ly into the Komar superpotential by taking^, for as was done by Pirani with the timelike vierbein vector n(o)~ ^ * n n * s adaptation of Komar's formulation. The possib i l i ty of ameliorating conservation laws for energy and momentum was explored in a different way by Mi l ler , after he had recognized the limitations of his metric complex (1.20). Curiously enough, even after Komar had shown how to generalize mi l ler 's old complex into a fu l l y covariant structure involving preferred vector f ie lds , Waller himself turned to a not quite covariant formalism based on vierbeins which he wanted to be preferred vector f i e l d s . Waller started from a Lagrangian density This Lagrangian was studied independently in different contexts by Rosenfeld (1930), mailer (1961), Rayski (1961), and Kaempffer (1968). Like Einstein's Lagrangian (1.11) this Lagrangian lacks second order f ie ld derivatives, and is derived from the curvature scalar density by sp l i t t ing off a total divergence. It is a true world scalar density, but is not covariant with respect to the vierbein Lorentz group • Miller derived a canonical energy-momentum vierbein complex from the above Lagrangian (1961b, 1964), together with a superpotential t \ « A . - (1.37, These quantities are not scalars under vierbein rotations, as the vierbein is defining the reference system. Although the complex is not a true world tensor, the super-potential is since i t takes the form U*> v - - = t ' y f f i ^ ( 1 - M ) miller found that the integrals V S provide a satisfactory definit ion of total energy and momentum for asymptotically f lat space-times when the vierbeins are constrained to be asymptotically para l le l . Then the P^, transform as the components of a free vector with respect to asymptotically r ig id coordinate transforma-t ions. The total energy wi l l decrease monotonically when there is outwards radiation. Despite many-efforts, Mil ler was unable to obtain physically satisfying conditions to f ix the vierbein orientation everywhere and thus obtain an int r ins ic loca l -ization of gravitational energy in a general space-time. He f ina l l y concurred (1966) with a widespread opinion that 30 a genuine local izat ion of gravitational energy cannot be just i f ied unless indicated by new physical considerations. Pel legr ini and Plebanski (1961, 1962) apparently f i r s t recognized the possib i l i ty of extracting a completely world covariant conservation law from the Mil ler formalism. Energy and momentum are associated with the vierbein currents C where ^ and (t\ + t\) l\ * O • The superpotential takes the expl ic it form (1.41) It is somewhat similar in structure to the Ulinicour-Tamburino superpotential (1.29) which also has a bivector term. This conservation law was f i r s t derived from the f u l l variational formalism by Rodichev (1965) in the particular "harmonic" gauge kt^U'O* a n o * i n general by Frolov (1964, 1965). The coordinate transformations considered were sx? —*J<?* ^ v kk^S^C*) with the vierbein vectors serving as descriptors. This development w i l l be considered in detai l in Section 4. Frolov also derived several other possible expressions for energy and momentum vierbein currents.from different statements of the action pr inciple, including that of Pirani but multiplied by a factor i . 31 W h i l e t h e P e l l e g r i n i - P l e b a n s k i s u p e r p o t e n t i a l ( 1 . 4 1 ) i s p r o j e c t e d f rom M i l l e r ' s v i e r b e i n s u p e r p o t e n t i a l ( 1 . 3 8 ) a c c o r d i n g t o (J*** - U**. D a v i s and York (1969) have p r o p o s e d a d i f f e r e n t c o v a r i a n t g e n e r a l i z a t i o n where Ja might be a v e c t o r d e s c r i p t o r of a group o f m o t i o n s . In t h i s l a s t a p p r o a c h t h e r e i s a dependence on t h e two r e f e r e n c e s y s t e m s p r o v i d e d by t h e 7ft and t h e JW • Much o f t h e p r e s e n t i n v e s t i g a t i o n w i l l u l t i m a t e l y f o c u s on t h e P e l l e g r i n i - P l e b a n s k i e x p r e s s i o n ( 1 . 4 0 ) , w h i c h w i l l be c a l l e d t h e " c a n o n i c a l " v i e r b e i n energy and momentum c u r r e n t s t o u n d e r l i n e t h e f a c t t h a t t h e y do not c o n t a i n s e c o n d o r d e r d e r i v a t i v e s (hence t h e s u b s c r i p t "C'). The main t o o l s t o be used h e r e a r e L a g r a n g i a n v a r i a t i o n a l p r i n c i p l e s a p p l i e d t o v i e r b e i n s a s g r a v i t a t i o n a l "gauge f i e l d s " and some d i f f e r e n t i a l geometry a l s o a p p l i e d t o v i e r b e i n s . Some o t h e r a p p r o a c h e s t o t h e p r o b l e m of g r a v i t a t i o n a l energy w h i c h use d i f f e r e n t t o o l s a r e m e n t i o n e d c u r s o r i l y . These i n c l u d e t h e i m p o r t a n t H a m i l t o n i a n t e c h n i q u e s a p p l i e d t o g e n e r a l r e l a t i v i t y i n t h e m e t r i c f o r m u l a t i o n by D i r a c ( 1 9 5 9 a , 1959b) and A r n o w i t t , Deser and M i s n e r ( 1 9 6 0 , 1 9 6 1 , 1962) and i n t h e v i e r b e i n f o r m u l a -t i o n by S c h w i n g e r ( 1 9 6 3 a , 1 9 6 3 b ) , w h i c h i d e n t i f y t h e t o t a l energy measured a t s p a t i a l i n f i n i t y i n a s y m p t o t i c a l l y f l a t s p a c e s . There i s a l s o t h e b i m e t r i c f o r m a l i s m d e v e l o p e d by 32 N. Rosen ( 1 9 4 0 , 1 9 6 2 ) , P a p a p e t r o u (1948) and o t h e r s w h i c h i n t r o d u c e s an u n p h y s i c a l f l a t s e c o n d m e t r i c t o o b t a i n f o r m a l c o v a r i a n c e t h r o u g h s u b t r a c t i o n o f a f f i n e c o n n e c t i o n s . Synge ( 1 9 5 9 , 1960) has p r o p o s e d i n t e g r a l c o n s e r v a t i o n laws based on t h e Riemann t e n s o r and t w o - p o i n t f u n c t i o n s . A l l o f t h e s e t e n t a t i v e s s h a r e t h e one common t h r e a d t h a t s u p p l e m e n t a r y s t r u c t u r e s o f some k i n d a r e b rought i n t o g e n e r a l r e l a t i v i t y f o r t h e d i s c u s s i o n of g r a v i t a t i o n a l e n e r g y . A n g u l a r momentum i n s p e c i a l r e l a t i v i t y g e n e r a t e s r i g i d L o r e n t z r o t a t i o n s of t h e c a r t e s i a n c o o r d i n a t e s y s t e m s about some o r i g i n . The c o n s e r v e d a n g u l a r momentum d e n s i t y s p l i t s i n t o o r b i t a l and s p i n c o m p o n e n t s , t h e l a t t e r d e p e n d i n g on t h e homogeneous L o r e n t z group r e p r e s e n t a t i o n r e a l i z e d by t h e f i e l d v a r i a b l e s . The o r b i t a l a n g u l a r momentum t e n s o r d e n s i t y i s i d e n t i c a l l y d i v e r g e n c e l e s s when t h e energy-momentum t e n s o r i s s y m m e t r i c . Landau and L i f s h i t z (1951) r e q u i r e d symmetry o f t h e i r g r a v i t a t i o n a l e n e r g y -momentum complex f o r t h i s r e a s o n . Bergmann and Thomson (1953) f o r m u l a t e d a i r a r a g u l a r c o n s e r v a t i o n law v a l i d even w i t h an a s y m m e t r i c e n e r g y -momentum c o m p l e x , u s i n g a h y b r i d m i x t u r e of r i g i d c o o r d i n a t e r o t a t i o n s and v i e r b e i n r o t a t i o n s . 33 A c o v a r i a n t s p i n c o n s e r v a t i o n law was d e r i v e d i n d e p e n d e n t l y by K i b b l e (1961) and Sc iama ( 1 9 6 1 ) . T h i s c o n s e r v a t i o n law i s a s s o c i a t e d w i t h i n v a r i a n c e under g e n e r a l v i e r b e i n r o t a t i o n s , q u i t e d i s t i n c t f rom w o r l d c o o r d i n a t e t r a n s f o r m a t i o n s . The v i e r b e i n s r o t a t e i n d e p e n d e n t l y about t h e p o i n t s o f t h e s p a c e - t i m e m a n i f o l d t o w h i c h t h e y a r e a t t a c h e d , u n l i k e t h e r i g i d r o t a t i o n of a g l o b a l c o o r d i n a t e s y s t e m about some s e l e c t e d o r i g i n . T h e r e f o r e no t e r m c o r r e s p o n d i n g t o o r b i t a l a n g u l a r momentum a p p e a r s . The s i m i l a r t r e a t m e n t s o f K i b b l e and Sc iama d e v i a t e somewhat f rom s t a n d a r d g e n e r a l r e l a t i v i t y , i n t h a t t h e v i e r b e i n s p i n c u r r e n t s o f m a t t e r a r e c o u p l e d t h r o u g h f i e l d e q u a t i o n s t o a dynamic v i e r b e i n c o n n e c t i o n i n g e n e r a l e q u i v a l e n t t o an a s y m m e t r i c a f f i n e c o n n e c t i o n . The g r a v i t a t i o n a l s p i n c o n s e r v a t i o n law has a l s o been d e r i v e d by R o d i c h e v (1965) and F r o l o v ( 1 9 6 4 , 1965) i n t h e e n t i r e l y c o n v e n t i o n a l t h e o r y , where t h i s c o n s e r v a t i o n law seems r a t h e r t r i v i a l . These t o p i c s a r e d i s c u s s e d i n d e t a i l i n S e c t i o n 4 . 34 2. G e n e r a l Symmetr ies and C o n s e r v a t i o n Laws a . I n v a r i a n c e s and I d e n t i t i e s L o c a l or d i f f e r e n t i a l c o n s e r v a t i o n laws a r i s e f rom t h e i n v a r i a n c e s of a s e t of f i e l d e q u a t i o n s . The i n f i n i t e s i m a l t r a n s f o r m a t i o n s of t h e f i e l d v a r i a b l e s t o be c o n s i d e r e d a r e symmetry t r a n s f o r m a t i o n s w h i c h means t h a t t h e y do not a l t e r t h e f i e l d e q u a t i o n s . I f t h e c o n t e n t of t h e f i e l d e q u a t i o n s i s condensed i n t o an a c t i o n p r i n c i p l e , t h e f i e l d e q u a t i o n s w i l l be i n v a r i a n t i n fo rm under a g i v e n t r a n s f o r m a t i o n i f t h e a c t i o n i n t e g r a l i s i n v a r i a n t or e q u i v a l e n t l y t h e a s s o c i a t e d L a g r a n g i a n d e n s i t y i s m o d i f i e d by a t most t h e a d d i t i o n of a t o t a l d i v e r g e n c e . C o n s i d e r a f i e l d ^ ^ a n d a L a g r a n g i a n d e n s i t y jfC^&^XAt) w h i c n s p e c i f i c a l l y i s a s c a l a r d e n s i t y and c o n t a i n s f i e l d d e r i v a t i v e s up t o second o r d e r . A c t u a l l y t h e n o t a t i o n *f w i l l be t a k e n t o i n c l u d e t h e components of a l l f i e l d s e n t e r i n g i n t o £ and i t i s not r e q u i r e d t h a t a l l t h e s e f i e l d s s a t i s f y f i e l d e q u a t i o n s . The symmetry t r a n s f o r m a t i o n s a c t i n g on t h e f i e l d v a r i a b l e s a r e assumed t o b e l o n g t o a c o n t i n u o u s g r o u p . S p a c e - t i m e t r a n s f o r m a t i o n s y i e l d i n g t h e may be i n c l u d e d . The " t o t a l v a r i a t i o n " or v a r i a t i o n i n f o r m of t h e f i e l d components i s (2.1) 35 Here $ S ^Ox-O-^O*) (2-2) is the "coordinate var iat ion". The dist inction between the total and coordinate variations is that between the comparison of f i e ld values at the same coordinate point or at the same world point respectively, before and after the transformation. This dist inction only matters when space-time transformations enter to shift the coordinate label l ing of world points. The operation $ commutes with part ia l d i f ferent iat ion. The condition that the f ie ld equations be invariant in form under the transformations is which follows immediately from the required invariance of the action integral . Here Then the basic identity (see Bergmann and Schi l ler , 1953i Trautman, 1962) is obtained through integration by partst u r<^  + cx ,\ = o , ( 2 . 4 ) - c^&*<&*^*4+&,™>' (2-6) is the "generating density" (Bergmann, 1958). When the Lagrangian contains only f i r s t order f ie ld derivatives, 3 6 the generating density reduces to When the f i e l d equations hold for a l l the^f^, or when for non-trivial transformations, a local conservation law holds immediately! C \ A - O • (2.9) The generating density depends on the arbitrary parameters of the symmetry group which actually enter into the r^ f^ . Useful conservation laws may be obtained.from (2.9) when this arbitrariness can be eliminated through a detailed consideration of the specific symmetry group. If the group is a p-parameter Lie group, then (2.10) = Ta A B where the E*' (a • l,...,p) are the infinitesimal constant group parameters and the ~J~«. are the corresponding generators in the representation(s) realized by the . Then a local conservation law may be obtained from (2.9) through the elimination of the constant* (see Section 2c). Conversely the invariance group may be an in f in i te dimensional "function" group whose transformations depend 37 on p arbitrary functions of position £ The general coordinate transformation group to be considered in Section 2b and the gauge groups of Section 2c are examples. The general inf initesimal transformation wi l l be assumed to be %<eA= VA«£*0*)- "V** £'c~)p. ( 2 .n ) Then (2.4) yields a succession of identit ies corresponding to the necessary vanishing of the coeff icients of £*C>c.) and i t s derivatives. If in particular i t is assumed that the £ * ^ v , vanish on the boundary of the region of integration of the action Integral and i t is not assumed yet that the f ie ld equations hold, then the "generalized Bianchi ident i t ies" of the theory are obtained di rect ly i ^ & ^ > - ° <2-12' If the in f in i te function group has anr parameter Lie subgroup (constant parameters), "strong conservation laws" hold independently of the f i e ld equations. When the position dependence of the function parameters is separated according to S"6c^)ei (^/>->o (2.i3) then f < f * = £*(f*4 *Aa _ ^ and C X Z £*C\ 38 The fundamental identity (2.4) becomes Upon insertion of the generalized Bianchi identit ies (2.12) the strong conservation law is obtainedt [C\- fg4i;*"x.]/\ =o {2.1S) This implies the existence of antisymmetric "superpotentials" since the strong conservation law holds ident ical ly . The statement of a superpotential is a statement of a strong conservation law. 39 b. Space-time Transformations The covariance group of general re lat iv i ty is the infinite-dimensional group of general coordinate transformations. It possesses an in f in i ty of one-parameter subgroups, each generated by some vector f i e ld • Any one-parameter group of point transformations induces a differentiable vector f i e ld which is tangent to the trajectories of the group. Conversely, given the vector f i e l d , such a group exists at least local ly (see Trautman, 1964, page 88). The vector f i e id ^ generates the point displacements, determining both a direction and a scale. A coordinate transformation may be associated with a point transformation of such a group by taking the new coordinate sys tem^ = r Gx?) t o b e that obtained by "dragging along" by the point transformation. When the parameter <f0>e) of the transformation is inf in i tes imal , the coordinate transformation is ^ H ^ = (2.1?) When £ is constant, the group is a one-parameter Lie group. The total variation in any geometric object f i e ld ^ under this transformation is given by the Lie derivative (see Yano, 1956) taken with respect to = £ r ^ ^ - ^ ; — {2.18) 40 The Lie derivative in (2.18) includes the "dragging along" (physical point displacement) required to compare the f ^ a t the same coordinate point before and after the coordinate transformation. When the ^ constitute a tensor f i e l d , where the are the generators of the general linear group in the corresponding representation (see B.S.DeWitt, 1963). The total variations of a scalar f i e l d , a contra-variant vector f i e l d , a covariant vector f i e l d , and the metric tensor are respectively = ~ ^ ( 2 ' 2 0 ) f f v = ~ ^ n - f v ^ ' " (2'22) 5 ^ ^ - - fht&p- friixf/cr (2.23) In this way the Lie derivative of a tensor f ie ld describes the deformation of the induced by a transformation oVa^-' ^ is a tensor f ie ld of the same type as . The commutator of the Lie d i f ferent ia l operators with respect to two vector f ie lds and f j, acting on any tensor f ie ld is 41 (2.24) where and are d i f ferent ia l operators defined by This relation gives the "bracket" of two one-parameter groups, and follows directly from (2.19) and the commutation relations of the general linear group f F > , / * £ « X F*, - F\ (2.26) If r l inearly independent vector f ie lds fa. are tangent to the trajectories of a r-parameter Lie group of transformations, where the cC^ c = ""^ -T«^  a r e ^ n e stnocture constants of the group. The structure constants are tensors under transformations of the basis of the Lie algebra (2.28). The inf initesimal transformations of the Lie grouptaare 4 2 A t e n s o r o r a f f i n e c o n n e c t i o n i s i n v a r i a n t u n d e r t h e s e t r a n s f o r m a t i o n s i f Under quite general conditions (Yano, 1956, Chapter III) there exists some linear geometric object which is invariant under a given Lie group of transformations. For example a one-parameter Lie group of transformations is the invariance group t r i v i a l l y of i ts generating vector f i e l d , since r In particular an r-parameter group is a group of motions or isometries i f £jff-& under i ts transformations Then from (2.23) and each such £ a is a K i l l i ng vector. The significance of the transformations of a group of motions - £ J"a is that they leave distances and indeed the entire geometry invariant. A Rlemannian space-time that has some symmetry may possess up to ten independent global K i l l ing vectors, corresponding to the dimension of i ts group of motions. Ten K i l l ing vectors exist i f the space-time is of constant curvature. The group of motions of Minkowski space-time is the Poincare group. The subgroup of anr-parameter group of motions that leaves a point unchanged is the isotropy group at that point, and wi l l be of dimension s ^ 6 . If there is a unique transformation carrying a given point to any adjacent point, the group is "simply transit ive" with s a 0 and i « 4. When the descriptors J of a group of motions are of constant length <jf*§<*» the trajectory of each point undergoing transformation is geodetic and the transformations are cal led "translations". K i l l ing vectors when they exist determine geometrically preferred directions to within a constant basis transformation. In their absence one is forced to use other c r i te r ia for determining reference vectors or else relinquish well defined local conservation laws of energy, momentum, and angular momentum in general re lat iv i ty except when a group of motions is present. Consider the case when there is a group of motions. A matter f i e ld with Lagrangian^/^(<f^) ¥*hf)ff*e) i s P r e s e n t and sat is f ies i ts f i e ld equations, but the gravitational f i e ld is taken to be non-dynamic ( i ts f i e ld equations are not enforced). The Noether identity (2.4) becomes since l^1* - O * &<%<yJ - 0 and E° is c o n s t a n t . The conserved quantities corresponding to transit ive transformations are canonical energy and momentum currents 7 * (2.33) The canonical energy and momentum currents assume the familiar form of the canonical energy-momentumttensor in special re lat iv i ty when the coordinate system is matched to the vector descriptors ya — * 6 j . Conservation laws for angular momentum are similarly obtained when the group of motions has an isotrbpy subgroup. In general, when a group of motions may not be present, the parameters'^,) wi l l be taken to be arbitrary functions of posit ion. The Bianchi identit ies (2.12) are The arbitrariness of the derivatives of £ a also precipitates a number of identit ies from the general* Noether identity beyond (2.32), including the connection between the canonical energy and momentum currents and the currents projected from the symmetric energy-momentum tensor appearing in (2.34)0(Trautman, 1962). This general procedure of decomposition of the Noether identity wi l l be applied to the combined Lagrangian of matter and gravitational f ie lds in Section 4. In general the following relations are determined expl ic i t ly (/j* * - u:x c. Gauge Fields and Currents When the symmetry transformations are those of a Lie group (2.36) with constant inf initesimal parameters £. and with coordinate transformations expressly excluded for now, the conservation law obtained from a Lagrangian €^/<r) and (2.4) is B (2.37) This is a weak conservation law, holding only when the f ie ld equations already hold. If i t is to be required that differences in gauge at different points be unobservable, the restr ict ion to r ig id gauge transformations ( constant) must be relaxed. The f i e ld ^ ^ i s then given f u l l gauge freedom through allow-ing transformations that vary with posit ion, = T / » (2.38) while maintaining the invariance of modified f i e ld equations. This requires the generalization of the gauge group into an infinite-dimensional group generated by the original n© group. The generalized gauge invariance may be achieved by modifying the Lagrangian through the derivative coupling of "compensating" or gauge f ie lds The gauge-covariant derivatives are written (using a dot) which are invariant under (2.38) when B^u transforms as = £l Cu f £ * > • ^ > (2 . «o) The variation in the gauge f i e ld is gauge-covariant (see B.S.DeWitt, 1963). Here the C">ic are the structure constants of the original Lie group, which give the algebra of i t s generators [T±)ToJ •=• C"i4e Tct (2.41) When a l l derivatives of the ^ a p p e a r i n g in the Lagrangian are replaced by gauge-covariant derivatives zee*, zc^j^.j xxwi^bi) the new f i e l d equations wi l l be invariant in form under position-dependent transformations as desired. Since there is a new set of f ie lds /T^, in £ which are not yet dynamically active (lacking a free Lagrangian), the current conservation law wi l l be modified. The identity (2.4) is now 47 Using the f i e l d equations ^4 ~ ^ » ( 2 » 3 9 ) t (2.40) and the arbitrariness of the parameters, one arrives at a v vpartial conservation law^ j \ C*>*C- BCA ~ 0 - JX«*\ ( 2 . 4 2 ) where the currents ^ - • - & are only gauge-covariantly divergenceless. This is of course very reminiscent of the situation in general relativ-i t y . A curvature tensor in group space may be defined from the commutator of gauge-covariant derivatives ^ . v - S f V 5 -F'x, T^e<€s (2.44) This "internal curvature" or f i e ld strength tensor is F V = B\i*-B%»- B^B^ <2-'5> It may be used to construct an invariant free Lagrangian^ for the gauge f i e l d . The f i e ld equations of the gauge f i e ld w i l l be §B*> W9A (2.46) Except in the case of electromagnetism where the gauge group is abelian (CC*j^ ca 0), these f ie ld equations are non-linear and the gauge-covariant divergence condition 48 (2.42) tor the source currents is not a true local conservation law. In order to regain a differential conservation law which is correctly a partial divergence, i t is necessary to add an extra gauge current to the source current vT« so that (J\ + - O (2.47) The current Jft is interpreted as the current of the non-abelian gauge f i e l d i t s e l f . If i t is assumed that then the Noether identity (2.4) and (2.40) give (see for example Kaempffer, 1965, page 182) and + j\ = f / W / v (2.49) Unlike J * , is not "gauge-covariant. This is also true of the energy-momentum complex in general r e l a t i v i t y . But the form of j^ is well defined because the f i e l d tensor (2.45) appears as superpotential, whereas in the theory of gravity there is no such evidently unique expression. The prototype gauge f i e l d is the vector potential of electromagnetism. Other gauge fields have been associated speculatively with the currents associated with SU2 and SU3 in strong interactions (Yang and mills, 1954t 49 Sakurai, I960, 1 9 6 8 ) . The term "Yang-mills group" is sometimes used to decribe gauge groups in general. As was anticipated in the ideas of Weyl (1929, 1950) , the c lass ica l theory of gravitation also f i t s into the above scheme when i t is extended to include space-time transformations (Utiyama, 1956) . This requires the vierbein formalism to be described next. 50 3. Vierbein Formalism a. Fundamentals The gravitational f i e ld is expl ic i t ly displayed in gauge f ie ld form when the vierbeins (also cal led four-legs or tetrads) already introduced are used as gravitational potentials. The four vierbein vector f i e l d s ^ themselves perform the function of gauge f ie lds associated with general coordinate transformations (see Kibble, 1961), while six additional gauge f ie lds (vierbein connection) must be introduced to ensure invariance under arbitrary position dependent Lorentz rotations of the vierbeins (Utiyama, 1956). The sixteen vierbein components A*°r wi l l be taken to be the primary gravitational f i e ld variables. As wi l l be discussed in the next section, the twenty-four components of the vierbein connection may either be regarded as fixed combinations of the f i r s t derivatives of the /Af or else allowed independent dynamical freedom. The f i r s t alternative yields a formulation of general re lat iv i ty that is entirely equivalent to the orthodox metric treatment. The second alternative leads to modifications in the presence of spinning matter. The vierbein formalism is based on the relationship between the space-time "world" manifold and the orthogonal axes a t any p o i n t ( l o c a l i n e r t i a l f r a m e ) t o w h i c h p h y s i c a l 51 q u a n t i t i e s may be r e f e r r e d . The o r t h o g o n a l d i r e c t i o n s of t h e l o c a l f r a m e s ( v i e r b e i n d i r e c t i o n s ) w i l l be l a b e l l e d by l a t i n i n d i c e s i , j , k , . . . w h i l e greek i n d i c e s w i l l be r e s e r v e d f o r s p a c e - t i m 8 ( " w o r l d " ) c o o r d i n a t e s . The c h o i c e of l o c a l i n e r t i a l f rame i s o n l y u n i q u e up t o a homogeneous L o r e n t z t r a n s f o r m a t i o n , h e r e c a l l e d a v i e r b e i n r o t a t i o n . A c c o r d i n g l y , v i e r b e i n s may be used t o g i v e a s p e c i f i c a t i o n of d i r e c t i o n s a t each p o i n t i n d e p e n d e n t l y of t h e c o o r d i n a t e l a b e l l i n g of p o i n t s . A n o t h e r use of t h e v i e r b e i n f o r m a l i s m i s t h e d e f i n i t i o n of s p i n o r f i e l d s i n g e n e r a l r e l a t i v i t y , w h i c h a r i s e n a t u r a l l y as s p i n o r r e p r e s e n t a t i o n s of t h e v i e r b e i n L o r e n t z g r o u p . As m e n t i o n e d on page 2 7 , a v i e r b e i n i s a s e t of f o u r m u t u a l l y o r t h o g o n a l u n i t v e c t o r s 7l/r ( k = ( 0 ) , ( 1 ) , ( 2 ) , ( 3 ) ) w h i c h a t some p o i n t d e f i n e a l o c a l i n e r t i a l f r a m e . In g e n e r a l t h e v i e r b e i n c a n n o t be t h e b a s i s e v e r y w h e r e of a h o l o n o m i c c o o r d i n a t e s y s t e m . The s i x t e e n components may be r e g a r d e d as t r a n s f o r m a t i o n c o e f f i c i e n t s f rom h o l o n o m i c w o r l d c o o r d i n a t e s t o n o n - h o l o n o m i c c o o r d i n a t e s o f t h e l o c a l i n e r t i a l f r a m e . The i n v e r s e t r a n s f o r m a t i o n i s g i v e n by t h e c o v a r i a n t v i e r b e i n components /i^f r e c i p r o c a l t o t h e • These c o v a r i a n t components w i l l be used as g r a v i t a t i o n a l f i e l d v a r i a b l e s . T h e i r d e t e r m i n a n t ( J a c o b i a n of t h e t r a n s f o r m a t i o n f rom l o c a l i n e r t i a l f rame t o w o r l d c o o r d i n a t e s ) i s 52 which by convention wi l l be posit ive. The metric in the local frames is the Minkowski metric, and the following orthogonality relations holdi hk'*r bfr* ^ * V ( 3 . 3 ) / V /«> • = (3 .4 ) - ^ ( 3 . 5 ) The restr ict ion to orthonormal vierbeins wi l l be assumed in the following, although it is not necessary. In part icular , a "null vierbein" may be formed as Null vierbeins are useful in the study of radiative f ie lds , and are most directly connected to the spinor formalism (see Newman and Penrose, 1962? Trautman, 1964). The vierbein components of any world tensor are projected from i t according to and constitute the corresponding tensor realizations of the vierbein Lorentz group. 5 3 When the vierbeinsare Lorentz rotated with an arbitrary dependence upon posit ion, = RHeM^ (3,B> The Lorentz group orthogonality conditions holds Rk* = £JL ( 3 . 9 ) The metric tensor and i t s functionsssuch as the affine connection and the Riemann tensor are of course invariant under vierbein Lorentz transformations. In the vierbein formulation of gravitation i t is desirable that the complete formalism be covariant with respect to world coordinate transformations, with any non-covariance shifted onto the selection of vierbein frames. In general f u l l covariance under both groups is unattainable for the three-index quantities from which conserved complexes and currents are constructed in gravitational theory. In the vierbein treatment unlike the metric formulation there is a simple third rank tensor l inear in the f i r s t derivatives of the potentials, in close analogy to the electromagnetic f ie ld tensor. Consider the vierbein transcription of (2.27) S/Ug/Y-UrA/n)= GL,jkk6 (3.io) 54 Hers G-'jk ~ PtCtftitt - V ^ j V ( 3 , U ) wi l l be cal led the "structure coeff ic ients" (G.Rosen, 1962). These objects are world scalars but transform inhomogeneously under the vierbein Lorentz group. They are not constants except in special vierbein frames in special spaces admitting transit ive groups of motion. The term "object of anholonomicity" is often used for the &L,jk 1" the l i terature . The world tensors formed from the structure coeff icients r k I k / K C r ~ h ftj — *> ^ / V (3.12) and Q^jyf - ^tfj (3.13) w i l l be cal led "structure tensors". Under vierbein rotations G%-+ G% + ^ V r t * < / ? % (3 .U ) -faR*LR"*\« J An " int r ins ic derivative" (Eisenhart, 1926, page 98) o f some quantity A may be defined as and (3.11) may be written in analogy to (2.28) as A coordinate transformation may be parametrized as ^ - * ^ v = ^ + £kc^-)^. ( 3 - 1 7 ) In analogy to the transformation behaviour of a general gauge f i e ld (2.40) the covariant vierbein components transform as $ ~ ~ ^ / > r - " - ^ ^ / b r " ^ " (j^jj£<y ( 3 . 1 3 } b. Vierbein Covariant Derivatives It is necessary to have a specif ication for the differentiat ion of vierbein quant i t ies^ ,v ... that is covariant under both world coordinate transformations and vierbein Lorentz transformations. This requires the introduction of a vierbein connection pij which is to be the gauge f i e ld corresponding to the gauge group generated by the vierbein Lorentz group. The total variation of a world contravariant vector under an inf initesimal paral le l displacement from P to P' is which defines the world covariant derivativeA*jr • The affine connection is assumed to be symmetric for now» so that r*e%f=r*/rof • T n e local parallelism of the 56 displacement means that the transformation is local ly isometric, so that Sp^^z O and hence fff/iV^O* Now i f /A is replaced in (3 .19) by a vierbein vector h/f , the paral lel - transferred vierbein wi l l in general be rotated with respect to that already at P' . /»/ - - $r«? = - e** /u* ( 3 . 2 0 ) where to f i r s t order ALj = SCj + €cj , <£cj ~ - Eji . ( 3 . 2 i ) The inf initesimal rotation of the vierbein wi l l be linear in the inf initesimal paral le l displacement, and hence (see Sciama, 1961) hf, " Sr*" = a«'h* = B« f» ( 3 . 2 2 ) introducing the vierbein connection B^^,--B^^,- Therefore V/* + r*v V - Bit *jV A,* = O <3-«> Then by antisymmetrization, since P''**^ - > or in terms of the "rotation coeff ic ients" &L),k5- hf?B1^ Bcj>*-— Bihj = t r o j * (3 .24) and also uniquely l?Lj,/r- 4;(&i>)k~ &),ik ~ (rtr, j) (3 .25) 57 The following useful relations hold h Lr)S = B (3.26) The geometrical interpretation of the "rotation coeff ic ients" (see Eisenhart, 1926, page 99) is that B\.yk is the rate of Lorentz rotation in the ( i j ) plane looking in the (k) direction of vierbeins towards immediately adjacent vierbeins. These components describe the layout of the vierbein la t t ice around each point. Bfix/)}(o) and are acceleration and angular velocity components respectively of an observer whose time direction is that of the local vierbein frame. A general vierbein quantity (where A , B t . . . refer to groups of vierbein indices according to the tensor or spinor representation of the vierbein Lorentz group realized by the quantity under consideration) wi l l transform under an inf initesimal vierbein rotation according to £ Q A - ^1** A U + * QB <3.29) Here the group generators are the "Lorentz spin matrices" Afa in the representation realized by the Q . For a vierbein scalar, /\k£_— O » for a vierbein vector, 58 and in the case of a Dirac spinor, (3.31) The spin commutation relations are [Acj,Ata>]~qu Ajk + ^jkAu~r(}jiALk-i{LkAj4. (3.32) The formalism of Section 2c applies direct ly , with the gauge group generated by the vierbein Lorentz group through allowing a position dependence of the parameters £ ^ and with the antisymmetric pairs of vierbein indices ( i j ) as gauge indices. The vierbein connection transforms as (3.33) with the structure constants of the Lorentz group contained in (3.32) ensuring agreement with (2.40). The vierbein covariant derivative is QV = <?V -i B**y A«±,A8 Q* <3-'«> If the quantity mixes vierbein and world tensor indices, terms in/""v^ wi l l be included to guarantee overall covariance under coordinate and vierbein trans-formations. Then (3.23) becomes 5 9 c . Dynamic V i e r b e i n C o n n e c t i o n S i n c e t h e v i e r b e i n c o n n e c t i o n 8 ^ p e r f e c t l y f u l f i l l s t h e f u n c t i o n of a gauge f i e l d , i t i s p l a u s i b l e t o go f u r t h e r and g i v e i t d y n a m i c a l f reedom on an e q u a l b a s i s w i t h t h e / i j ^ a s g r a v i t a t i o n a l f i e l d v a r i a b l e s . T h i s was p r o p o s e d i n d e t a i l by K i b b l e ( 1 9 6 1 ) and Sc iama ( 1 9 6 1 ) . I t i s an echo of an e a r l i e r s u g g e s t i o n by P a l a t i n i , who f i r s t p r o p o s e d t r e a t i n g t h e components of t h e a f f i n e c o n n e c t i o n as a s e t of i n d e p e n d e n t f i e l d v a r i a b l e s whose form i s t o be r e g a i n e d f rom t h e f i e l d e q u a t i o n s . I f t h e v i e r b e i n c o n n e c t i o n i s t o be a dynamic v a r i a b l e i n a " P a l a t i n i " f o r m u l a t i o n , t h e d e f i n i t i o n ( 3 . 2 5 ) must be r e l a x e d s i n c e i t f i x e s t h e u ^ as a l i n e a r c o m b i n a t i o n of t h e f i r s t o r d e r d e r i v a t i v e s of t h e hwhich a r e a l r e a d y dynamic f i e l d v a r i a b l e s . T h i s d e f i n i t i o n f o l l o w e d d i r e c t l y f rom t h e a s s u m p t i o n t h a t t h e a f f i n e c o n n e c t i o n i s s y m m e t r i c , and t h e r e f o r e t h i s a s s u m p t i o n / " " ^ y — f^^f^^ must a l s o be s u s p e n d e d . The dynamic v i e r b e i n c o n n e c t i o n w i l l be denoted t o d i s t i n g u i s h i t f rom 5^,^ a l r e a d y d e f i n e d i n ( 3 . 2 5 ) more r e s t r i c t i v e l y . The dynamdc v i e r b e i n c o n n e c t i o n has t h e f o r m 60 (3.36) is the torsion tensor whose specif ic form wi l l emerge from the solution of the f ie ld equations. In the theory of Kibble and Sciama, the connection is dynamically determined to be symmetric everywhere except where spinning matter may local ly engender a torsion tensor. to spin currents and to energy and momentum currents. It w i l l be considered in Section 4 as a f o i l to the orthodox theory in the derivation of conserved currents and their relationships. However the standard theory with i t s Riemannian connection wi l l always be assumed in the following except where stated otherwise. d. Curvature Tensor A vierbein curvature tensor is defined from the commutator of double vierbein covariant derivatives From (3.27) this is just the vierbein projection of the Riemann tensor This theory assigns essentially equal status (3.37) (3.39) 61 The Blanch! identi t ies sat is f ied by the Riemann tensor are written in vierbein notation as The contracted Bianchi identi t ies are GX*-\ = O (3.41) "here ^CR\'£^R) = /T7~V (3.42) Therefore the energy and momentum currents based on the symmetric energy-momentum tensor of matter (3.43) sat isfy the vierbein covariant divergence relat ion, which is not an exact local conservation law, = T efX -f- Gk,&* 1 -The last step above follows from the symmetry of The Bianchi identit ies also yield IRXV)ij*y ~ / j ' / ? * t y » —hi*'RXj.f (3.45) which is similar to the f i e ld equations of " internal" gauge f i e l d s . 62 e . V i e r b e i n G r a v i t a t i o n a l L a g r a n q i a n s The dynamics o f t h e g r a v i t a t i o n a l f i e l d a r e assumed t o be c o n t a i n e d i n a L a g r a n g i a n d e n s i t y t h a t l e a d s t o f i e l d e q u a t i o n s o f not h i g h e r t h a n s e c o n d o r d e r i n t h e v i e r b e i n f i e l d . J^q. w i l l be r e q u i r e d t o r e f l e c t t h e i n v a r i a n c e s of t h e t h e o r y under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s and g e n e r a l v i e r b e i n r o t a t i o n s . The o n l y q u a n t i t y w h i c h i n c l u d e s f i e l d d e r i v a t i v e s o n l y up t o s e c o n d o r d e r and w h i c h i s a t r u e w o r l d s c a l a r d e n s i t y and a t r u e v i e r b e i n s c a l a r i s t h e c u r v a t u r e s c a l a r d e n s i t y , so t h e most d i r e c t c h o i c e o f L a g r a n g i a n d e n s i t y i s The E i n s t e i n g r a v i t a t i o n a l c o n s t a n t K i s i n c l u d e d i n o r d e r t o make t h e d i m e n s i o n a l i t y e x p l i c i t . A d i v e r g e n c e •fc(&"i)f« m a y 0 8 9 P m f r o m ( 3 . 4 6 ) t o o b t a i n a f i r s t o r d e r v i e r b e i n L a g r a n g i a n ( R o s e n f e l d , 1930» m i l l e r , 1961» K a e m p f f e r , 1968) w h i c h was m e n t i o n e d i n S e c t i o n l b . T h i s i s a w o r l d s c a l a r d e n s i t y i n v a r i a n t under o n l y r i g i d v i e r b e i n r o t a t i o n s . Second o r d e r d e r i v a t i v e s may a l s o be a v o i d e d by a s s i g n i n g i n d e p e n d e n t v a r i a t i o n a l and dynamic f reedom t o t h e v i e r b e i n c o n n e c t i o n . The " P a l a t i n i " L a g r a n g i a n d e n s i t y 63 fo rmed f r o m t h e c u r v a t u r e s c a l a r d e n s i t y i s The d i v e r g e n c e ^(^A^f^can be d ropped t o o b t a i n a n o t h e r " P a l a t i n i " L a g r a n g i a n d e n s i t y a n a l o g o u s t o (3.47) (3.49) 2K 64 4 . Vierbein Energy, momentum and Spin Currents a . General Considerations It mas emphasized in Section la that there is an extremely wide kinematic ambiguity in the choice of possible expressions for the conserved currents of general re la t i v i t y , whether they are derived from a Lagrangian or simply guessed. Such expressions are without much content unt i l a reference system is fixed by some supplementary geometrical structure. In the absence of any clear direction from the dynamics of the theory, i t seems that a choice between the kinematically allowed expressions for the currents must be based upon considerations of covariance and of s impl ic i ty . When a given expression for the currents is combined with a complementary geometric reference structure, other more physical tests become possible and may be suff ic ient to remove any residual arbitrar iness. It is necessary to make a basic i n i t i a l decision regarding the type of geometric supplemental structure that wi l l be used. From now on, only reference systems based on vierbeins wi l l be considered. This choice is made for the following reasonst (1) Arbitrary contravariant vector f ie lds serve naturally as the descriptors of coordinate transformations which are generated by corresponding generalized "energy" and 65 "momenta". By allowing only unit vector f ie lds which are legs of a vierbein as descriptors, the parameters' of the displacements generated at each point are normalized to the "proper time" of an observer occupying the local vierbein frame. The vierbein descrip-tors f ix the direction and the scale of the displacements. (2) The vierbein frame is a convenient orthogonal basis for the energy and momentum currents. It is an essential basis for the spin currents which are associated with invariance under the vierbein Lorentz group. (3) The vierbein gauge group provides very conveniently the geometrical latitude required for wide classes of supplementary reference structures. (4) The vierbein currents can be formulated to be completely covariant under world coordinate transformations, (50 Since vierbeins constitute a natural set of gravita-t ional f i e l d variables, their use as a basis for the currents entails a minimal redundancy in the formalism even after the vierbein gauge has been fixed by supplementary conditions. Everything f i t s into a Lagrangian formulation in which time and space directions are not treated di f ferent ly . 66 Nora the transformation properties of the currents w i l l be restr icted. The total energy and momentum currents of matter and gravitational f ie lds w i l l be required to have the form (4.1) where each constituent transforms as a contravariant vector density under general coordinate transformations. The matter contribution ~T\ - l^4~J~^/*'tTansforms as a vierbein vector under arbitrary vierbein rotations, but the gravitational contribution ~{_\ only transforms as a vierbein vector under constant vierbein rotations. The total energy and momentum components formed by integration over some space-l ike hypersurface wi l l transform as a vierbein vector under constant vierbein rotations. Similarly the total spin currents *£- * S^/C£ + S^kjt ( 4 . 3 ) wi l l be required to transform as contravariant vector densities under general coordinate transformations and as an antisymmetric vierbein tensor under constant vierbein Lorentz transformations. A l l ten total currents w i l l be chosen to be divergenceless, and wi l l be associated with superpotentialsi 67 &\/x-o , 2^a = o («•«) Expl ic i t expressions for these currents wi l l be derived by subjecting a total Lagrangian embracing matter and gravitational f ie lds to vierbein rotations and coordinate transformations with vierbein descriptors, and applying the variational procedure of Section 2 a . Matter f ie lds in the absence of gravitational f ie lds wi l l be considered f i r s t . It is convenient to start with the spin currents. b. Vierbein Spin Currents The invariance of the matter and gravitational f i e l d equations under the vierbein gauge group of homgeneous Lorentz transformations leads to local conservation laws for six "vierbein spin currents". They are intimately associated with the vierbein energy and momentum currents. The absence of any orbital angular momentum partner to the spin currents follows from the nature of the transformations considered. Each of the vierbeins rotates around the world point to which i t is attached, in contrast to the r ig id gyration of a global 68 coordinate system about a single point to which moments elsewhere refer . 1) Matter Spin Currents The vierbein components of some f ie ld transform as under a Lorentz rotation of the vierbeins. The/l^ are the homogeneous Lorentz group generators in the appropriate representation. The transformation is inf in i tes imal . The Lagrangian density of this f i e ld is f i r s t assumed to be invariant only under vierbein rotations with constant parameters £~ ~ — This i s a minimum requirement. The gauge f i e ld provided by the vierbein connection is not included here. Then the structure coeff ic ients must vanish, the vierbeins must be regarded as referring everywhere to a single global iner t ia l frame, space-time is f l a t , and there is no gravitational f i e l d . The vierbein f ie ld i t se l f must appear in the be a world scalar density since coordinate systems have not been restr icted. (4.7) Lagrangian density ^^,f1v, A^) i n o r d e r t h a t J t The variation in the above Lagrangian density under an inf initesimal vierbein rotation is ( 4 . 8 ) where the matter f i e ld equations =• 0 apply, and -r\ = <)A ( 4.9) are the canonical energy and momentum currents of the matter f i e l d which wi l l be discussed in Section 4 c . The vierbein spin currents of the matter f i e ld ^ ^ a r e and satisfy the divergence condition SX**/\ = T^KA - HjLk (4.11) The matter spin currents defined above are not conserved because of the asymmetry of the canonical energy-momentum tensor. Now the above Lagrangian density wi l l be modified to be invariant under position dependent vierbein rotations through the insertion of the vierbein connection through the prescription of Section 3 b The vierbein-covariant derivative is An "external" gravitational f i e l d may now be present. 70 The variation of the vierbein gauge invariant Lagrangian density under an inf initesimal vierbein Lorentz transformation is Assuming the matter f i e ld equations and taking leads to the following relations (Kibble, 1961) on account of the arbitrariness of and i t s derivatives. S\jL-34? A ^ > A P ^ f ^ - ^ ^ y (4.13) 2) Total Vierbein Spin Currents The vierbein spin currents wi l l now be derived when matter and gravitational f ie lds are dynamically act ive. The conventional theory of gravity with a Riemannian connection wi l l be considered f i r s t . The gravitational Lagrangian is chosen to be so that the total Lagrangian density is Under an inf initesimal vierbein rotation 71 the total Lagrangian density transforms as 2 6 3 2 ^ WAJ where the respective f ie ld equations have been assumed S*** ~ (4'10) The total spin current which is conserved is (4.12) Now the term is expressible as a certain combination of the matter spin currents. Since and IK. one has ^ _ ^ ^ ^A^Aii/ <5Jl^  l { \ i v (4.20) and inversely S = —hkv<)/<~> (4.21) Then the matter spin currents are cancelled out, so that J>^k^ - -/- ^A>4^6- (4.22) _ - A 72 F o r t h i s g r a v i t a t i o n a l L a g r a n g i a n d e n s i t y * k[8\ + h S - ( 4 . 2 3 ) so One i s l e f t w i t h a t o t a l s p i n c u r r e n t l i n e a r i n t h e s t r u c t u r e c o e f f i c i e n t s o f t h e v i e r b e i n f i e l d and c o n t a i n i n g no m a t t e r c o n t r i b u t i o n a t a l l . Indeed t h i s l o c a l c o n s e r v a t i o n law i s r a t h e r t r i v i a l b e c a u s e a s u p e r p o t e n t i a l e x i s t s i d e n t i c a l l y ( 4 . 2 5 ) X so t h a t S^/h£/X - ^^fie/\ ~ O ( 4 . 2 6 ) and l/X^ = -frC^tS-ASAe*)]. ( 4 . 2 7 ) T h i s s i t u a t i o n i s a d i r e c t r e s u l t o f t h e a s s u m p t i o n t h a t t h e s p a c e - t i m e i s a s t r i c t l y R i e m a n n i a n m a n i f o l d and o f t h e c o n s e q u e n t n o n - d y n a m i c c h a r a c t e r o f t h e c o n n e c t i o n . 3) S p i n C u r r e n t s and Dynamic V i e r b e i n C o n n e c t i o n The r o l e o f t h e v i e r b e i n s p i n c u r r e n t s i n t h e " P a l a t i n i " f o r m u l a t i o n o f K i b b l e #1961) and Sc iama (1961) i s now r e v i e w e d . The g r a v i t a t i o n a l L a g r a n g i a n d e n s i t y w i t h d y n a m i c a l l y i n d e p e n d e n t v i e r b e i n c o n n e c t i o n w i l l be t a k e n t o be t h e c u r v a t u r e s c a l a r d e n s i t y f i r s t 73 The t o t a l L a g r a n g i a n d e n s i t y i s t h e n /T - Z J W , fv , 4 v, A v;) A % A whose v a r i a t i o n under an i n f i n i t e s i m a l v i e r b e i n r o t a t i o n 29) where t h e r e s p e c t i v e f i e l d e q u a t i o n s have been imposed &£r s 0 $ £ r ^ O =r 0 (4.30) and T A % = £ • ( 4 . 3 1 ) The r e l a t i o n s w h i c h f o l l o w f rom ( A . 2 9 ) because of t h e a r b i t r a r i n e s s o f £°J and i t s d e r i v a t i v e s a r e 21*/he/A = (S*k* + S~X/c*)/A = O (4<32) / j , ( 4 . 3 3 ) where 5 ^ = £ ( ^ 4 ^ ~ +t>M6<L,6 -A/A%) ( 4 . 3 4 ) These r e l a t i o n s may a l s o be d e r i v e d s t a r t i n g f rom t h e a l t e r n a t i v e g r a v i t a t i o n a l L a g r a n g i a n d e n s i t y •dciGArti - M j f y - ft*} A'v-AWd) ( 4 . 3 6 ) 74 w h i c h d i f f e r s b y a d i v e r g e n c e f r o m t h e c u r v a t u r e s c a l a r d e n s i t y . I n t h i s c a s e t h e v a r i a t i o n t a k e s t h e d i f f e r e n t f o r m b u t e x a c t l y t h e s a m e r e l a t i o n s a s a b o v e e m e r g e f o r t h e s p i n c u r r e n t s . T h e u n i q u e n e s s o f t h e d e f i n i t i o n o f t h e s p i n c u r r e n t s i n t e r m s o f t h e s u p e r p o t e n t i a l (4.35) i s a c o n s e q u e n c e o f t h e f i e l d e q u a t i o n s o f t h e v i e r b e i n c o n n e c t i o n . T h e t o r s i o n p a r t o f t h e d y n a m i c v i e r b e i n c o n n e c t i o n Aij>k = ^ Gi>)k+Tctj/r)- (fyk + Jjti*)-(Gbij *7ht)'3] (3 .35) w i l l n o w b e e x p r e s s e d i n t e r m s o f t h e m a t t e r s p i n c u r r e n t s f o l l o w i n g K i b b l e (1961). E q u a t i o n (8.33) e x p a n d s t o O n s u b t r a c t i n g o u t t h e t r a c e S >kt ~ T^-M t h e t o r s i o n t e n s o r i s f o u n d t o b e f4.40) T h e n t h e g e n e r a l v i e r b e i n c o n n e c t i o n i s Alj,fr - Bijttr-h +*[ik$'')4 "IjfrS^J (4.41) 75 When t h e v i e r b e i n c o n n e c t i o n assumes i t s t o r s i o n - f r e e fo rm t h e g r a v i t a t i o n a l s p i n c u r r e n t s r e d u c e t o t h e p r e v i o u s r e s u l t <£.^ ~ ^~^/*r-€ • ( 4 . 2 4 ) I t may be remarked t h a t when an e l e c t r o m a g n e t i c f i e l d i s p r e s e n t a l o n g w i t h t h e dynamic v i e r b e i n c o n n e c t i o n ( 4 . 4 1 ) , t h e r e q u i r e m e n t o f e l e c t r o m a g n e t i c gauge i n v a r i a n c e f o r c e s t h e d e f i n i t i o n o f t h e e l e c t r o -m a g n e t i c f i e l d t e n s o r t o be =• A — A/?/<Y and not F^^ A x ^ - A , ^ o r F^y = I%ACy-K*A*<r C l e a r l y gauge f i e l d s w i t h d e r i v a t i v e c o u p l i n g must not e n t e r each o t h e r ' s f r e e L a g r a n g i a n s i f t h e t h e o r y i s t o c o n s i s t e n t . T h i s has t h e c o n s e q u e n c e t h a t t h e e l e c t r o -m a g n e t i c f i e l d has v a n i s h i n g s p i n c u r r e n t s * 76 c. Vierbein Energy and Momentum Currents 1) Matter Energy and Momentum Currents The vierbein energy and momentum currents of a matter f i e ld described by a Lagrangian density wi l l be derived. The invariance of this Lagrangian under general coordinate transformations and arbitrary vierbein rotations wi l l be assumed. It wi l l also be assumed that the connection is Riemannian so that ~7~L>jk - O • The gravitational f i e ld wi l l not yet be allowed dynamical freedom. /AA The f i e ld \ w i l l be allowed to realize dist inct simultaneous world and vierbein tensor (or spinor) representations* Thus the index A may represent a certain mixture of world and vierbein indices. Under the coordinate transformation generated by a vierbein f ie ld with inf initesimal parameters c. , the f ie lds transform as (4.42) = - +F\At><e *Ak*ekn and fAK * - £*/°r + G^u* (*•«) The variation in the Lagrangian density is 77 s& * $& + & « * + ^ ^ & ^ A ^ ( 4 - 4 4 ) After applying the matter f i e ld equations, the following identi t ies are obtained as usual through the arbitrariness of £ ^ and i ts successive derivatives so that the coeff ic ients of each must vanish. The "symmetric matter energy and momentum currents" are now defined as They constitute the source term of Einstein's equations, and the symmetry T/fce = Tkk is obvious. The "canonical r r matter energy and momentum currents" are next defined and in general "J\jt, ±\&k ' The above relationships are now consolidated. Equation (4.46) becomes -r\ = r\ +(&.r\\L'<ee- , , ls £. j. « VJ^j, * <?/<V/ (4.50) 78 From (4.47) the bracketted term in (4.50) is antisymmetric A - r A so 1~S 4t\ ~ 7T &/\ C ^ ° ) (4.51) Then the part ia l conservation law (4.45) is expressed as Tc\(\ + Tr \G-ks\ = 7///A + $~<A which is the contracted Bianchi identity. The above treatment simplif ies considerably i f the f i e l d variables A transform as world scalers so that the generators vanish. This means simply that vierbein projections must be taken of any world tensor matter variables. Then T ' • r« ...» If&Xt* = i f ^ / A ( 4 ' 5 4 ) An expression for <y/^ *» mas derived in Section 4b Hence ^ - V - 1% V <«•••> The structure in the divergence term reduces to precisely 79 the quantity introduced by Belinfante (1939) and Rosenfeld (1940) to symmetrize the canonical energy-momentum tensor in special re lat iv i ty u/hen BCjtk = O • It is here derived from more fundamental considerations. 2) Gravitational Energy and Momentum Currents Consider the f i r s t order vierbein Lagrangian density, in the absence of matter f ie lds so - & ) Ai,^) = &(6y'r 8*A< -6** 6%) ( 3 > 4 7 ) The following identit ies are drawn in the usual fashion from the variation under infinitesimal coordinate transformations parametrized as - ( & > " > ( 4 - 5 8 ) d/f - _ i & (4.59) The "canonical gravitational energy and momentum currents" are defined Z J t = nM. { 4 . 6 0 ) = ASS* + ^ ^ + ( ? ' ' x & ^ Q * % M 80 The s u p e r p o t e n t i a l has t h e fo rm The above e x p r e s s i o n s a r e a l l w o r l d t e n s o r s . They t r a n s f o r m as v i e r b e i n v e c t o r s under c o n s t a n t v i e r b e i n r o t a t i o n s , but depend s t r o n g l y on t h e r e l a t i v e o r i e n t a t i o n of t h e v i e r b e i n s w i t h p o s i t i o n . The above v a r i a t i o n a l d e r i v a t i o n o f t h i s l o c a l c o n s e r v a t i o n law i s due t o F r o l o v (1964) and i n t h e p a r t i c u l a r gauge A^t^-dto R o d i c h e v ( 1 9 6 5 ) . I t s e x i s t e n c e may a l s o be i n f e r r e d d i r e c t l y f rom t h e f i e l d e q u a t i o n s t h r o u g h t h e o b s e r v a t i o n ( P e l l e g r i n i and P l e b a n s k i , 1962) t h a t i s a n t i s y m m e t r i c because o n l y a p p e a r s i n / £ ^ ( 3 . 4 7 ) i n t h e a n t i s y m m e t r i c s t r u c t u r e t e n s o r . The t e r m " c a n o n i c a l " i s used f o r t h e s e c u r r e n t s b e c a u s e no o t h e r s u p e r p o t e n t i a l w i l l y i e l d v i e r b e i n energy and momentum c u r r e n t s w i t h o u t s e c o n d o r d e r v i e r b e i n f i e l d d e r i v a t i v e s . The c a n o n i c a l g r a v i t a t i o n a l energy and momentum c u r r e n t s a l s o u n i q u e l y s a t i s f y t h e r e l a t i o n & ' ° • (4.62) The s u p e r p o t e n t i a l can be e x p r e s s e d i n t e r m s o f t h e g r a v i t a t i o n a l s p i n c u r r e n t s ( 4 . 2 4 ) as 81 V\V = (V* * S^* ~ 3"A*) <«•«> c-i n c l o s e a n a l o g y t o ( 4 . 5 6 ) . Indeed t h e L a g r a n g i a n ( 3 . 4 7 ) and t h e c a n o n i c a l energy and momentum c u r r e n t s a r e q u a d r a t i c i n t h e components o f t h e g r a v i t a t i o n a l s p i n c u r r e n t s , w h i c h a r e l i n e a r i n t h e s t r u c t u r e c o e f f i c i e n t s . Thus t h e L a g r a n g i a n d e n s i t y may be f o r m a l l y w r i t t e n ^HUi^y^S^ +ZOt,Jk*k,jC-Si*ist*j)j ( 4 . 6 4 ) w h i c h i s somewhat s i m i l a r t o t h e F e r m i c u r r e n t - c u r r e n t i n t e r a c t i o n L a g r a n g i a n . S i m i l a r s t r u c t u r e s a l s o i n v o l v i n g t h e m a t t e r s p i n c u r r e n t s a r i s e i n t h e K i b b l e - S c i a m a t h e o r y and may p o s s i b l y have more s i g n i f i c a n c e t h e r e . F r o l o v (1964) a l s o d e r i v e d a d i f f e r e n t s e t o f v i e r b e i n c u r r e n t s f rom t h e L a g r a n g i a n ( 3 . 4 6 ) based on t h e c u r v a t u r e s c a l a r d e n s i t y w i t h s u p e r p o t e n t i a l Ui" - A 6V* ( 4 - 6 5 ) 82 3) Energy and Momentum C u r r e n t s i n K i b b l e - S c i a m a Theory The v i e r b e i n energy and momentum c u r r e n t s w i l l be d e r i v e d i n t h e c o n t e x t o f t h e K i b b l e - S c i a m a f o r m a l i s m i n w h i c h t h e v i e r b e i n c o n n e c t i o n i n c l u d e s t o r s i o n t e r m s l i n e a r i n t h e m a t t e r s p i n c u r r e n t s . F i r s t t h e energy and momentum c u r r e n t s s o f a f i e l d w h i c h may engender s p i n c u r r e n t s w i l l be c o n s i d e r e d . F o r s i m p l i c i t y t h e p u r e l y v i e r b e i n components o f w i l l be t r e a t e d as b a s i c v a r i a b l e s , so t h e ^ ^ t r a n s f o r m a s w o r l d s c a l a r s . The L a g r a n g i a n d e n s i t y i s £ / X * i < ( A 1 > r > t * > A % ) and t h e g r a v i t a t i o n a l v i e r b e i n and v i e r b e i n c o n n e c t i o n f i e l d e q u a t i o n s a r e not i m p o s e d . The t o t a l v a r i a t i o n s i n t h e f i e l d s a f t e r an i n f i n i t e s i m a l c o o r d i n a t e t r a n s f o r m a t i o n h i ' * a r e A f t e r a p p l y i n g t h e m a t t e r f i e l d e q u a t i o n s and i n s e r t i n g t h e m a t t e r s p i n c u r r e n t s S c j - ^ £ AlyS^^-^^s t h e v a r i a t i o n a l i d e n t i t i e s f o u n d a r e te +£S*$(-A%xlJ+AQ«,\ +A*'*/, A/J ( 4 # 6 7 ) 8 3 The energy and momentum currents in this case are T\ * f'W? +£S\ij ACX* They do not coincide with the canonical matter energy and momentum currents of the conventional s t r i c t l y Riemannian formulation except when the matter spin currents are absent. These currents are the source terms of the modified Einstein equations of Kibble and Sciama. T\ = - i G\ - ±(R\ -UXR) (4.70) and are in general asymmetric. ^ ~T~4Ar • Since the covariant divergence of a vector density 3^ takes the form j\x = j*lx+j irx<x-J ir\<--3\-j<Tx,A (4.7D when there is torsion, the part ial conservation law contained in (4 .67) may be written (Kibble, 1961) = T \ . \ + T X * T < A < + T \ T * ^ + £ S x , C j R < - J M ( 4 - 7 2 ) - O Similarly the non-conservation of the matter spin currents is expressed ^ 84 A s e t o f g r a v i t a t i o n a l c u r r e n t s f o r t h e K i b b l e -Sc iama t h e o r y a r e n e x t d e r i v e d f r o m t h e L a g r a n g i a n The i d e n t i t i e s c o r r e s p o n d f u l l y t o ( 4 . 5 7 ) , ( 4 . 5 8 ) and ( 4 . 5 9 ) . The c o n s e r v e d c u r r e n t s a r e t\'hl& — ^JA^i&%y + A(\(?\ev~A6V,6(j\^] ( 4 . 7 4 ) These c u r r e n t s a r e c l e a r l y t h e g e n e r a l i z a t i o n o f t h e c a n o n i c a l g r a v i t a t i o n a l energy and momentum c u r r e n t s ( 4 . 6 0 ) o f t h e c o n v e n t i o n a l t h e o r y t o t h e K i b b l e - S c i a m a t h e o r y . F r o l o v (1964) has shown t h a t t h e L a g r a n g i a n dens i t y ^ (A V, 6 V^>, A CJ-r) ( 3 . 4 9 ) w i t h s u p e r p o t e n t i a l ( 4 . 7 5 ) 3 . 4 8 ) based on t h e c u r v a t u r e s c a l a r d e n s i t y l e a d s t o t h e s u p e r p o t e n t i a l ( 4 . 7 6 ) 85 d. G e n e r a l E x p r e s s i o n s f o r C o n s e r v e d V i e r b e i n C u r r e n t s In t h e p r e c e d i n g S e c t i o n s 4b and 4c c e r t a i n e x p r e s s i o n s f o r v i e r b e i n s p i n and energy and momentum c u r r e n t s were d e r i v e d f rom p a r t i c u l a r s t a t e m e n t s o f t h e a c t i o n p r i n c i p l e . D i f f e r e n t but e q u i v a l e n t L a g r a n g i a n d e n s i t i e s w i l l l e a d t o d i f f e r e n t c u r r e n t s . The e x t e n t o f t h i s k i n e m a t i c a m b i g u i t y f o r t h e t o t a l c u r r e n t s w i l l now be d e t e r m i n e d , c o m b i n i n g t h e t r a n s f o r m a t i o n r e q u i r e m e n t s o f S e c t i o n 4a w i t h a p o s t u l a t e o f s i m p l i c i t y . I t w i l l be assumed t h a t t h e s u p e r p o t e n t i a l s a r e c o n s t r u c t e d s o l e l y f rom g r a v i t a t i o n a l v a r i a b l e s ( v i e r b e i n s ) o f t h e l o w e s t p o s s i b l e d i f f e r e n t i a l o r d e r . The g e n e r a l fo rm o f t h e s u p e r p o t e n t i a l f o r t h e energy and momentum c u r r e n t s i s t h e n most s i m p l y l i n e a r i n t h e s t r u c t u r e c o e f f i c i e n t s . where a , b and c a r e c o n s t a n t s t o be d e t e r m i n e d by o t h e r a r g u m e n t s . S i m i l a r s t r u c t u r e s have been c o n s i d e r e d by S b i y t o v (1965) and by D a v i s and York ( 1 9 6 9 ) . The two most i n t e r e s t i n g s p e c i a l i z a t i o n s o f ( 4 . 7 7 ) seem t o t h e s u p e r p o t e n t i a l o f P i r a n i (1959) w i t h a = 0 , b » 2 , and c • 0t and t h e " c a n o n i c a l " s u p e r p o t e n t i a l o f P e l l e g r i n i and P l e b a n s k i (1962) w i t h a = 1 , b = 0 , and K —' ( 4 . 7 7 ) 86 The most s i m p l e s p i n s u p e r p o t e n t i a l i s t h e s i m p l e b i v e c t o r d e n s i t y d e r i v e d i n S e c t i o n 4b Vii- («.7B, u n i q u e l y t o w i t h i n a m u l t i p l i c a t i v e c o n s t a n t . The s t r u c t u r e o f t h e g r a v i t a t i o n a l s p i n c u r r e n t s i s t h e n J**. = £ ~-r\^G&,kc) («-79) l i n e a r i n t h e r o t a t i o n o r e q u i v a l e n t l y t h e s t r u c t u r e c o e f f i c i e n t s a t t h i s l o w e s t l e v e l o f c o m p l e x i t y . At t h e n e x t l e v e l o f c o m p l e x i t y t h e s u p e r -p o t e n t i a l s wou ld i n c l u d e t e r m s l i n e a r i n t h e Riemann t e n s o r . In p a r t i c u l a r t h e " s p i n " s u p e r p o t e n t i a l wou ld i n c l u d e some c o m b i n a t i o n o f t h e f o l l o w i n g t e r m s i W h i l e t h e c o n s e r v a t i o n , l a w s f o l l o w i n g f rom ( 4 . 8 0 ) a r e an i m m e d i a t e e x p r e s s i o n o f t h e B i a n c h i i d e n t i t i e s , t h e r e i s no j u s t i f i c a t i o n f o r an i n t e r p r e t a t i o n as " s p i n " y e t . Such a c o n n e c t i o n would e x i s t i n a t h e o r y whose g r a v i t a t i o n a l L a g r a n g i a n d e n s i t y i n c l u d e s t e r m s q u a d r a t i c i n t h e Riemann t e n s o r . C o n s i d e r f o r example 87 A " P a l a t i n i " v a r i a t i o n a l f o r m u l a t i o n w i l l be used t o show r e l a t i o n s h i p s between t h e a r b i t r a r y c o e f f i c i e n t s i n ( 4 . 8 0 ) and i n ( 4 . 8 1 ) . The Riemann t e n s o r i s The L a g r a n g i a n d e n s i t y = /£^> (A V, ALJ°TJ A V°rjs) i s now s u b j e c t e d t o an a r b i t r a r y i n f i n i t e s i m a l v i e r b e i n r o t a t i o n , and when t h e f i e l d e q u a t i o n s & -o , SC. *o h 0 l d r \ V ' ' I The c o n t r i b u t i o n o f t h e L a g r a n g i a n t o t h e t o t a l s p i n i s c l e a r l y , and t o t h e t o t a l s u p e r p o t e n t i a l Hence a = 2 a ' , b * ^ b * . and c « c ' i s t h e c o r r e l a t i o n between ( 4 . 8 0 ) and ( 4 . 8 1 ) c o n t a i n e d i n t h i s q u a d r a t i c a c t i o n p r i n c i p l e . The p a r t i c u l a r c h o i c e o f c o n s t a n t s a = i ^ a and c = ^ - a i s i n t e r e s t i n g b e c a u s e t h e n ( 4 . 8 0 ) i s j u s t t h e Uleyl t e n s o r w h i c h c h a r a c t e r i z e s t h e f r e e g r a v i t a t i o n a l f i e l d . S z e k e r e s (1966) has c o n s i d e r e d c l o s e l y r e l a t e d p a r t i a l c o n s e r v a t i o n laws a l s o based on ' ( 4 . 8 5 ) w h i c h a r e u s e f u l i n t h e s t u d y of g r a v i t a t i o n a l f i e l d s p r o p a g a t i n g i n m a t t e r . 88 The above d i g r e s s i o n i n d i c a t e s t h a t t h e e x p r e s s i o n s ( 4 . 7 7 ) and ( 4 . 7 8 ) a r e s u f f i c i e n t l y g e n e r a l f o r t h e s u p e r p o t e n t i a l s i n t h e c o n v e n t i o n a l t h e o r y whose f i e l d e q u a t i o n s a r e of t h e s e c o n d o r d e r and whose L a g r a n g i a n i s l i n e a r i n t h e Riemann t e n s o r . More c o m p l i c a t e d s t r u c t u r e s w i l l not be c o n s i d e r e d f u r t h e r . The s u p e r p o t e n t i a l s were r e q u i r e d t o t r a n s f o r m as s e c o n d rank a n t i s y m m e t r i c t e n s o r d e n s i t i e s under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s . The s p i n s u p e r p o t e n t i a l t r a n s f o r m s as an a n t i s y m m e t r i c v i e r b e i n t e n s o r under g e n e r a l v i e r b e i n r o t a t i o n s « R^^) h*°r ^ V% - A*' Vcj • <*-86» The energy-momentum s u p e r p o t e n t i a l ( 4 . 7 7 ) t r a n s f o r m s as a v i e r b e i n v e c t o r o n l y under c o n s t a n t v i e r b e i n r o t a t i o n s , and i n g e n e r a l as and a s s o c i a t e d c u r r e n t s have t h e u n i q u e p r o p e r t y o f d e p e n d i n g o n l y on a s i n g l e member o f t h e v i e r b e i n ; so t h a t " e n e r g y " i s i n d e p e n d e n t o f t h e s p a t i a l v i e r b e i n o r i e n t a t i o n . The " c a n o n i c a l " s u p e r p o t e n t i a l t r a n s f o r m s as 89 RSCfc tik^?+ tfMM/tffy =KSuZ * ASM/* A/V^ - ^ a / The l a s t s t e p uses ^ ^^--R^^k w h i c h f o l l o w s f rom t h e L o r e n t z o r t h o g o n a l i t y c o n d i t i o n ( 3 . 9 ) . C l e a r l y t h e c a n o n i c a l energy and momentum c u r r e n t s o f t h e g r a v i t a t i o n a l f i e l d w i l l depend s t r o n g l y on a l l v i e r b e i n d i r e c t i o n s . Indeed t h e s t r u c t u r e o f t h e inhomogeneous te rm i n (4 .88) i m p l i e s t h a t " e n e r g y " i s more s t r o n g l y s e n s i t i v e t o v a r i a t i o n s i n t h e s p a t i a l v i e r b e i n o r i e n t a t i o n s t h a n t o v a r i a t i o n s i n t h e t i m e l i k e v i e r b e i n d i r e c t i o n . But t h e symmetry o f t h i s t e r m w i l l be c r u c i a l i n t h e l a t e r d e m o n s t r a t i o n t h a t t h i s s e n s i t i v i t y o f t h e l o c a l d e n s i t i e s does not e x t e n d t o t h e i r i n t e g r a l s under s u i t a b l e a s y m p t o t i c c o n d i t i o n s . The d i v e r g e n c e o f t h e g e n e r a l s u p e r p o t e n t i a l (4 .77) c a n be s u b t r a c t e d f r o m t h e v i e r b e i n E i n s t e i n t e n s o r G\l — (J^Iv = ~ (4 .89) t o o b t a i n t h e g e n e r a l e x p r e s s i o n f o r t h e v i e r b e i n g r a v i t a t i o n a l energy and momentum c u r r e n t s . N<3 c h o i c e o f c o e f f i c i e n t s i n (4 .77) w i l l y i e l d s y m m e t r i c components 90 5. V i e r b e i n R o t a t i o n Gauge The u t i l i t y of the v i e r b e i n fo rmal i sm r e s i d e s l a r g e l y i n the c l e a r s e p a r a t i o n of the l a b e l l i n g of p o i n t s and the s e l e c t i o n of o r t h o g o n a l d i r e c t i o n s at each p o i n t . When t h i s f o r m a l s i m i s a p p l i e d to the t reatment of s p a c e -t ime c o n s e r v a t i o n laws , c o v a r i a n c e wi th r e s p e c t to wor ld c o o r d i n a t e t r a n s f o r m a t i o n s can be ob ta ined at the expense of c o v a r i a n c e wi th r e s p e c t to g e n e r a l v i e r b e i n r o t a t i o n s . But the r e s u l t i n g e x p r e s s i o n s c o n t a i n no p h y s i c a l i n f o r m a t i o n u n t i l a c h o i c e of d i r e c t i o n s h h a s been made by f i x i n g the v i e r b e i n r o t a t i o n gauge. Observab les can be.lfofcmetiofrom q u a n t i t i e s which a re not g a u g e - c o v a r i a n t on ly when the gauge can be f i x e d i n an i n t r i n s i c and i n v a r i a n t manner. A v i e r b e i n f i e l d a s s o c i a t e d wi th i n t r i n s i c p r e f e r r e d d i r e c t i o n s i s i t s e l f o b s e r v a b l e . The s e l e c t i o n of p r e f e r r e d d i r e c t i o n s at each p o i n t may be made i n a v a r i e t y of g e o m e t r i c a l or p h y s i c a l ways. Geometr ic p r e f e r r e d d i r e c t i o n s a re d e f i n e d by the s t r u c t u r e of the s p a c e - t i m e , by the c u r v a t u r e as the " p r i n c i p a l " d i r e c t i o n s of the R i c c i t e n s o r or the Weyl tensor and by s p a c e - t i m e symmetr ies as K i l l i n g v e c t o r s when t h e r e i s a group of m o t i o n s . It i s o f t e n u s e f u l to t i e v i e r b e i n r e f e r e n c e systems to s p e c i a l cu rves or h y p e r s u r f a c e s by gauge c o n d i t i o n s . A few at tempts have been made u s u a l l y i n the con tex t of u n i f i e d f i e l d t h e o r i e s to i n t e r p r e t the s i x r o t a t i o n a l degrees e f freedom o f u n i q u e l y determined v i e r -91 b e i n s as a p h y s i c a l f i e l d . C o n v e n t i o n a l l y and i n t h e v i e r b e i n gauge f o r m u l a t i o n o f g r a v i t a t i o n t h e s e d e g r e e s of f reedom a r e r e g a r d e d as p h y s i c a l l y s p u r i o u s . The p r e f e r r e d v i e r b e i n s d e t e r m i n e d by t h e i r r e d u c i b l e p a r t s o f t h e Riemann t e n s o r w i l l be c o n s i d e r e d f i r s t . These v i e r b e i n s a r e t h o s e i n w h i c h e i t h e r t h e R i c c i t e n s o r o r t h e Weyl t e n s o r t a k e s a u n i q u e a l g e b r a i c " c a n o n i c a l " f o r m . They a r e f u n c t i o n s s o l e l y o f t h e m e t r i c t e n s o r ; and t h e r e f o r e t h e i r L i e d e r i v a t i v e w i t h r e s p e c t t o any K i l l i n g v e c t o r j m u s t v a n i s h s i n c e ^^/f ~ O as i s w e l l known. I t s h o u l d be p o i n t e d out t h a t t h e c o n s i d e r a t i o n o f t h e s e s p e c i a l v i e r b e i n s as p h y s i c a l q u a n t i t i e s r a t h e r t h a n s o l e l y g e o m e t r i c e n t i t i e s r u n s c o u n t e r t o t h e gauge f i e l d i n t e r p r e t a t i o n used up t o now. i n w h i c h t h e i d e a o f an u n r e s t r i c t e d gauge group o f v i e r b e i n r o t a t i o n s i s f u n d a m e n t a l . Here t h a t gauge i s c o m p l e t e l y p i n n e d down and t h e group i s r a t h e r i r r e l e v a n t These v i e r b e i n s wou ld seem t o be of more i n t e r e s t i n a f o r m u l a t i o n o f g r a v i t a t i o n b u i l t e n t i r e l y f rom i n t r i n s i c q u a n t i t i e s ( o b s e r v a b l e s , however o b s c u r e ) r a t h e r t h a n one c o n s t r u c t e d f rom gauge p o t e n t i a l s ( n o n - o b s e r v a b l e s ) , i f a t h e o r y o f t h e f o r m e r t y p e c a n be a s s e m b l e d and f o u n d s a t i s f a c t o r y . 92 An i n c e n t i v e f o r e x a m i n i n g t h e p o s s i b i l i t y o f u s i n g t h e p r i n c i p a l v i e r b e i n s o f t h e Weyl t e n s o r and o f t h e R i c c i t e n s o r as d e s c r i p t o r s i n v i e r b e i n c o n s e r v a t i o n laws comes fcom t h e o b s e r v a t i o n ( P i r a n i , 1957) t h a t a h y p e r s o c f a c e ^ o r t h o g o n a l K i l l i n g d i r e c t i o n i s s i m u l t a n e o u s l y a p r i n c i p a l d i r e c t i o n of t h e Weyl t e n s o r and of t h e R i c c i t e n s o r . When t h e d i r e c t i o n i s t i m e l i k e , t h e s p a c e - t i m e i s s t a t i c by d e f i n i t i o n and energy i s w e l l d e f i n e d as t h e g e n e r a o t o r o f d i s p l a c e m e n t s i n t h i s d i r e c t i o n . S i n c e a g e n e r a l s p a c e - t i m e ( s u f f i c i e n t l y c o m p l i c a t e d , w i t h m a t t e r p r e s e n t ! ) has c o m p l e t e l y d e f i n e d p r i n c i p a l Weyl and R i c c i v i e r b e i n s , and no K i l l i n g v e c t o r s a t a l l , i t seems p o s s i b l e t h a t t h e s e p r i n c i p a l v i e r b e i n s c o u l d be used t o c o n s t r u c t p h y s i c a l l y i n t e r e s t i n g c o n s e r v e d c u r r e n t s . The i n t e r e s t of s u c h c u r r e n t s w o u l d be o b v i o u s i f t h e y were a b l e t o p r o v i d e some c o n t i n u a t i o n o f l o c a l i z e d energy i n t o r e g i o n s w i t h o u t s y m m e t r y . The g e o m e t r i c a l e l e m e n t s d e f i n e d by t h e R i c c i and Weyl t e n s o r s w i l l now be d i s c u s s e d b r i e f l y . The R i c c i t e n s o r i s t h a t p a r t o f t h e Riemann t e n s o r w h i c h i s d i r e c t l y c o u p l e d t o l o c a l energy-momentum and s t r e s s o f m a t t e r . From a g e o m e t r i c a l p o i n t o f v iew i t i s e q u i v a l e n t t o t h e s y m m e t r i c m a t t e r energy-momentum t e n s o r when E i n s t e i n ' s f i e l d e q u a t i o n s h o l d . The R i c c i t e n s o r o r d q u i v a l e n t l y t h e m a t t e r t e n s o r d e t e r m i n e up t o f o u r d i s t i n c t p r i n c i p a l d i r e c t i o n s w h i c h a r e i t s e i g e n v e c t o r s . The t e n s o r i s i n c a n o n i c a l fo rm when p r o j e c t e d on to t h e p r i n c i p a l v e c t o r s . 93 The c l a s s i f i c a t i o n o f t h e R i c c i t e n s o r (more p r e c i s e l y , i t s t r a c e l e s s p a r t ) has been a n a l y z e d i n d e t a i l by E i s e n h a r t ( 1 9 2 6 ) , P l e b a n s k i (1964) and P e t r o v ( 1 9 6 6 ) . The most s i m p l e c a n o n i c a l t y p e i s t h a t i n w h i c h t h e t e n s o r i s c o m p l e t e l y d i a g o n a l i n a p r i n c i p a l v i e r b e i n . The i n d e f i n i t e n e s s o f t h e s p a c e - t i m e m e t r i c f o r c e s t h e c o n s i d e r a t i o n o f o t h e r t y p e s w i t h n u l l and complex e i g e n v e c t o r s . P l e b a n s k i (1964() showed t h a t t y p e s w i t h complex e i g e n v e c t o r s a r e not a d m i s s i b l e as energy-momentum t e n s o r s a t l e a s t f o r m a c r o s c o p i c m a t t e r whose energy d e n s i t y must be p o s i t i v e - d e f i n i t e i n a l l f r a m e s . The r e m a i n i n g p h y s i c a l l y p o s s i b l e t y p e s have p r i n c i p a l v e c t o r s w h i c h can be c o l l e c t e d i n t o o r t h o n o r m a l or n u l l v i e r b e i n s . Then i n t h e p r i n c i p a l f rame f l u i d (no s t r e s s e s ) d e t e r m i n e s a s i n g l e e i g e n v e c t o r w h i c h i s t h e f o u r - v e l o c i t y o f t h e c o n s t i t u e n t p a r t i c l e s . T h i s i s t r u e a l s o f o r an i d e a l f l u i d w i t h i s o t r o p i c s t r e s s e s . When a n i s o t r o p i c s t r e s s e s e x i s t , t h e s p a t i a l members o f t h e p r i n c i p a l v i e r b e i n a r e d e t e r m i n e d . Of c o u r s e , whenever t h e s i t u a t i o n i s more complex ( s i n g l e f l u i d w i t h heat f l u x , two f l u i d s , o r s e v e r a l f i e l d s p r e s e n t ) t h e t i m e l i k e p r i n c i p a l v e c t o r i s no l o n g e r t a n g e n t t o t h e w o r l d - l i n e s o f f l u i d e l e m e n t s f o r m i n g but p a r t o f t h e t o t a l m a t t e r t e n s o r . no summation over " i ( 5 . 2 ) The energy-momentum t e n s o r o f an i n c o h e r e n t 94 An o b v i o u s major use o f t h e R i c c i p r i n c i p a l v i e r b e i n i s i n t h e a n a l y s i s o f t h e energy-momentum t e n s o r s o f v a r i o u s p o s t u l a t e d m a t t e r f i e l d s . T h i s v i e r b e i n d i s p l a y s t h e g e o m e t r i c s t r u c t u r e of t h e m a t t e r f i e l d v i e w e d c l a s s i c a l l y as a p e c u l i a r f l u i d i n i t s own g r a v i t a t i o n a l f i e l d . As shown by G. Rosen (1959) f o r t h e E i n s t e i n - M a x w e l l f i e l d , t h e R i c c i p r i n c i p a l v i e r b e i n may be used t o i l l u m i n a t e and s i m p l i f y " R a i n i c h " c o n d i t i o n s f o r t h e p u r e l y g e o m e t r i c d e s c r i p t i o n of t h e g r a v i t a t i n g m a t t e r f i e l d . The c o n d i t i o n s f o r m e r l y imposed on t h e R i c c i t e n s o r may be t r a n s l a t e d i n t o s i m p l e r c o n d i t i o n s on t h e R i c c i e i g e n v a l u e s and on t h e s t r u c t u r e c o e f f i c i e n t s o f t h e p r i n c i p a l v i e r b e i n . Of c o u r s e t h e s t r u c t u r e c o e f f i c i e n t s a r e c o m p l e t e l y e q u i v a l e n t t o t h e r o t a t i o n c o e f f i c i e n t s and t o t h e components o f t h e s p i n c u r r e n t s s i n c e S J ,jk = (yi,j(c f- f^ijCj^k* ~ "X^kG^jjt . The f r e e e l e c t r o m a g n e t i c f i e l d o n l y p a r t i a l l y d e t e r m i n e s a R i c c i p r i n c i p a l v i e r b e i n . The f i e l d t e n s o r (and a c c o r d i n g l y t h e M a x w e l l t e n s o r as w e l l ) has two p r i n c i p a l n u l l d i r e c t i o n s . These n u l l d i r e c t i o n s d e t e r m i n e two e i g e n b i v e c t o r s , one c o n t a i n i n g them and t h e o t h e r o r t h o g o n a l ( s e e L . W i t t e n , 1 9 6 2 ) . The p r i n c i p a l v i e r b e i n of t h e M a x w e l l ( R i c c i ) t e n s o r s p l i t s i n t o p a i r s o f v e c t o r s w i t h i n each b i v e c t o r and f r e e t o r o t a t e w i t h i n t h e m . Two o f t h e p o s s i b l e s i x s p i n c u r r e n t s a r e u n i q u e l y d e f i n e d , one f o r each e i g e n b i v e c t o r . The s t a t e m e n t of t h e s e c u r r e n t s i s e q u i v a l e n t t o R o s e n ' s (1959) c o n d i t i o n s on t h e s t r u c t u r e 95 c o e f f i c i e n t s , but i s o b t a i n e d more d i r e c t l y and i n d e e d t r i v i a l l y by t a k i n g t h e d i v e r g e n c e o f t h e e i g e n b i v e c t p r s . In t h e d e g e n e r a t e c a s e of a n u l l e l e c t r o m a g n e t i c f i e l d w h i c h o n l y has a s i n g l e p r i n c i p a l n u l l d i r e c t i o n , c o n d i t i o n s on t h e s t r u c t u r e c o e f f i c i e n t s e x p r e s s t h e g e o d e s i c and s h e a r - f r e e q u a l i t y of t h e r a y s ( S z e k e r e s , 1 9 6 6 ) . A " R a i n i c h " g e o m e t r i z a t i o n has been g i v e n f o r s c a l a r f i e l d s by Kuchar ( 1 9 6 3 ) . Here t h e r e i s but a s i n g l e p r i n c i p a l v e c t o r o f t h e energy-momentum t e n s o r , w h i c h i s p r o p o r t i o n a l t o t h e g r a d i e n t of t h e f i e l d and i s t h e r e f o r e t a n g e n t t o a n o r m a l c o n g r u e n c e i m p o s i n g c o n d i t i o n s ( 5 . 9 ) on t h e s t r u c t u r e c o e f f i c i e n t s . The energy-momentum t e n s o r of c l a s s i c a l n e u t r i n o f i e l d s has been d i s c u s s e d by U la inwr ight ( 1 9 7 1 ) . Here t h e n u l l d i r e c t i o n d e f i n e d by t h e n e u t r i n o c u r r e n t i s most i m m e d i a t e t o p h y s i c s . The i d e a b e h i n d a l l t h e s e examples i s t h a t t h e g e o m e t r i c e x p r e s s i o n of o b s e r v a b l e s may u l t i m a t e l y f u r t h e r t h e advancement and u n i f i c a t i o n o f p h y s i c s . The R i c c i p r i n c i p a l v i e r b e i n i s i n d e t e r m i n a t e i n t h e vacuum i n E i n s t e i n ' s t h e o r y . But any m a t e r i a l o b s e r v e r o r h i s a p p a r a t u s must l o c a l l y d e t e r m i n e a p r i n c i p a l v i e r b e i n by e x i s t i n g . " T e s t p a r t i c l e s " a r e an u n p h y s i c a l e x t r a p o l a -t i o n of r e a l m a t t e r i n w h i c h t h e r e f e r e n c e v i e r b e i n i s d e f i n e d w h i l e t h e i n c o n v e n i e n c e of c o n s i d e r i n g o r even c o m p u t i n g t h e r e a c t i o n on t h e m e t r i c i s s h u n t e d a s i d e . 96 The a c t o f m e a s u r i n g t h e g r a v i t a t i o n a l f i e l d ( o r a n y t h i n g a t a l l ) f i r s t p r o v i d e s a r e f e r e n c e s y s t e m , w h i c h i s t h e R i c c i principal v i e r b e i n o f t h e o b s e r v e r t o g e t h e r w i t h any mat far. f i e l d s a t h i s l o c a t i o n . DeWit t (1962) has e m p h a s i z e d t h e i m p o r t a n c e o f m a t e r i a l s y s t e m s i n p r o v i d i n g some bas&s for fcbe a n a l y s i s 6 f g r a v i t a t i o n a l f i e l d measurements and quantum u n c e r t a i n t i e s . P o s s i b l y t h e v i e r b e i n c u r r e n t s d i s c u s s e d i n C h a p t e r 4 may be of some use i n t h e deve lopment o f a quantum t h e o r y o f g r a v i t a t i o n when f i x e d by t h e p r i n c i p a l v i e r b e i n of t h e m a t t e r t e n s o r . W h i l e t h e R i c c i t e n s o r i s i n t i m a t e l y a s s o c i a t e d w i t h t h e l o c a l m a t t e r t e n s o r , t h e Weyl t e n s o r i s t h a t p a r t o f t h e Riemann t e n s o r d i r e c t l y c o n n e c t e d t o t h e p r o p a g a t i o n o f t h e f r e e g r a v i t a t i o n a l f i e l d . P e n r o s e (1966) has a t t r i b u t e d t o t h e Weyl t e n s o r an e s s e n t i a l l y a s t i g m a t i c f o c u s s i n g e f f e c t on n u l l r a y s i n c o n t r a s t t o t h e pu re a n a s t i g m a t i c f o c u s s i n g e f f e c t of t h e R i c c i t e n s o r , and throughn,an o p t i c a l a n a l o g y d i s p l a y e d t h e n o n l o c a l i t y o f g r a v i t a t i o n a l energy by s h o w i n g t h a t e f f e c t i v e p o s i t i v e a n a s t i g m a t i c f o c u s s i n g may be e f f e c t e d by t h e Weyl t e n s o r o v e r a r e g i o n . T h i s n o n l o c a l i t y w i l l be d e m o n s t r a t e d a g a i n i n S e c t i o n 6 ( c ) . D e s p i t e t h e a p p a r e n t f u t i l i t y o f t r y i n g t o l o c a l i z e g r a v i t a t i o n a l energy u s i n g e i g e n - e l e m e n t s of t h e Weyl t e n s o r w h i c h embodies t h e n o n l o c a l i t y of t h a t e n e r g y , i t i s s t i l l p o s s i b l e t h a t t h e r e may be some use f o r i n v a r i a n t v i e r b e i n c u r r e n t s o f t h e Weyl t e n s o r . 97 The c l a s s i f i c a t i o n o f t h e Weyl t e n s o r may be a p p r o a c h e d by t r e a t i n g i t as a s y m m e t r i c s e c o n d rank t e n s o r i n t h e s i x - d f m e n s i o n a l b i v e c t o r s p a c e . T h i s s p a c e i s t h e group s p a c e o f t h e v i e r b e i n r o t a t i o n g r o u p . The s i x e i g e n v a l u e s and e i g e n b i v e c t o r s o f t h e Weyl t e n s o r a r e f o u n d d i r e c t l y , and i t i s d i s p l a y e d i n " c a n o n i c a l " fo rm i n t h e p r i n c i p a l v i e r b e i n w h i c h i s d e t e r m i n e d by t h e i n t e r s e c t i o n s o f t h e s u r f a c e s d e f i n e d by t h e b i v e c t o r s ( P e t r o v ? 1 9 6 1 , 1 9 6 6 ) . The Weyl t e n s o r a l s o d e t e r m i n e s up f o u r d i s t i n c t n u l l e i g e n v e c t o r s ( D e b e v e r , 1 9 5 9 ; P e n r o s e , 1 9 6 0 ) , i n a n a l o g y t o t h e two n u l l e i g e n v e c t o r s o f a g e n e r a l e l e c t r o m a g n e t i c f i e l d . The most g e n e r a l Weyl t e n s o r i s P e t r o v t y p e I ( a l g e b r a i c a l l y g e n e r a l ) w h i c h c o m p l e t e l y d e t e r m i n e s a p r i n c i p a l v i e r b e i n . Types I I and I I I c o n t a i n a p a r a m e t e r w h i c h may be s e t t o some f i x e d v a l u e t o m a t h e m a t i c a l l y d e t e r m i n e a u n i q u e p r i n c i p a l v i e r b e i n ( E h l e r s and K u n d t , 1 9 6 2 ) , but t h i s n o r m a l i z a t i o n of t h e f i e l d i n t e n s i t y does not seem p h y s i c a l l y j u s t i f i a b l e as a g e n e r a l p r e s c r i p t i o n . Type D ( d e g e n e r a t e t y p e I, f o r example S c h w a r z s c h i l d m e t r i c ) o n l y f i x e s two n u l l d i r e c t i o n s ? and t y p e N ( d e g e n e r a t e t y p e I I , f o r example p l a n e waves) o n l y f i x e s one h u l l d i r e c t i o n . As p o i n t e d out by P o l i s h c h u k ( 1 9 7 0 ) , t h e r e i s a smooth t r a n s i t i o n fcom t h e t i m e l i k e p r i n c i p a l v e c t o r of t y p e I t o t h e f i r s t p r i n c i p a l n u l l v e c t o r o f t y p e s I I and I I I , i n t h a t t h e c a n o n i c a l f o r m s a r e t h e same when r e f e r r e d 98 t o t h i s f i r s t p r i n c i p a l v e c t o r . T h i s s u g g e s t s t h a t an i n t e r e s t i n g g e n e r a l i z e d energy s u p e r p o t e n t i a l s h o u l d o n l y depend upon t h e one p r i n c i p a l v e c t o r , w h i c h i m m e d i a t e l y r e s t r i c t s c o n s i d e r a t i o n t o t h e K o m a r - P i r a n i s u p e r p o t e n t i a l w h i c h i s t h e c u r l o f t h e d e s c r i p t o r v e c t o r . T h i s same s u p e r p o t e n t i a l w i l l a l s o be s i n g l e d out by t h e example of t h e S c h w a r z s c h i l d m e t r i c ( s e e S e c t i o n 6 ( a ) ) as t h e o n l y p o s s i b l e e x p r e s s i o n t o be combined w i t h t h e Weyl p r i n c i p a l v i e r b e i n . One has t e n t a t i v e l y t h e r e f o r e Type I QX= -2f,(u^-U^)iv , © A / A - t f (5.3) Types I I and I I I Q X - - 2 J > ( k / c " ' X ) j » (5,4) where # and Tt a r e t h e r e s p e c t i v e f i r s t p r i n c i p a l v e c t o r s . Now (6.4) i s not w e l l d e f i n e d because t h e r e i s no c o n d i t i o n t o g i v e t h e s c a l e o f t h e n u l l v e c t o r ~]< . These a r e p r e c i s e l y P i r a n i ' s (1957) pure r a d i a t i o n t y p e s , and f o r them t h e Weyl c u r r e n t (6.4) , becomes g a u g e - d e p e n d e n t In t h e c a s e o f t h e pu re p l a n e waves of S e c t i o n 6 ( c ) , t h e o n l y c l e a r p r o p e r t y o f i s t h a t i t must be p a r a l l e l t o t h e p r o p a g a t i o n v e c t o r . In t h i s c a s e t h e r e i s no f u r t h e r i n v a r i a n t s t r u c t u r e w h i c h c o u l d p r o v i d e a l o c a l i z e d energy c u r r e n t . The a s s o c i a t i o n between (6.3) and (6.4) and g r a v i t a t i o n a l energy seems t e n u o u s a t b e s t . However (6.3) does p r o v i d e a f o r m a l d e f i n i t i o n of a l o c a l i n v a r i a n t c u r r e n t w h i c h may r e a s o n a b l y be 99 i d e n t i f i e d as an energy c u r r e n t i n t h e v e r y s p e c i a l c a s e o f a s t a t i c s p a c e t i m e when t h e r e i s a h y p e r s u r f a c e - o r t h o g o n a l K i l l i n g v e c t o r , a l t h o u g h not so i n t h e s l i g h t l y l e s s r e s t r i c t i v e c a s e of , a s t a t i o n a r y s p a c e t i m e where t h e t i m e l i k e K i l l i n g v e c t o r and p r i n c i p a l Uleyl v e c t o r need not c o i n c i d e . The s p i n c u r r e n t s a l s o a r e c o m p l e t e l y f i x e d by an a l g e b r a i c -a l l y g e n e r a l Uleyl t e n s o r , v e r y d i r e c t l y s i n c e t h e Uleyl e i g e n b i s e c t o r s may be used i m m e d i a t e l y i n t h e s p i n s u p e r p o t e n t i a l . The s p i n c u r r e n t components a r e o f c o u r s e e q u i v a l e n t t o t h e r o t a t i o n c o e f f i c i e n t s of t h e p r i n c i p a l Uleyl v i e r b e i n , and a r e t h i r d o r d e r i n v a r i a n t s of t h e m e t r i c . Whether t h e c o n s e r v e d v i e r b e i n c u r r e n t s a r e a u s e f u l way t o p r e s e n t t h e i n v a r i a n t s t r u c t u r e o f t h e Weyl t e n s o r i s u n c l e a r . I t may be n o t e d t h a t Bergmann and Komar (1959) have a l r e a d y used t h e e i g e n v a l u e s o f t h e Weyl t e n s o r i n an a t t e m p t t o s e t up a u s e f u l c o m p l e t e s e t o f o b s e r v a b l e s w h i c h wou ld u n d e r l i e a quantum t h e o r y of g r a v i t a t i o n . 100 T h e r e have been a number o f a t t e m p t s t o g i v e n o n - g r a v i t a t i o n a l f i e l d s t h e r o l e of s e l e c t i n g a p r e f e r r e d v i e r b e i n f r a m e , as d i s t i n c t f rom t h e R i c c i p r i n c i p a l v i e r b e i n . One of E i n s t e i n ' s e a r l y u n s u c c e s s f u l a t t e m p t s t o e r e c t a v i a b l e u n i f i e d f i e l d t h e o r y (1928) t r i e d t o r e l a t e t h e s i x d e g r e e s of f reedom of v i e r b e i n r o t a t i o n t o t h o s e of t h e e l e c t r o m a g n e t i c f i e l d . M i l l e r ( 1 9 6 1 b , 1964a) sought u n s u c c e s s f u l l y t o p h y s i c a l l y j u s t i f y some c o n d i t i o n *€cfc - ~ f^c ~0 t h a t wou ld f i x t h e v i e r b e i n gauge up t o a c o n s t a n t r o t a t i o n . He t r i e d t o f i n d such a c o n d i t i o n e i t h e r w i t h i n t h e t h e o r y of g r a v i t a t i o n i t s e l f ( f o r e x a m p l e , B ^^/f9O ) o r e l s e i n some u n i f i c a t i o n w i t h e l e c t r o m a g n e t i s m . P e l l e g r i n i and P l e b a n s k i r e c o g n i z e d t h a t such a c o n d i t i o n i f i t e x i s t s s h o u l d be f o u n d i n t h e f i e l d e q u a t i o n s , ahd showed how t h i s c o u l d be done by a d d i n g some r a t h e r a r b i t r a r y e x t r a t e r m s t o t h e L a g r a n g i a n . Such a p p r o a c h e s c o n f l i c t w i t h ahy, i n t e r p r e t a t i o n o f g r a v i t a t i o n as a gauge f i e l d . O t h e r a u t h o r s have t r i e d t o a t t a c h s p e c i a l s i g n i f i c a n c e t o p a r t i c u l a r gauge c o n d i t i o n s , much as Fock (1959) has a d v o c a t e d t h e harmon ic c o o r d i n a t e c o n d i t i o n R o d i c h e v (1965) has used t h e " h a r m o n i c gauge" c o n d i t i o n ( somet imes c a l l e d " q u a s i - h a r m o n i c " ) 1 0 1 w h i c h i s i n o b v i o u s a n a l o g y t o t h e L o r e n t z gauge i n e l e c t r o m a g n e t i s m and t o ( 5 . ) . The h a r m o n i c gauge c o n d i t i o n p r o v i d e s a c o n s i d e r a b l e s i m p l i f i c a t i o n o f t h e g r a v i t a t i o n a l f i e l d e q u a t i o n s and c o n s e r v e d v i e r b e i n c u r r e n t s s i n c e Cf'^O* B l J t t h i s c o n d i t i o n can o n l y f i x f o u r o f s i x d e g r e e s of f r e e d o m , w h i l e t h e harmon ic c o o r d i n a t e c o n d i t i o n f i x e s a l l f o u r d e g r e e s o f c o o r d i n a t e a r b i t r a r i n e s s when s u p p l e m e n t e d w i t h s u i t a b l e boundary c o n d i t i o n s . The c o n s i d e r a t i o n of a s y m p t o t i c boundary c o n d i t i o n s f o r r a d i a t i v e m e t r i c s i n S e c t i o n 6 ( b ) w i l l f i n d t h e harmonic c o o r d i n a t e s q n d i t i o n t o be more c l e a r l y u s e f u l t h a n t h e h a r m o n i c gauge c o n d i t i o n . Of c o u r s e t h e g l o b a l a p p l i c a b i l i t y o f subh c o n s t r a i n t s i s not o b v i o u s i n g e n e r a l . O f t e n u s e f u l v i e r b e i n gauges may be s e t a c c o r d i n g t o t h e s p e c i a l p r o p e r t i e s of some c o n g r u e n c e of c u r v e s . These p r o p e r t i e s may be e x p r e s s e d as c o n d i t i o n s on t h e r o t a t i o n c o e f f i c i e n t s . The s t a n d a r d d e f i n i t i o n s may be f o u n d i n somet imes d i f f e r e n t g u i s e i n t h e r e f e r e n c e s c i t e d , and a r e a p p l i e d be low t o a t i m e l i k e c o n g r u e n c e w i t h t a n g e n t u n i t v e c t o r lX — 7?®) • 1) The c o n g r u e n c e i s g e o d e t i c : U Ufjr 5 5 O «* = B%W * O ( 5 . 7 ) 2) The c o n g r u e n c e i s e x p a n s i o n l e s s ( h a r m o n i c gauge i n one d i r e c t i o n ) K Q v 102 The c o n g r u e n c e i s h y p e r s u r f a c e - n o r m a l or i r r o t a t i o n a l ( s e e E i s e n h a r t , 1926) G(%i<*)* GC°\C0G)= GC°l>(30) " ° ( 5 . 9 ) E q u i v a l e n t l y 1 4 ° ^ ~ U*r - $H • When c o n g r u e n c e i s a l s o g e o d e t i c , When h y p e r s u r f a c e i s " m i n i m a l " , t h e c o n g r u e n c e i s n o r m a l and e x p a n s i o n l e s s A^ iTj*f - ~ B*(Ph6 ~ S c h w i n g e r ( 1 9 6 3 a ) has used t h e " t i m e gauge" A%*0 (<<~ />2,*) o r e q u i v a l e n t l y hk°-0 Qr^Q),0t),(?)) ( 5 . 1 0 ) w h i c h i s a combined c o o r d i n a t e and v i e r b e i n gauge c o n d i t i o n l o c k i n g t h e w o r l d t i m e c o o r d i n a t e t o some t i m e i i k e n o r m a l c o n g r u e n c e . The c o n g r u e n c e i s r i g i d ( E h l e r s and K u n d t , 1 9 6 2 ) , o r s h e a r - f r e e ^ ( 5 . 1 1 ) and e x p a n s i o n l e s s B*#9,6 ~ O-The c o n g r u e n c e i s a group c o n g r u e n c e ( s e e E i s e n h a r t , 1926) when f^^^/,{%, i s a K i l l i n g v e c t o r . The c o n g r u e n c e i s r i g i d and a l s o s a t i s f i e s When t h e c o n g r u e n c e i s a l s o g e o d e t i c , ~JC - c o n s t a n t and t h e m o t i o n s a r e " t r a n s l a t i o n s " . When t h e c o n g r u e n c e n o r m a l , t h e Riemann t e n s o r i s c o m p l e t e l y d i a g o n a l i z e d i n some v i e r b e i n f rame w i t h /,%= U<* ( P i r a n i , 1 9 5 7 ) . 103 The t a n g e n t v e c t o r t o a c o n g r u e n c e i s a R i c c i p r i n c i p a l v e c t o r i f ( 6 . 1 3 ) F e r m i t r a n s p o r t o f a v i e r b e i n a l o n g a c u r v e t o w h i c h t V o r s ^ r i s t a n g e n t i s d e f i n e d by ( s e e S y n g e , 1 9 6 0 , f o r a n o t h e r e s s e n t i a l l y e q u i v a l e n t d e f i n i t i o n ) » BWi>,p> - = B°>0),W - O ( 6 . 1 « ) I f t h e c u r v e i s a g e o d e s i c t h i s r e d u c e s t o p a r a l l e l t r a n s p o r t w i t h An i n e r t i a l o b s e r v e r o c c u p i e s a p a r a l l e l - t r a n s p o r t e d f rame w h i c h f e e l s no r o t a t i o n s o r a c c e l e r a t i o n s . An i n v a r i a n t v i e r b e i n may be d e f i n e d f o r a n o n - g e o d e t i c c u r v e by t h e F r e n e t - S e r r e t c o n d i t i o n s ( s e e Synge , 1960) w h i c h r e q u i r e Bo)tJ)>&) =• B ~ $co)Q)>p) = o (6,15) when 8to)U),{0) , B(IK2)&) a " d BfaW/M a r e t n B p r i n c i p a l c u r v a t u r e s of t h e c o n g r u e n c e . 1 0 4 6 . P h y s i c a l Examples a . S c h w a r z s c h i l d M e t r i c The S c h w a r z s c h i l d s o l u t i o n f o r t h e g r a v i t a t i o n a l f i e l d o f an i s o l a t e d s p h e r i c a l l y s y m m e t r i c g r a v i t a t i n g body i s c e n t r a l t o t h e main e x p e r i m e n t a l t e s t s o f g e n e r a l r e l a t i v i t y . As t h e b a s i c model f o r g r a v i t a t i o n a l mass , i t i s a f i r s t t e s t f o r any d e s c r i p t i o n o f g r a v i t a t i o n a l e n e r g y . I t i s t o o d e g e n e r a t e a s o l u t i o n t o be much more t h a n a f i r s t t e s t . The S c h w a r z s c h i l d m e t r i c i s ( 6 . 1 ) where t o t h e t t i m e l i k e K i l l i n g v e c t o r The t i m e c o o r d i n a t e i s s c a l e d o f t h i s s t a t i c f i e l d . The r a d i a l c o o r d i n a t e i s a l u m i n o s i t y d i s t a n c e so t h e a r e a o f a s p h e r i c a l s u r f a c e r » c o n s t a n t , t » c o n s t a n t i s ^ 7 7 tZ • An i m m e d i a t e c h o i c e o f v i e r b e i n i s Iff) O W f o 2. •3 — o -01 o o o c o t O W o o 0 r*he t ( 6 .2 ) In t h i s f rame t h e Riemann (Wey l ) t e n s o r i s c o m p l e t e l y d i a g o n a l and i s d i s p l a y e d as P e t r o v t y p e 0 w i t h e i g e n v a l u e T h i s i s a p r i n c i p a l v i e r b e i n o f t h e Weyl t e n s o r . However t h e t y p e D Weyl t e n s o r o n l y f i x e s i t s 105 p r i n c i p a l v i e r b e i n up t o a r o t a t i o n i n t h e (2K3) p l a n e and a s p e c i a l L o r e n t z t r a n s f o r m a t i o n i n t h e (0}{1) p l a n e . The t i m e l i k e v i e r b e i n v e c t o r hp) i n ( 6 . 2 ) i s d i s t i n g u i s h e d by i t s t a n g e n c y t o t h e h y p e r s u r f a c e -o r t h o g o n a l g l o b a l K i l l i n g d i r e c t i o n . T h i s p r o p e r t y i s s t r o n g enough t o e n s u r e t h a t i s a p r i n c i p a l v e c t o r o f t h e vacuum Uleyl t e n s o r , but does not make i t a u n i q u e Uleyl p r i n c i p a l v e c t o r . The n o n v a n i s h i n g s t r u c t u r e c o e f f i c i e n t s o f t h e v i e r b e i n ( 6 . 2 ) a r e h and t h e components o f t h e s t r u c t u r e t e n s o r a r e ( 6 . 4 ) The g e n e r a l v i e r b e i n s u p e r p o t e n t i a l f o r t h e energy c u r r e n t i s The c o r r e s p o n d i n g g e n e r a l i z e d t o t a l energy i s e v a l u a t e d o v e r a s p h e r i c a l s u r f a c e of i n f i n i t e r a d i u s k~ (\^C/pM^4fi ( 6 . 6 ) From ( 6 . 2 ) , ( 6 . 3 ) and ( 6 . 4 ) ( 6 . 7 ) 106 In t h e l i m i t f - > « « • j tf / so C / f * <J7rJ = 0_ i^<V^ + c.(ct,vvj&,ce) ( 6 . 8 ) S i n c e t h e p h y s i c a l energy a s s o c i a t e d w i t h t h e S c h w a r z s c h i l d m e t r i c i s j u s t i t s a c t i v e g r a v i t a t i o n a l mass Pg>;-Al > t h e c o e f f i c i e n t s o f ( 6 . 8 ) must be c h o s e n a - b = 2, c a 0 , i f t h i s v i e r b e i n f rame i s t o be used f o r t h e c o m p u t a t i o n o f e n e r g y . In p a r t i c u l a r P i r a n i ' s s u p e r -p o t e n t i a l a * c a 0 , b « - 2 , i s c o m p a t i b l e w i t h t h i s c h o i c e o f v i e r b e i n gauge w h i l e t h e " c a n o n i c a l ' * s u p e r -p o t e n t i a l a « - c a 1 , b « 0 , i s n o t . A n o t h e r f a i r l y i m m e d i a t e c h o i c e of v i e r b e i n f o r t h e S c h w a r z s c h i l d m e t r i c ( 6 . 1 ) i s 3 C ( 6 . 9 ) Here t h e t i m e l i k e v i e r b e i n v e c t o r i s unchanged f rom ( 6 . 2 ) . but t h e s p a t i a l v e c t o r s now c o n f o r m a s y m p t o t i c a l l y t o a r e c t a n g u l a r r a t h e r t h a n s p h e r i c a l o r i e n t a t i o n . The n o n - v a n i s h i n g components o f t h e s t r u c t u r e t e n s o r a r e ( 6 . 1 0 ) = G\u = £(f-i - I) In t h i s gauge U?'^*[«rt$ +'&«i£)^j^0+rig)) (6 . i i ) 107 As t-* oo , # — » / and K f ~ ^ ^ ) — * A l T h e r e f o r e fy} ^ / - J i f ( a - ^ - c ) « • T h i s c h o i c e o f v i e r b e i n gauge i s c o m p a t i b l e w i t h a wide r a n g e o f p o s s i b l e s u p e r p o t e n t i a l s , i n c l u d i n g a g a i n t h e P i r a n i e x p r e s s i o n and t h i s t i m e a l s o t h e " c a n o n i c a l " e x p r e s s i o n . In t h e c h a r g e d R e i s s n e r - N o r d s t r o m m e t r i c , where )f-l—^-4-S^^ r e p l a c e s ft - (— i n t h e uncharged S c h w a r z s c h i l d m e t r i c , Pc*) — /^\ w i t h any a c c e p t a b l e p r e s c r i p t i o n . The above d i s c u s s i o n i s u n c h a n g e d , e x c e p t t h a t Ag» i s now a n o n - u n i q u e p r i n c i p a l d i r e c t i o n o f t h e R i c c i t e n s o r as w e l l as a n o n - u n i q u e p r i n c i p a l d i r e c t i o n o f t h e Weyl t e n s o r and a u n i q u e K i l l i n g d i r e c t i o n . The above c o n s i d e r a t i o n s show t h a t t h e S c w a r z s c h i l d m e t r i c does not e x c l u d e t h e p o s s i b i l i t y o f u s i n g p r e f e r r e d d i r e c t i o n s o f t h e Riemann t e n s o r t o f i x s u p e r p o t e n t i a l s fo rmed f o a m a n d / , $ ^ g » a l t h o u g h s u c h a p r e s c r i p t i o n i s not e n t i r e l y unambiguous i n t h i s c a s e dme t o t h e d e g e n e r a c y o f t h e p r i n c i p a l v i e r b e i n s . The P i r a n i s u p e r p o t e n t i a l ^Z^XTJL^ *S °? more i n t e r e s t h e r e because i t i s c o m p l e t e l y d e t e r m i n e d by t h e t i m e l i k e K i l l i n g v e c t o r . The i n a d m i s s i b i l i t y o f a s p h e r i c a l s p a t i a l o r i e n t a t i o n o f t h e v i e r b e i n s i n t h e " c a n o n i c a l c u r r e n t s " i s a m a n i f e s t a t i o n o f t h e Bauer (1918) d i f f i c u l t y d i s c u s s e d 108 i n S e c t i o n l b . However t h e r e c t a n g u l a r o r i e n t a t i o n i s a c c e p t a b l e , and t h i s i s c l o s e l y l i n k e d t o t h e e x i s t e n c e o f a s y m p t o t i c K i l l i n g d i r e c t i o n s w h i c h d e f i n e p r e f e r r e d r e c t a n g u l a r f r a m e s a t i n f i n i t y . The d i s c u s s i o n of a s y m p t o t i c symmetry c o n d i t i o n s and t h e r e s u l t i n g u n i q u e s e l e c t i o n of t h e " c a n o n i c a l " s u p e r p o t e n t i a l w i l l be d e f e r r e d u n t i l r a d i a t i o n f i e l d s have been i n c l u d e d . The i n t e g r a t e d " s p i n " components a r e A l l s i x components t r i v i a l l y v a n i s h i n t h e " r e c t a n g u l a r " ( 6 . 1 2 ) v i e r b e i n f rame ( 6 . 9 ) , w h i l e S&O) d i v e r g e s q u a d r a t i c a l l y i n t h e " s p h e r i c a l " v i e r b e i n f rame ( 6 . 2 ) . 109 b. A s y m p t o t i c a l l y F l a t R a d i a t i v e M e t r i c s I t i s r e a s o n a b l e t o expec t t h a t a good d e f i n i t i o n of g r a v i t a t i o n a l energy s h o u l d y i e l d w i t h o u t a m b i g u i t y t h e t o t a l energy i n a s p a c e - t i m e w i t h an a s y m p t o t i c a l l y f l a t m e t r i c . I f a bounded s o u r c e i n such a s p a c e - t i m e i s e m i t t i n g g r a v i t a t i o n a l r a d i a t i o n , t h e n t h e t o t a l energy measured at r e t a r d e d t i m e s h o u l d r e m a i n w e l l d e f i n e d and s h o u l d d e c l i n e m o n o t o n i c a l l y i n t h e a b s e n c e of any i n w a r d r a d i a t i o n . T h i s n o t i o n was f i r s t put on an i n t u i t i v e l y sound b a s i s i n t h e c o n t e x t of a c t u a l r a d i a t i v e s o l u t i o n s by B o n d i and S a c h s , who f o u n d t h a t t h e t o t a l energy d e f i n e d i n a n a l o g y t o t h e S c h w a r z s c h i l d mass can o n l y d e c r e a s e as t h e s y s t e m r a d i a t e s . J . G o l d b e r g (1962) u s i n g E i n s t e i n ' s g r a v i t a t i o n a l complex and M i l l e r (1964) 1965) o s i n g L i h i s e o w n v i e r b e i n complex c o r r o b o r a t e d t h i s r e s u l t , u s i n g a s y m p t o t i c a l l y r e c t a n g u l a r c o o r d i n a t e s and v i e r b e i n f r a m e s . l ) A s y m p t o t i c S o l u t i o n of B o n d i and Sachs B o n d i , van der Burg and Metzner (1962) found t h e a s y m p t o t i c m e t r i c s u r r o u n d i n g i n s u l a r m a t t e r s y s t e m s w i t h o u t w a r d g r a v i t a t i o n a l r a d i a t i o n a l l o w e d , s u b j e c t t o g l o b a l r e s t r i c t i o n s of a x i a l and r e f l e c t i o n symmetry . Sachs (1962) f o u n d a more g e n e r a l a s y m p t o t i c s o l u t i o n w i t h o u t any r e s t r i c t i o n t o a x i s y m m e t r y . These s o l u t i o n s e x c l u d e i n w a r d r a d i a t i o n , and a r e s e t i n an a s y m p t o t i c a l l y f l a t b a c k g r o u n d m e t r i c . 1 1 0 The a p p r o x i m a t e s o l u t i o n s of B o n d i and a s s o c i a t e s and o f Sachs were found by f i r s t s e t t i n g up c o o r d i n a t e s y s t e m s matched t o t h e f u t u r e p o i n t i n g n u l l g e o d e s i c r a y s of t h e g r a v i t a t i o n a l r a d i a t i o n f i e l d , a p p l i c a b l e t o a s y m p t o t i c r e g i o n s w e l l away f r o m t h e bounded s o u r c e s . The c o o r d i n a t e s a r e t u , a r e t a r d e d t i m e whose g r a d i e n t v e c t o r :~ u/x i s t a n g e n t t o t h e n u l l r a y s i :=• h* , a l u m i n o s i t y d i s t a n c e a l o n g t h e r a y s ( t h e 2 - s u r f a c e du * dr * 0 has a r e a •^/T/'2 ) i ^x^-G and — <j> , " s p h e r i c a l " a n g u l a r c o o r d i n a t e s c o n s t a n t a l o n g t h e r a y s . Boundary c o n d i t i o n s ( S a c h s , 1962) a r e s e t t o e n s u r e E u c l i d e a n t o p o l o g y o f t h e r e g i o n " n e a r i n f i n i t y " , t o e x h i b i t t h e a s y m p t o t i c f l a t n e s s i n t h e l i n e e lement so t h a t </r* * e/cs* -&24^ - A ^ ' f X ^ ^ ' ) and t o impose an o u t g o i n g r a d i a t i o n c o n d i t i o n . Incoming r a d i a t i o n i s e x c l u d e d by r e q u i r i n g t h e m e t r i c c o e f f i c i e n t s ( s a y "cT ) t o have t h e power s e r i e s e x p a n s i o n i n /"""' T h i s i s e q u i v a l e n t t o S o m m e r f e l d ' s r a d i a t i o n c o n d i t i o n £z[(*'')ll]u*c»«fi.*/ = O , ( 6 . 1 4 ) but i n c l u d e s a somewhat r e s t r i c t i v e a s s u m p t i o n o f a n a l y t i c -i t y . The n u l l h y p e r s u r f a c e s u = c o n s t a n t p r o v i d e a I l l s p l i t t i n g of t h e E i n s t e i n f i e l d e q u a t i o n s a p p r o p r i a t e f o r t h e " c h a r a c t e r i s t i c i n i t i a l v a l u e p r o b l e m ( s e e S a c h s , 1964). G i v e n two i n i t i a l d a t a f u n c t i o n s 9 o n one of t h e s e h y p e r s u r f a c e s , t h e f i e l d e q u a t i o n s may be i n t e g r a t e d t o y i e l d t h e f u t u r e fo rm o f t h e a s y m p t o t i c f i e l d up t o f i v e f u n c t i o n s o f i n t e g r a t i o n : t h e r e a l "mass a s p e c t " ( B o n d i , 1 9 6 2 ) / V | = / f ( c / , t h e complex " a n g u l a r momentum d i p o l e moment a s p e c t " f\J~ h^Cfj&> » a n d t n e complex "news f u n c t i o n " C/0= C/0CU)&>4)* Sachs* a p p r o x i m a t e s o l u t i o n i s , t o second o r d e r , "ST where II C = C' + C e" 'I (6.15) (6.16) d e n o t e s t h e s p l i t i n t o r e a l and i m a g i n a r y p a r t s , and (6.18) 1 1 2 The a x i s y m m e t r i c m e t r i c o f Bond i and h i s c o l l a b o r a t o r s (1962) i s o b t a i n e d f rom ( 6 . 1 5 ) by e l i m i n a t i n g any dependence on t h e a z i m u t h a l a n g u l a r c o o r d i n a t e <f> f rom t h e m e t r i c c o e f f i c i e n t s and s e t t i n g a l l t h e i m a g i n a r y p a r t s o f t h e f u n c t i o n s i n ( 6 . 1 6 ) e q u a l t o z e r o . I t i s now c o n v e n i e n t t o t r a n s f o r m t o an a s y m p t o t i c a l l y r e c t a n g u l a r s y s t e m o f c o o r d i n a t e s , i n p r e p a r a t i o n f o r t h e s e l e c t i o n o f a v i e r b e i n f rame used by m i l l e r ( 1 9 6 4 , 1 9 6 5 ) . The B o n d i c o o r d i n a t e s 'ft*/)/', e, <f^ a r e r e p l a c e d by new c o o r d i n a t e s j£? {.'t.) sx*,^j a c c o r d i n g t o £ = 6/ -hk - ksJn&cos<f> ( 6 1 1 9 ) The d i s t i n c t i o n between p r i m e d " r e c t a n g u l a r " c o o r d i n a t e s and unpr imed B o n d i c o o r d i n a t e s w i l l be m a i n t a i n e d c a r e f u l l y i n t h e f o l l o w i n g d i s c u s s i o n . The t r a n s f o r m a t i o n c o e f f i c i e n t s a r e now a s s e m b l e d i n t o a s e t o f q u a n t i t i e s hfy,* , = f OJ Sine cox<(>> SihOri*}, <^as£>J 1 ( 6 . 2 0 ) rrP,*kte ={o> cos6>c«r<f> c e r e - s ; » e j * ~~ IK? <-113 The M i n k o w s k i m e t r i c t e n s o r w i l l be used here only t o r a i s e and l o w e r i n d i c e s o f k and *y>L , so t h a t t h e f o l l o w i n g o r t h o g o n a l i t y c o n d i t i o n s h o l d » The p a r t i a l d e r i v a t i v e o f a q u a n t i t y may be e x p r e s s e d (6.22) C l e a r l y t h e f i r s t t e r m t a k e s p r e c e d e n c e i n a power s e r i e s e x p a n s i o n i n -j^ , a p p e a r i n g one o r d e r l o w e r . The m e t r i c t e n s o r may be w r i t t e n d i r e c t l y as a power s e r i e s e x p a n s i o n i n t h e " r e c t a n g u l a r " c o o r d i n a t e s * g^y + /W*>(v>e>j>) + B*£'(u Os (6 23) where Q n means a q u a n t i t y t h a t i s l e s s t h a n $ f o r some f u n c t i o n $((S>&>$) and s u f f i c i e n t l y l a r g e r . From ( 6 . 1 5 ) and ( 6 . 2 0 ) t h e c o e f f i c i e n t s o f t h e e x p a n s i o n a r e (6.24) , + X '(^.fc+fityM,.) + + fo) 114 2) P a r t i c u l a r V i e r b e i n Gauges The v i e r b e i n gauge i s now t i e d t o t h e c o o r d i n a t e s y s t e m t h r o u g h t h e s y m m e t r i c gauge ( m i l l e r , 1964) ( 6 . 2 6 ) A l s o V ' - & - J . AF' ( t f - j . AS'ArV+O, T h i s gauge c h o i c e i s p u r e l y a m a t t e r o f c o n v e n i e n c e , w i t h i n t h e b r o a d r e q u i r e m e n t t h a t t h e v i e r b e i n l a t t i c e be a s y m p t o t i c a l l y r e c t a n g u l a r . The v i e r b e i n gauge and a p p r o p r i a t e boundary c o n d i t i o n s w i l l be d i s c u s s e d i n d e t a i l l a t e r . F i r s t t h e g e n e r a l v i e r b e i n s u p e r p o t e n t i a l ( 4 . 7 7 ) w i l l be c a l c u l a t e d f o r t h e above c h o i c e of v i e r b e i n gauge i n o r d e r t o e s t a b l i s h c o n t a c t w i t h t h e r e s u l t s o f Bond i and S a c h s . T h i s c a l c u l a t i o n i s a s l i g h t e x t e n s i o n of t h a t used by N a i l e r ( 1 9 6 4 , 1965) t o a p p l y h i s v i e r b e i n complex ( 1 . 3 7 ) t o t h e same p r o b l e m . Indeed m a i l e r ' s r e s u l t s g u a r a n t e e t h a t t h e v e r y c l o s e l y r e l a t e d c a n o n i c a l v i e r b e i n energy and momentum c u r r e n t s w i l l y i e l d t h e e x p e c t e d r e s u l t , s i n c e t h e s u p e r p o t e n t i a l s a r e r e l a t e d by U ^ - A J T O ^ . oz mW-rji+o, • The s t r u c t u r e t e n s o r i s ( 6 . 2 7 ) 115 The g e n e r a l v i e r b e i n s u p e r p o t e n t i a l f o r t h e energy and momentum c u r r e n t s i s U ? - $ B \ * IG*,X'+ c(A2>G*y< -k^(?'\)] (4 . 7 7 ) From ( 6 . 2 7 ) and ftijifc = - C-j*Ck-- C*,Cj) > -AXU„^' y Ah, eg*'- P^Jc"* 0 W * A*yI* - A*** +k"'(&x'>o -JAsA^.-^A^A^/o] -kx -4^A'",,-$A"s.4»-/*)l 116 In c a l c u l a t i n g ( 6 . 2 8 ) t h e i m m e d i a t e c o n s e q u e n c e s of ( 6 . 2 4 ) f / V * O A**' = O ( 6 . 2 9 ) were u s e d . Now t h e t o t a l g e n e r a l i z e d energy and momentum i s found by i n t e g r a t i o n o v e r a c l o s e d 2 - s u r f a c e du = dr a dt a 0 near r a d i a l i n f i n i t y ft - J ( 6 . 3 0 ) s i n c e r*sJ»0AoiJj. From ( 6 . 2 9 ) and t h e o r t h o g o n a l i t y r e l a t i o n s ( 6 . 2 1 ) * 0, ( 6 . 3 1 ) S u b s t i t u t i o n o f ( 6 . 2 0 ) , ( 6 . 2 4 ) , and ( 6 . 2 5 ) t h e n g i v e s ( 6 . 3 2 ) 117 The t o t a l g e n e r a l i z e d energy-momentum i s a c c o r d i n g l y = ^ j J (J°jJ kv *-74**& < / / ( 6 . 3 3 ) The f i r s t t e r m w i t h * £ and ^ - CoJ i s t h e m o n o t o n i c a l l y d e c r e a s i n g mass i n t e g r a l o f B o n d i and S a c h s , w h i c h i s t h e p h y s i c a l p a r t o f The " s u p p l e m e n t a r y " f i e l d e q u a t i o n Al/. - - ^ - ( C V ' + ^ + f'cfit6>+Jtu,)h ( 6.34) i m p l i e s t h a t s ^ ^ ^ ^ ^ ^ 4&<sf</ must d e c r e a s e m o n o t o n i c a l l y b e c a u s e o b v i o u s l y j = < ? , ( 6 . 3 5 ) and a l s o r e g u l a r i t y c o n d i t i o n s a t t h e c o o r d i n a t e p o l e s ( B o n d i , 1962» S a c h s , 1962) i m p l y ^j^&s0 ~0 and t h e r e f o r e _ 6*0 &*o ( 6 . 3 6 ) 118 In o r d e r t o e l i m i n a t e t h e u n p h y s i c a l a r b i t r a r i n e s s i n ( 6 . 3 3 ) i t i s n e c e s s a r y t h a t t h e c o n t r i b u t i o n s o f a l l but t h e f i r s t p h y s i c a l t e r m v a n i s h i n g e n e r a l . These l a s t t h r e e te rms do not have t h e m o n o t o n i c d e c r e a s i n g p r o p e r t y . The l a s t t e r m e v i d e n t l y v a n i s h e s i n g e n e r a l i f and o n l y i f (a-6-hc) O • The s e c o n d and t h i r d a l s o v a n i s h when (<? — 6 + c~J=£) , s i n c e t h e n and o n l y t h e n may i n t e g r a t i o n by p a r t s be a p p l i e d t o o b t a i n i n t e g r a n d s t h a t a r e p u r e g r a d i e n t s i n t h e v a r i a b l e of i n t e g r a t i o n . The d e s i r e d i n t e g r a l fo rms a r e 4>=o o and s i n c e r A»*^>-/ = TT TT (/??k+t'c*t*)k*. - r'tot»J)**,e 4<s>*j^),z 4a *o 6*0 * ( 6 . 3 8 ) The c o n d i t i o n s t h a t ( 6 . 3 3 ) becomes (sine*, fr-#7r) ( 6 . 3 9 ) a r e (*-£^cJ~ -2 and + O , o r - c: c- /. The p r e f e r r e d v i e r b e i n s u p e r p o t e n t i a l f o r t h e v i e r b e i n gauge ( 6 . 2 6 ) i s t h e n * *L (6<40) The d e g e n e r a c y i n G and 6 w h i c h f i r s t a p p e a r e d i n ( 6 . 3 1 ) i s a p e c u l i a r i t y o f t h e p a r t i c u l a r ( s y m m e t r i c ) v i e r b e i n g a u g e . T h i s f o l l o w s f rom t h e o b s e r v a t i o n t h a t whenever t h e m a t r i x o f v i e r b e i n components i s s y m m e t r i c hk*r~ 119 and , t ? * y l * " " & j * V t o f i r s t o r d e r . A l t h o u g h t h i s s y m m e t r i c gauge i s o f t e n v e r y c o n v e n i e n t when t h e m e t r i c t e n s o r i s a l r e a d y g i v e n i n some weak f i e l d a p p r o x i m a t i o n , no r e a s o n has yet a p p e a r e d t o p r e f e r i t o v e r o t h e r a s y m p t o t i c a l l y r e c t a n g u l a r v i e r b e i n gauges f o r t h e c o m p u t a t i o n o f e n e r g y . I f t h e s y m m e t r i c gauge were t o be t a k e n so s e r i o u s l y , P i r a n i ' s energy s u p e r p o t e n t i a l us - G^ <«•«> would be d i s q u a l i f i e d a l r e a d y . A n o t h e r v i e r b e i n gauge w i l l now be g i v e n i n w h i c h P i r a n i ' s e x p r e s s i o n does y i e l d t h e B o n d i - S a c h s m a s s . T h i s v i e r b e i n i n B o n d i c o o r d i n a t e s and t o second o r d e r i s (6.42) 120 The o n l y component o f t h e s t r u c t u r e t e n s o r w h i c h c o n t r i b u t e s t o P i r a n i ' s energy s u p e r p o t e n t i a l t o s e c o n d o r d e r i s Cr°^.oi = 1^ . Then UV) = *~~G -i- O, ( 6 . 4 3 ) A" and as d e s i r e d t h e energy i s p ( 0 ) « ( 6 . 4 4 ) However t h e v i e r b e i n ( 6 . 4 2 ) does not y i e l d t h e t o t a l momentum components ( 6 . 3 9 ) b e c a u s e o f t h e " s p h e r i c a l " o r i e n t a t i o n of t h e s p a t i a l v i e r b e i n d i r e c t i o n s . A s y m p t o t i c r e c t a n g u l a r i t y may be o b t a i n e d by a v i e r b e i n r o t a t i o n ( 6 . 4 5 ) and i n t h e m o d i f i e d gauge Pje =j'c/t\~ ^^^rt4^e></e</ft ( 6 . 4 6 ) T h i s r e s u l t e s t a b l i s h e s ' t h a t a gauge does e x i s t i n w h i c h P i r a n i ' s energy f o r m u l a t i o n ( w h i c h i s a s p e c i a l i z a t i o n o f K o m a r ' s a p p r o a c h ) g i v e s p h y s i c a l l y e x p e c t e d r e s u l t s . But t h e r e seems t o be n o t h i n g d i s t i n c t i v e about t h i s gauge o t h e r t h a n t h e r e s p e c t a b i l i t y i t p r o v i d e s t o P i r a n i ' s s u p e r p o t e n t i a l i n t h i s e x a m p l e . The t i m e l i k e v i e r b e i n v e c t o r c e r t a i n l y i s not a u n i q u e Uleyl p r i n c i p a l v e c t o r . I t i s not p a r a l l e l - p r o p a g a t e d a l o n g t h e r a y s . r \ 0 O o y 0 0 ~ 0 0 0 121 The above examples i n d i c a t e t h e need f o r a s t r o n g e r and more g e n e r a l a p p r o a c h i f a u n i q u e s u p e r p o t e n t i a l i s t o be s e l e c t e d w h i c h w i l l convey u s e f u l p h y s i c a l i n f o r m a t i o n about t h e s e r a d i a t i v e m e t r i c s . I n s t e a d o f c o n s i d e r i n g i n d i v i d u a l v i e r b e i n gauges w h i c h meet c e r t a i n boundary c o n d i t i o n s o f " a s y m p t o t i c r e c t a n g u l a r i t y " , a l l p o s s i b l e gauges w i t h i n p r o p e r l y s t a t e d a s y m p t o t i c c o n d i t i o n s s h o u l d be c o n s i d e r e d a t o n c e . The boundary c o n d i t i o n s w i l l be c o n s i d e r e d n e x t , f o l l o w e d by an e x a m i n a t i o n o f t h e t r a n s f o r m a t i o n p r o p e r t i e s o f t h e g e n e r a l s u p e r p o t e n t i a l under t r a n s f o r m -a t i o n s r e s p e c t i n g t h o s e c o n d i t i o n s . 3 ) A s y m p t o t i c V i e r b e i n Gauge The a s y m p t o t i c f l a t n e s s o f t h e s p a c e - t i m e i s e x p r e s s e d i n R***>46 ~ O , o r more i n f o r m a t i v e l y i n t h e " p e e l i n g p r o p e r t y " o f t h e Riemann t e n s o r ( S a c h s , 1962) R*r*,rs - fa 6 > M H M ( U * * * 4 ) + * > < / > ) + 9 / - A 2 h * ( 6 . 4 7 ) The c o e f f i c i e n t s o f t h e e x p a n s i o n a r e o f t h e i n d i c a t e d P e t r o v t y p e s . Types " N " and " I I I " a r e t h e f a r and s e m i -f a r r a d i a t i o n f i e l d s , w h i c h a r e f u n c t i o n s of t h e news f u n c t i o n C/0 and i t s d e r i v a t i v e s , w h i l e f y p e " I I " c o n t a i n s a d i r e c t c o n t r i b u t i o n f r o m t h e mass a s p e c t A\ . T y p i c a l components o f t h e t y p e N f i e l d a r e AoZtCg •= Cfoe, /Qo2,oz « c"toc ( 6 . 4 8 ) 1 2 * The v i e r b e i n gauge i s a s y m p t o t i c a l l y r e c t a n g u l a r i f £i<> k — O ( 6 . 4 9 ) w h i c h i s o n l y p o s s i b l e when a s y m p t o t i c f l a t n e s s a l r e a d y h o l d s . The v i e r b e i n gauge c o n d i t i o n s h o u l d a l s o i n c l u d e an o u t w a r d s r a d i a t i o n c o n d i t i o n i n o r d e r t o r e a c h t h e m o n o t o n i c a l l y d e c r e a s i n g energy-momentum ( 6 . 3 9 ) T h i s energy-momentum i s measured a t " f u t u r e n u l l i n f i n i t y " (r*-^ oo , "t —*• °«* and some f i n i t e r e t a r d e d t i m e U ) and a c t u a l l y P(0)(u) r e p r e s e n t s t h e energy measured a t s p a t i a l i n f i n i t y {k—><*> • some "t, ) l e s s a l l energy r a d i a t e d b e f o r e r e t a r d e d t i m e U . The energy measured a t s p a t i a l i n f i n i t y i s a c o n s t a n t o f t h e m o t i o n ( s e e A r n o w i t t . Oeser and (Tlisnerf 1 9 6 1 ) . ( V i l e r (1964) d i d not e x p l i c i t l y s t a t e a r a d i a t i o n c o n d i t i o n f o r t h e v i e r b e i n gauge , but imposed a s y m p t o t i c r e c t a n g u l a r i t y and used t h e s y m m e t r i c gauge ( 6 . 2 6 ) t o t i e t h e v i e r b e i n gauge t o a c o o r d i n a t e f rame a l r e a d y c o n t a i n i n g B o n d i ' s o u t w a r d r a d i a t i o n c o n d i t i o n ( 6 . 1 3 ) . From ( 6 . 2 4 ) , ( 6 . 2 5 ) and ( 6 . 2 7 ) i t f o l l o w s t h a t M i l l e r ' s v i e r b e i n ( 6 . 2 6 ) a c t u a l l y c o n t a i n s t h e f o l l o w i n g c o n d i t i o n s ( 6 . 5 0 ) 123 s i n c e AC'*' fo,srO f A\°%' - O and These c o n d i t i o n s a r e not m a i n t a i n e d under a p o s i t i o n v a r y i n g v i e r b e i n r o t a t i o n p r e s e r v i n g fej§cj,k ~ O-with s €ij , e\),* o s i n c e di^t%i—^ B\)iY't €C)IV' I n o r d e r t h a t t h e t r a n s f o r m e d v i e r b e i n s t i l l c o n f o r m t o an o u t w a r d s r a d i a t i o n c o n d i t i o n , t h e a l l o w e d v i e r b e i n r o t a t i o n p a r a m e t e r s may be r e q u i r e d t o s a t i s f y h w h i c h i s e q u i v a l e n t t o a Sommerfe ld c o n d i t i o n used by Waller ( 1 9 6 4 ) . The t r a n s f o r m e d r o t a t i o n c o e f f i c i e n t s a r e and s a t i s f y Bcjti'k*' = Oz , and (Btj>r'+ Br^k* - fo B*j*r ~ 02 (6.53) The f i n a l c o n d i t i o n o b t a i n e d i s a l s o s a t i s f i e d by t h e t i m e l i k e member o f t h e v i e r b e i n ( 6 . 4 2 ) . 124 The c o n d i t i o n ( 6 . 5 3 ) may be found by a somewhat d i f f e r e n t r o u t e d e v e l o p e d i n t h e m e t r i c f o r m u l a t i o n by Trautman ( 1 9 5 8 . 1 9 6 2 ) . C o n s i d e r a combined v i e r b e i n gauge and c o o r d i n a t e s y s t e m s u c h t h a t l n k c < = +- O, ( 6 . 5 4 ) and /j'V^ * A/^ f where /V*V « O/ . T h i s i s a Sommerfe ld o u t w a r d r a d i a t i o n c o n d i t i o n . Then G'k.«* = Hk«b ~ Hk,k«+Oi and g C j i I L « ^ (H±+fc)- kc(tfa+ fafcj-fyp Q> ( 6 . 5 5 ) The r a d i a t i o n c o n d i t i o n on t h e v i e r b e i n gauge i s (By f fcjj/c*- fee B°j^ - Oi i f t h e c o n d i t i o n on t h e c o o r d i n a t e s y s t e m i s k*(Hn + H**)-k*H'6~ Oi ( 6 - 5 6 ) w h i c h i s t h e weakened h a r m o n i c c o o r d i n a t e c o n d i t i o n k^f^kt s O*. u s e d by Trautman* The v i e r b e i n a s y m p t o t i c gauge c o n d i t i o n ( 6 . 5 3 ) i s c o v a r i a n t under c o o r d i n a t e t r a n s f o r m a t i o n s . I t i s s t r u c t u r a l l y v e r y s i m i l a r t o t h e W i n i c o u r - T a m b u r i n o ( 1 9 6 5 ) d e s c r i p t o r p r o p a g a t i o n e q u a t i o n + Ssi-clM"- A>r>v • O (1.28) As r e c o g n i z e d by t h e s e a u t h o r s , t h i s v e c t o r e q u a t i o n o n l y p r o v i d e s c o n d i t i o n s on t h e l i g h t cone on t h e d e r i v a t i v e s 125 oft t h e d e s c r i p t o r s . These c o n d i t i o n s a r e t h e p r o j e c t i o n o f K i l l i n g ' s e q u a t i o n s on t h e o u t g o i n g n u l l h y p e r s u r f a c e . C o n s i d e r t h e d e c o m p o s i t i o n of ( 6 . 5 3 ) i n t e r m s o f a n u l l v i e r b e i n ( 3 . 6 ) a d a p t e d t o t h e p r o p a g a t i o n v e c t o r f i e l d Ic w i t h fix* - + ™« fcf — t^&/> ~ t,j> ( 6 . 5 7 ) t h e c o n d i t i o n s a r e No s u c h c o n d i t i o n s on t h e r o t a t i o n c o e f f i c i e n t s a r e imposed o f f t h e l i g h t cone i n any p l a n e c o n t a i n i n g t h e i n w a r d n u l l — > v e c t o r A » . I t i s t o be n o t e d t h a t t h e p r o p a g a t i o n e q u a t i o n ( 1 . 2 6 ) c a n n o t i n g e n e r a l h o l d e x a c t l y f o r v i e r b e i n v e c t o r s b e c a u s e t h e t r a n s p o r t /f"/W;y * Ay J *v - k^fv,^ does not p r e s e r v e o r t h o n o r m a l i t y r e l a t i o n s a l o n g t h e r a y s . The a s y m p t o t i c v i e r b e i n gauge c o m p a t i b l e w i t h t h e o u t g o i n g r a d i a t i o n c o n d i t i o n s i s •£ll,tr ' °' ( 6 . 5 9 ) The meaning o f t h i s gauge i s t h a t t h e v i e r b e i n f rame c o n f o r m s a t f u t u r e n u l l i n f i n i t y w i t h t h e t r a n s l a t i o n d e s c r i p t o r s o f t h e B o n d i - r O e t z n e r - S a c h s g r o u p » ( s e e S a c h s , 1 9 6 4 ) . T h i s a s y m p t o t i c symmetry group c o n t a i n s t h o s e 126 c o o r d i n a t e t r a n s f o r m a t i o n s w h i c h p r e s e r v e Sachs* (1962) boundary c o n d i t i o n s . W i t h t h e c o n v e n t i o n s w h i c h have been used i n t h i s s e c t i o n ( 1 . 2 8 ) i s s a t i s f i e d by t h e d e c r i p t o r s o f t h e B o n d i - M e t z n e r - S a c h s group (Tambur ino and W i n i c o u r , 1 9 6 6 ) . V i e r b e i n v e c t o r s s a t i s f y i n g ( 6 . 5 9 ) a l s o a s y m p t o t i c a l l y s a t i s f y ( 1 . 2 8 ) t o s e c o n d o r d e r . I t w i l l be r e q u i r e d t h a t t h e v i e r b e i n s s a t i s f y ( 6 . 5 9 ) i n t h e a s y m p t o t i c a l l y f l a t r a d i a t i v e m e t r i c . T h i s l e a v e s t h e o p t i o n s open o f e i t h e r a c c e p t i n g a l l p o s s i b l e v i e r b e i n s t h a t f i t t h e s e r a t h e r w ide boundary c o n d i t i o n s and t h e n l o o k i n g f o r a u n i q u e s u p e r p o t e n t i a l , o r e l s e s e a r c h i n g f o r more nar row c o n d i t i o n s w i t h i n ( 6 . 5 9 ) t h a t w i t h a c o m p a t i b l e s u p e r p o t e n t i a l may p r o v i d e some l o c a l i z a t i o n o f e n e r g y . The f i r s t a p p r o a c h w i l l t u r n out t o be more f r u i t f u l . The l a t t e r o p t i o n c a l l s f o r t h e i m p o s i t i o n o f t i g h t e r gauge c o n d i t i o n s w h i c h may be g l o b a l j u s t as ( 6 . 5 9 ) o r l o c a l . G l o b a l gauge c o n d i t i o n s such as t h e h a r m o n i c gauge w i t h boundary c o n d i t i o n s and K o m a r ' s (1962) " s e m i - K i l l i n g " c o n d i t i o n a l s o w i t h boundary c o n d i t i o n s a p p l i e d t o a v i e r b e i n v e c t o r htoClPtifi-*- Lf'j;*)- O , h(ofit ~ O ( 6 . 6 1 ) - O may be i m p o s e d . Both v i e r b e i n examples ( 6 . 2 6 ) and ( 6 . 4 2 ) 127 o b e y t h e h a r m o n i c g a u g e t o s e c o n d o r d e r , a n d ( 6 . 4 2 ) a l s o s a t i s f i e s t h e s e m i - K i l l i n g c o n d i t i o n t o t h e same o r d e r . T h e r e seems t o be no c l e a r j u s t i f i c a t i o n f o r t h e g e n e r a l a p p l i c a t i o n o f t h e s e a d d i t i o n a l r e s t r i c t i o n s o v e r ( 6 . 5 9 ) , a n d on t h e i r own t h e y a r e c e r t a i n l y n o t s u f f i c i e n t t o e n f o r c e ( 6 . 5 9 ) . The h a r m o n i c g a u g e i s o n l y m a i n t a i n e d u n d e r t h o s e a s y m p t o t i c a l l y r i g i d v i e r b e i n r o t a t i o n s t h a t a r e c o n s t r a i n e d by I?Sit>c/o ~ 0%. r a t h e r t h a n /c^So^/o' Of » anc* t h e s e m i - K i l l i n g c o n d i t i o n i s e v e n more r e s t r i c t i v e . 4) Weyl P r i n c i p a l V i e r b e i n A p u r e l y l o c a l l y d e t e r m i n e d i n v a r i a n t v i e r b e i n i s s u p p l i e d i n g e n e r a l by t h e Weyl t e n s o r . I t seems c l e a r f r o m t h e e x a m p l e s d e v e l o p e d f r o m t h e v i e r b e i n s ( 6 . 2 ) a n d ( 6 . 4 2 ) t h a t o n l y t h e t i m e l i k e p r i n c i p a l Weyl v e c t o r m i g h t be o f any i n t e r e s t h e r e , a n d t h e n o n l y i n c o n j u n c t i o n w i t h P i r a n i ' s s u p e r p o t e n t i a l ( 6 . 4 1 ) a s d i s c u s s e d on p age 98. T h e s p a c e l i k e p r i n c i p a l d i r e c t i o n s must be r e j e c t e d b e c a u s e t h e y c a n n o t c o n f o r m t o any a s y m p t o t i c a l l y r e c t a n g u l a r f r a m e , s i n c e one o f them must l i e i n t h e o u t w a r d s d i r e c t i o n f r o m t h e b o u n d e d s o u r c e . A u n i q u e t i m e l i k e p r i n c i p a l v e c t o r i s n o t o b t a i n e d u n t i l one f o l l o w s t h e a s y m p t o t i c e x p a n s i o n o f t h e Weyl t e n s o r t o t h e Y T t e r m w h i c h i s P e t r o v t y p e I n o n d e g e n e r a t e . T h i s t i m e l i k e v e c t o r w i l l l i e v e r y c l o s e t o t h e l i g h t c o n e . T h e t y p e N p u r e r a d i a t i o n t e r m i s d o m i n a n t a t g r e a t d i s t a n c e s , and d e t e r m i n e s o n l y a s i n g l e n u l l p r i n c i p a l v e c t o r f r o m w h i c h c u r r e n t s s u f f i c i e n t l y w e l l d e f i n e d t o l o c a l i z e any a n a l o g u e t o e n e r g y c a n n o t be c o n s t r u c t e d . 128 I t seems i m p l a u s i b l e t h a t h i g h e r t e r m s i n t h e Riemann t e n s o r w h i c h a r e e x t r e m e l y weak p e r t u r b a t i o n s a t g r e a t d i s t a n c e s f rom t h e s o u r c e s h o u l d be a b l e t o d e t e r m i n e u s e f u l c u r r e n t s i n t h e f a r z o n e , and i t w i l l be shown l a t e r t h a t t h e r a d i a t i v e f i e l d a l o n e c a n n o t p r o v i d e a v a l i d l o c a l i z a t i o n o f e n e r g y . 5 ) V i e r b e i n Gauge C o v a r i a n t T o t a l Energy-Momentum The c i r c u m s t a n c e s under w h i c h t h e t o t a l g e n e r a l i z e d v i e r b e i n energy-momentum t r a n s f o r m s under p o s i t i o n v a r y i n g v i e r b e i n r o t a t i o n s as a t r u e v i e r b e i n v e c t o r w i l l now be i n v e s t i g a t e d . The g e n e r a l v i e r b e i n s u p e r p o t e n t i a l f o r energy and momentum c u r r e n t s i s From ( 4 . 8 7 ) and ( 4 . 8 8 ) t h i s c o m b i n a t i o n t r a n s f o r m s under v i e r b e i n gauge t r a n s f o r m a t i o n s as * ' ' (6.62) i s t h e g e n e r a l i z e d K r o n e c k e r d e l t a . S i n c e d o ^ i r ^ e where e *cj*£ i s t h e p e r m u t a t i o n symbol ( € o t 3 L 3 ~ I ) » a n d a l s o /T' ecj/c» €«v«r h?4/ Ak*4*s ( 6 . 6 3 ) 129 -/ b e c a u s e A = ctetCh?) » t n e l a s t term i n (6«62) becomes ^ ( 6 . 6 4 ) The i n t e g r a l o f t h i s t e r m o v e r a c l o s e d s u r f a c e i s ^ ( 6 . 6 5 ) = -s^f (h\ e.&r» /U^L)/€ R *y"€ W r The l a s t s t e p i n v o l v e s an a p p l i c a t i o n o f S t o k e ' s theorem o v e r a c l o s e d 2 - s u r f a c e w h i c h i m p l i e s f o r a v e c t o r f i e l d Note t h a t hc/6\v i s t h e d u a l s u r f a c e e l e m e n t , w h i c h i s d u a l t o t h e t e n s o r e x t e n s i o n ( s e e Synge , 1 9 6 0 , page 42) o f a 2 - c e l l . I t i s n e x t shown t h a t t h e o n l y s u p e r p o t e n t i a l c o n t a i n e d i n ( 4 . 7 7 ) w h i c h i s c o m p a t i b l e w i t h t h e v i e r b e i n r a d i a t i o n gauge i s t h e c a n o n i c a l s u p e r p o t e n t i a l w i t h a = - c = 1 , b a 0 . T h i s means t h a t t h e c a n o n i c a l e n e r g y -momentum w i l l be t h e same p h y s i c a l e n t i t y when fa rmed f rom 130 any v i e r b e i n a l l o w e d by t h e v i e r b e i n r a d i a t i o n gauge ( 6 . 5 9 ) , and more r e s t r i c t i v e gauge c o n d i t i o n s a r e not 'or, s n e c e s s a r y . The s p e c i f i c gauge and c o o r d i n a t e s ( 6 . 1 9 ) o f M i l l e r w i l l be used as a c o n v e n i e n t s t a r t i n g p o i n t i o t h e r v i e r b e i n s w i t h i n t h e v i e r b e i n r a d i a t i o n gauge w i l l be r e a c h e d by a p e r t u r b i n g v i e r b e i n r o t a t i o n ( 6 . 6 6 ) The energy-momentum i n t e g r a l s (6.30) t r a n s f o r m under ( 6 . 6 6 ) a s T h i s f o l l o w s f r o m t h e o b s e r v a t i o n s t h a t U K KV"VZ as i s o b v i o u s f rom ( 6 . 2 8 ) and t h a t = 0? ( 6 . 6 8 ) E v i d e n t l y t h e v i e r b e i n energy-momentum t r a n s f o r m s as a t r u e v i e r b e i n v e c t o r w i t h i n t h e v i e r b e i n r a d i a t i o n gauge w i t h o u t a d d i t i o n a l r e s t r i c t i o n s o n l y when (a + c ) » 0 and b = 0 . Much s t r o n g e r and somewhat a r b i t r a r y gauge 131 c o n d i t i o n s a r e r e q u i r e d t o f i x any o t h e r s u p e r p o t e n t i a l . The s c a l e o f t h e p r e f e r r e d s u p e r p o t e n t i a l i s now d e t e r m i n e d d i r e c t l y f rom c o r r e s p o n d e n c e w i t h ( 6 . 3 9 ) and ( 6 . 4 0 ) , and t h e r e s u l t i s t h e c a n o n i c a l s u p e r p o t e n t i a l I t was a l r e a d y e s t a b l i s h e d by P l e b a n s k i '(>3>862i)!hthat t h i s s u p e r p o t e n t i a l does l e a d t o a s a t i s f a c t o r y energy-momentum d e f i n i t i o n i n r a d i a t i v e a s y m p t o t i c a l l y f l a t s p a c e s f rom e s s e n t i a l l y s i m i l a r c o n s i d e r a t i o n s . M i l l e r (1964) showed t h e g a u g e - i n v a r i a n c e o f t h e t o t a l energy-momentum d e r i v e d f r o m h i s v e r y c l o s e l y r e l a t e d v i e r b e i n complex under a s y m p t o t i c a l l y r i g i d v i e r b e i n r o t a t i o n s r e s p e c t i n g a Sommer fe ld r a d i a t i o n c o n d i t i o n , by a l e n g t h y e x p l i c i t c a l c u l a t i o n . A l l o f t h e s e r e s u l t s a g r e e . The p r e s e n t c o n t r i b u t i o n i s t o show t h a t t h e p r e f e r r e d c a n o n i c a l s u p e r p o t e n t i a l ( 6 . 6 9 ) may i n f a c t be d e r i v e d by demanding c e r t a i n t r a n s f o r m a t i o n p r o p e r t i e s o f i t s a s y m p t o t i c i n t e g r a l s w i t h i n a c l a s s o f gauge t r a n s f o r m a t i o n s c l e a r l y a d a p t e d t o t h e a s y m p t o t i c symmetry g r o u p . T h i s i s d i r e c t l y a s s o c i a t e d w i t h t h e n e c e s s a r y p r e s e n c e o f s e c o n d o r d e r f i e l d d e r i v a t i v e s i n t h e c u r r e n t s fo rmed f rom o t h e r s u p e r p o t e n t i a l s , w h i c h a r e t h e r e b y more s e n s i t i v e t o t h e a s y m p t o t i c o r i e n t a t i o n o f t h e v i e r b e i n s . 132 The v i e r b e i n r a d i a t i o n gauge ( 6 . 5 9 ) as a l r e a d y m e n t i o n e d has t h e e f f e c t o f b r i n g i n g t h e v i e r b e i n f rame a t f u t u r e n u l l i n f i n i t y i n t o c o n j u n c t i o n w i t h t r a n s l a t i o n d e s c r i p t o r s o f t h e B o n d i - f l l e t z n e r - S a c h s group ( s e e S a c h s , 1 9 6 4 , Fofr a d i s c u s s i o n o f t h i s g r o u p ) . T h i s a s y m p t o t i c symmetry group has a l s o i m p l i c i t l y d e f i n e d t h e t w o - s u r f a c e of i n t e g r a t i o n a t f u t u r e n u l l i n f i n i t y w h i c h i s spanned by B o n d i ' s a n g u l a r c o o r d i n a t e s 0 and <j> . In o r d e r t o c l e a r l y d i s t i n g u i s h t h e t o t a l energy and momenta, i t i s n e c e s s a r y t o r e q u i r e t h a t t h e a s y m p t o t i c v i e r b e i n r e f e r t o t h e same B o n d i f rame as t h e s u r f a c e of i n t e g r a t i o n . The f reedom of c o n s t a n t v i e r b e i n r o t a t i o n s has been r e m o v e d . At t h i s p o i n t i t may be n o t e d t h a t t h e r e i s no r e a l d i s t i n c t i o n between t h e p r e s e n t a p p r o a c h u s i n g t h e c a n o n i c a l v i e r b e i n c u r r e n t w i t h a s y m p t o t i c v i e r b e i n gauge and s u r f a c e of i n t e g r a t i o n a d a p t e d t o a B o n d i - M e t z n e r -Sachs f rame and M i l l e r ' s a p p r o a c h u s i n g t h e c l o s e l y r e l a t e d c a n o n i c a l v i e r b e i n complex w i t h c o o r d i n a t e s and v i e r b e i n gauge s i m i l a r l y c o n f o r m i n g t o t h e BMS g r o u p , e x c e p t i n f o r m a l c o v a r i a n c e , p r o p e r t i e s . The D a v i s - Y o r k s u p e r p o t e n t i a l ( 1 . 4 3 ) w h i c h b e l o n g s t o t h e same f a m i l y c o u l d a l s o be used e q u i v a l e n t l y w i t h BMS d e s c r i p t o r s p r o p a g a t e d by t h e U l i n i c o u r - T a m b u r i n o e q u a t i o n ( 1 . 2 8 ) , 133 The a s y m p t o t i c f o r m o f t h e c a n o n i c a l energy and momentum c u r r e n t s ( 4 . 6 0 ) i s r e a d i l y c a l c u l a t e d u s i n g M i l l e r ' s g a u g e . The c u r r e n t s a r e b i l i n e a r i n t h e s t r u c t u r e c o e f f i c i e n t s w h i c h i n t h i s gauge a r e g i v e n i n ( 6 . 2 7 ) , They a r e C A" A"* ( 6 . 7 0 ) 2 The p o s i t i v e d e f i n i t e n e s s o f t h e energy d e n s i t y i s m a n i f e s t . Under v i e r b e i n r o t a t i o n s r e s p e c t i n g t h e v i e r b e i n r a d i a t i o n gauge ( 6 . 5 9 ) t h e above c u r r e n t s t r a n s -fo rm t o ibbwest o r d e r as v i e r b e i n v e c t o r s + S i & M f l S f c f M ) , * ( 6 > 7 1 ) a s i s r e a d i l y seen u s i n g a rguments s i m i l a r t o t h o s e used i n t h e d e r i v a t i o n of ( 6 . 6 7 ) . Of c o u s e t h e s e q u a n t i t i e s a r e not u n i q u e l y d e f i n e d s i n c e t h e news f u n c t i o n i s not a g e o m e t r i c a l i n v a r i a n t . I t i s p e r h a p s w o r t h m e n t i o n i n g t h a t t h e " e n e r g y d e n s i t y " computed f rom P i r a n i ' s s u p e r p o t e n t i a l ( 6 . 4 1 ) u s i n g t h e v i e r b e i n ( 6 . 4 2 ) i s t°' = -2/lu = Zgg -l(<f'n+ fe*.*),* ( 6 . 7 2 ) w h i c h i s not p o s i t i v e d e f i n i t e . The above e x p l i c i t e x p r e s -s i o n a p p l i e s o n l y t o B o n d i ' s a x i s y m m e t r i c m e t r i c . 134 6) "Spin" Integrals The f ina l topic to be considered in this study of asymptotically f lat radiative metrics is the interpretation of the "spin" integrals 5 ij >* l / c j ^ k A v (6.73) associated with the identi t ies 1/^ . -A(AUJ-/,US) (6.74) Despite the manifest gauge-covariance of the superpotential ]/ \j 9these integrals are not gauge-covariant and indeed are very sensitive to the asymptotic vierbein orientation. Under vierbein rotations respecting the vierbein radiation gauge (6.59) the "spin" integrals transform quite wildly, with not even their convergence assured in general. They only transform as vierbein tensors under r ig id rotations and those non-rigid rotations which are not functions of the angular variables O and (j> . Ttfte ^nbegrals <S"*j were calculated using Mi l le r ' s vierbein (6.26). Five of the six Sij diverge l inearly when the metric is (6.15) without any symmetry restr ict ions, only 5c/^ j being even f i n i t e . However when the metric is restr icted to reflect the axial and ref lect ion symmetries of Bondi's solution, a l l six integrals vanish. This may be traced to the ref lect ion symmetry which is reflected in Mi l le r ' s vierbein gauge. 135 T h e g e o m e t r i c i n t e r p r e t a t i o n o f t h e " s p i n " i n t e g r a l s i s s i m p l e . T h e y a r e i n t e g r a l s o f d i r e c t e d a r e a p r o j e c t e d o n t o t h e r e f e r e n c e 2 - s u r f a c e o f i n t e g r a t i o n . I n f l a t s p a c e - t i m e t h e i n t e g r a l s v a n i s h b y c a n c e l l a t i o n w h e n t h e v i e r b e i n i s r e c t a n g u l a r El^lr-O a n d t h e s u r f a c e o f i n t e g r a t i o n i s c l o s e d , b u t d o n o t v a n i s h i n g e n e r a l i f t h e v i e r b e i n l a t t i c e i s n o t r e c t a n g u l a r . T h e r e i s c e r t a i n l y n o p h y s i c a l i n t e r p r e t a t i o n f o r t h e s p i n i n t e g r a l s w i t h i n t h e u n r e s t r i c t e d v i e r b e i n r a d i a t i o n g a u g e . P o s s i b l y a u s e f u l g l o b a l v i e r b e i n g a u g e c o n d i t i o n m a y b e t h e r e q u i r e m e n t t h a t a l l s i x o f t h e s e i n t e g r a l s v a n i s h o v e r s o m e r e g i o n . 136 c . P l a n e G r a v i t a t i o n a l Wave M e t r i c s The e x a c t s o l u t i o n s f o r t h e most s i m p l e c a s e s of p u r e g r a v i t a t i o n a l r a d i a t i o n a r e u s e f u l t o i l l u s t r a t e t h e f u n d a m e n t a l n o n l o c a l i t y of g r a v i t a t i o n a l e n e r g y . The i m p o r t a n c e o f t h e e x a c t p l a n e wave m e t r i c s i n t e r p r e t e d by B o n d i , P i r a n i and R o b i n s o n (1959) i s t h a t t h e y a r e an unambiguous example o f e l e m e n t a r y g r a v i t a t i o n a l waves v e r y a n a l o g o u s t o e l e c t r o m a g n e t i c p l a n e w a v e s . They a r e c h a r a c t e r i z e d by a f i v e - f o l d group of m o t i o n s w h i c h l e a v e s t h e o u t g o i n g n u l l h y p e r s u r f a c e s i n v a r i a n t . S i n c e t h e y c a n n o t be a s y m p t o t i c a l l y f l a t i n a l l d i r e c t i o n s , boundary c o n d i t i o n s such as t h o s e d i s c u s s e d i n t h e p r e v i o u s s e c t i o n c a n n o t be a p p l i e d . There have been s e v e r a l p r e v i o u s d i s c u s s i o n s o f energy t r a n s p o r t i n t h e s e m e t r i c s . B o n d i et a l (1959) and E h l e r s and Kundt (1962) have d i s c u s s e d q u a l i t i t a v e l y t h e a b s o r p t i o n o f energy f rom t h e s e waves by i d e a l i z e d m a t t e r whose r e a c t i o n on t h e f i e l d i s n e g l e c t e d . T rautman m e n t i o n e d (1959) t h a t E i n s t e i n ' s c a n o n i c a l complex and M i l l e r ' s m e t r i c complex b o t h v a n i s h i n t h e c o o r d i n a t e s y s t e m ( 6 . 7 5 ) below* a l t h o u g h i t i s known ( see Landau and L i f s h l t z , 1951j T r a a t m a n , 1962) t h a t c o o r d i n a t e f rames do e x i s t i n w h i c h t h e s e " e n e r g y d e n s i t i e s " a r e p o s i t i v e d e f i n i t e . An a t t e m p t was made by Kuchaf and Langer (1963) t o l o c a l i z e energy i n pu re g r a v i t a t i o n a l r a d i a t i o n m e t r i c s 137 using Miller's vierbein complex and his proposed gauge condition (1961), but they recognized the ambiguity following from the lack of gauge boundary conditions and succeeded mainly in finding vierbein gauges in which the "energy density" vanishes. It will be sufficient for present purposes to consider a subclass of the exact gravitational plane waves with fixed polarization. The metric may be expressed in the form (Jordan, Ehlers and Kundtj 1961) dsZ = c/^aCcr - CtZ(C4)</^ - 6 \ U ) ^ Z (6.75) where U is a retarded time, V an advanced time, and and ^  are transverse coordinates. The retarded time 6/ is restricted only by the requirement that be the propagation vector of the wave field. The only nontrivial field equation is The Riemann tensor is Petrov type N with amplitude ?^ = - j 2 f t is important to note that this amplitude is only defined up to a choice of scale for the null propagation vector field ~7c , and that this scale can only be fixed by some convention which amounts to selecting a Lorentz reference frame. Neither the symmetries nor the curvature structure of the gravitational plane wave can select any unique time axis locally, The apparent intensity of the field varies to different Lorentz observers. 138 F o r ease of i n t e r p r e t a t i o n a t i m e c o o r d i n a t e T £ i s i n t r o d u c e d by U - iz — 2 } AJ~ = iz + 2* > As j u s t e m p h a s i z e d t h i s t i m e c o o r d i n a t e w i l l not be a t a l l i n v a r i a n t b e c a u s e o f t h e c o m p l e t e f reedom of " n u l l r o t a t i o n s " w h i c h l e a v e t h e p r o p a g a t i o n d i r e c t i o n unchanged) and o n l y t h i s n u l l p r o p a g a t i o n d i r e c t i o n i s u n i q u e l y d e f i n e d by t h e f i e l d s t r u c t u r e . In t h e new c o o r d i n a t e s [ I O O O O - a 2 O O O O O [O O O -I o i 2 J (6.77) A t r i v i a l c h o i c e o f v i e r b e i n gauge i s o o o aiu) O O O Uu) O 0 O I (6.78) The n o n - v a n i s h i n g s t r u c t u r e c o e f f i c i e n t s a r e CL O ,ct)Co) - ~ Cr t0)C?) (6.79) The t i m e l i k e v i e r b e i n v e c t o r T^ey i s h y p e r s u r f a c e - o r t h o g o n a l and t a n g e n t t o a g e o d e t i c c o n g r u e n c e . The s p a c e l i k e members o f t h e v i e r b e i n a r e F e r m i - p r o p a g a t e d a l o n g th<fcs c o n g r u e n c e . The " e n e r g y d e n s i t y " c a l c u l a t e d f r o m t h e c a n o n i c a l v i e r b e i n s u p e r p o t e n t i a l i s (6.80) by t h e f i e l d e q u a t i o n ( 6 . 7 6 ) . i s 139 An a l t e r n a t i v e v i e r b e i n f o r t h e m e t r i c ( 6 . 7 7 ) 0 0 ^ 01 O Q O O 0 0 b O 0 0 0) 0) 2 z The n o n - v a n i s h i n g s t r u c t u r e c o e f f i c i e n t s a r e ( 6 . 8 1 ) ( 6 . 8 2 ) T h i s v i e r b e i n obeys t h e h a r m o n i c gauge c o n d i t i o n The t i m e l i k e v e c c b r i s h y p e r s u r f a c e - o r t h o g o n a l but not g e o d e t i c ? and t h e s p a c e l i k e v e c t o r s a r e a g a i n F e r m i -p r o p a g a t e d a l o n g t h e t i m e l i k e c o n g r u e n c e . The c a n o n i c a l v i e r b e i n " B n e r g y d e n s i t y " i s t > = -ite+t),, *m+$f) <6-83) w h i c h i s m a n i f e s t l y p o s i t i v e d e f i n i t e . In a r e g i o n where t h e f i e l d i s weak so t h a t and L d i f f e r o n l y i n f i n i t e s i m a l l y f rom u n i t y , t h e " e n e r g y d e n s i t i e s " ( 6 . 8 0 ) and ( 6 . 8 3 ) c o i n c i d e . W i t h 0,6^ I and t h e f i e l d e q u a t i o n 0 + i> O , a^~b so t h a t fe Z a ^ X L ( 6 . 8 4 ) 7? hr T h i s meshes p e r f e c t l y w i t h t h e f a r - f i e l d l i m i t o f B o n d i ' s a s y m p t o t i c a l l y f l a t r a d i a t i v e m e t r i c c o n s i d e r e d i n t h e 140 p r e v i o u s s e c t i o n , w i t h B o n d i ' s news f u n c t i o n d£ t a k i n g t h e p l a c e o f CL o r b h e r e . The " v i e r b e i n r a d i a t i o n gauge" ( 6 . 5 9 ) a p p l i e d a s y m p t o t i c a l l y t h e r e may a l s o be a p p l i e d i n a weak f i e l d a p p r o x i m a t i o n h e r e . I f Q and A d i f f e r f rom u n i t y t o f i r s t o r d e r i n a s u i t a b l e p a r a m e t e r o f s m a l l n e s s , t h e n f o r e i t h e r v i e r b e i n ( 6 . 7 8 ) o r ( 6 . 8 1 ) Bij, k- - 0> (jfi)* + fry) k" - /h - 0& ( 6 • 5 9 ) where h e r e O h i n d i c a t e s t h e power t>6 w h i c h t h e p a r a m e t e r o f s m a l l n e s s a p p e a r s . T h i s i s t h e v i e r b e i n r a d i a t i o n gauge ( 6 . 5 9 ) , w h i c h as n o t e d e a r l i e r cannot be a p p l i e d e x a c t l y t o v i e r b e i n f i e l d s i n g e n e r a l b e c a u s e t h e o r t h o g o n a l i t y r e l a t i o n s c o o l d t n o t t h e n be p r e s e r v e d a l o n g t h e r a y s . The components o f b o t h v i e r b e i n s ( 6 . 7 8 ) and ( 6 . 8 1 ) a r e f u n c t i o n s o n l y o f t h e r e t a r d e d t i m e Lf , and i t i m m e d i a t e l y f o l l o w s t h a t t h e s e v i e r b e i n s a r e i n v a r i a n t under t h e p l a n e wave group o f m o t i o n s . I t a l s o f o l l o w s t h a t t h e s e v i e r b e i n s a r e p a r a l l e l - p r o p a g a t e d a l o n g t h e r a y s ' By**? f O . The w r i t e r does not b e l i e v e t h a t e i t h e r o f t h e s e v i e r b e i n s i s b e t t e r t h a n t h e o t h e r , but t h a t t h e i m p o r t a n t t h i n g i s t h e r e f l e c t i o n o f a l l e x i s t i n g symmetry ( g r o u p of m o t i o n s ) i n t h e r e f e r e n c e v i e r b e i n u s e d . I t i s now o f i n t e r e s t t o c o n s i d e r two v i e r b e i n s w h i c h do not r e f l e c t t h e c o m p l e t e p l a n e wave symmetry . I t i s s i m p l e s t t o t r a n s f o r m t o a n o t h e r c o o r d i n a t e 1 4 1 s y s t e m 5^^} def i n e d by s& = , y, AJ ' J U =- ( 6 . 8 5 ) Then t h e m e t r i c t a k e s R o b i n s o n ' s fo rm w i t h H = ACu) , A = & The f i e l d e q u a t i o n i s now HliY+ Hl& ~ O- ( 6 . 8 7 ) A g a i n d e f i n i n g a new t i m e c o o r d i n a t e by ~L' = CdL^*^'-•^e'' t h e m e t r i c becomes o H O -/ o o o o - / o H o o ( 6 . 8 8 ) One p o s s i b l e v i e r b e i n i s (')) O I O C O) 0 0 f 0 ( 6 . 8 9 ) The n o n - v a n i s h i n g components of t h e s t r u c t u r e t e n s o r a r e ro- 5 ( 6'9 0 ) The h a r m o n i c gauge i s s a t i s f i e d , but t h e c o n g r u e n c e t o w h i c h b(/>) i s t a n g e n t i s n e i t h e r h y p e r s u r f a c e - o r t h o g o n a l nor g e o d e t i c . The c a n o n i c a l " e n e r g y d e n s i t y " v a n i s h e s ! -t°'(p) = -^rCHf*' + = O • ( 6 . 9 1 ) 142 o o 1 0 O o J O o o (6.93) A more i n t e r e s t i n g v i e r b e i n i s wf J7=7/ Lk ^ t" J O I O  (6.92) (*){ o The s t r u c t u r e t e n s o r s a r e Some components of t h e c a n o n i c a l v i e r b e i n s u p e r p o t e n t i a l /)0'l'_ - J Hit! I P'*'a _ / _///2' / y ^ ' J " ^ a r e - , ^ ^ J ^ f , U w - £> ; and t h e c a n o n i c a l " e n e r g y d e n s i t y " i s ( u s i n g (6.87)) w h i c h i s n e g a t i v e d e f i n i t e f o r ^ and <y. not t o o l a r g e . I t i s remarked t h a t b o t h v i e r b e i n s (6.89) and (6.92) s a t i s f y t h e r e l a t i o n e x a c t l y . As r e p e a t e d e a r l i e r , t h i s p a r t o f t h e " v i e r b e i n r a d i a t i o n gauge" i s not g e n e r a l l y a d m i s s i b l e as an e x a c t c o n d i t i o n } and h e r e i t must be r e g a r d e d as a f e a t u r e o f t h o s e v i e r b e i n s w h i c h a r e s i m p l y r e l a t e d t o R o b i n s o n ' s m e t r i c ( 6 . 8 6 ) . The n a t u r a l b a s i s o f R o b i n s o n ' s c o o r d i n a t e s y s t e m i s p r o p a g a t e d a c c o r d i n g t o t h e U l i n i c o u r - T a m b u r i n o p r o p a g a t i o n e q u a t i o n (1.28J) as f o l l o w s r e a d i l y f rom r e l a t i o n s g i v e n by E h l e r s and Kundt (1962). Of c o u r s e t h e s e v i e r b e i n s 143 do not c o n f o r m w i t h t h e c o m p l e t e " v i e r b e i n r a d i a t i o n gauge" ( 6 . 5 9 ) w h i c h r e q u i r e s a l s o t h a t t h e r o t a t i o n c o e f f i c i e n t s a l l be s m a l l i n some s e n s e i t h e r o t a t i o n c o e f f i c i e n t s c o r r e s p o n d i n g t o ( 6 . 9 0 ) and ( 6 . 9 3 ) b low up a t l e a s t l i n e a r l y w i t h i n c r e a s i n g t r a n s v e r s e d i s t a n c e ^ 0Tf • The n o n l o c a l i t y of g r a v i t a t i o n a l energy i s a p p a r e n t f rom t h e examples d e p l o y e d a b o v e . The " e n e r g y d e n s i t i e s " ( 6 . 8 0 ) and ( 6 . 8 3 ) a r e b i l i n e a r i n Q and I , w h i c h a r e r e t a r d e d t i m e i n t e g r a l s o f t h e a m p l i t u d e o f t h e n u l l Riemann t e n s o r . Hence t h e energy a s s o c i a t e d w i t h a s i n g l e wave p u l s e c a n n o t be l o c a l i z e d i n t h e wave f r o n t , but c o u l d e q u a l l y w e l l be f o c m a l l y ahead o f t h e p u l s e or b e h i n d i t . T h i s f u n d a m e n t a l n o n l o c a l i t y has been d i s c u s s e d by P e n r o s e ( 1 9 6 6 ) . The " e n e r g y d e n s i t y " ( 6 . 9 4 ) i s l o c a l i z e d t o some e x t e n t , as i t i s q u a d r a t i c i n t h e a m p l i t u f l e of t h e Riemann t e n s o r . T h i s i s a c h i e v e d a t t h e f a t a l c o s t o f i n t r o d u c i n g a dependence on t h e s q u a r e d t r a n s v e r s e d i s t a n c e f rom some r a y ^.—^.•s: O w h i c h i s c o n t r a r y t o t h e symmetry o f t h e s e w a v e s . No p o i n t of a p l a n e wave f r o n t i s d i s t i n g u i s h e d , i n t r i n s i c a l l y f rom any o t h e r . The a p p e a r a n c e of a s q u a r e d l e n g t h f a c t o r c o m b i n i n g w i t h t h e s q u a r e d Riemann t e n s o r a m p l i t u d e i s o f c o u r s e f o r c e d by t h e d i m e n s i o n a l i t y of t h e Riemann t e n s o r . C l e a r l y a pu re g r a v i t a t i o n a l r a d i a t i o n f i e l d c a n n o t d e f i n e a u n i q u e energy c u r r e n t . 144 7 . C o n c l u d i n g Remarks The p h y s i c a l t e s t s a p p l i e d a g a i n s t the f o r m a l v i e r b e i n c u r r e n t c o n s e r v a t i o n laws have e l i m i n a t e d a l l but one e x p r e s s i o n f o r the energy c u r r e n t . T h i s i s the " c a n o n i c a l " energy c u r r e n t , which i s the P e l l e g r i n i - P l e b a n s k i a d a p t a t i o n of M i l l e r ' s v i e r b e i n complex . It owes i t s e x c l u s i v e a c c e p t a b i l i t y w i t h i n the assumpt ions of S e c t i o n 4(a) to the absence of second o rder f i e l d d e r i v a t i v e s . T h i s f e a t u r e o r i g i n a l l y l e d M i l l e r to i t . The term " c u r r e n t " here used may be somewhat o v e r - s u g g e s t i v e , s i n c e on ly i n t e g r a l s formed wi th s u i t a b l e boundary c o n d i t i o n s in a s y m p t o t i c a l l y f l a t s p a c e - t i m e s have any meaning . It seems u s e f u l to compare t h i s f o r m a l i s m wi th t h a t of Win icour (see h i s rev iew a r t i c l e , 1970) and Tamburino d i s c u s s e d on pages 2 5 - 2 6 . The c a n o n i c a l v i e r b e i n c u r r e n t i s drawn q u i t e d i r e c t l y from a Lagrang ian f o r m u l a t i o n of g e n e r a l r e l a t i v i t y e x p l o i t i n g gauge f i e l d a n a l o g i e s . It cannot g e n e r a l l y p r o v i d e any l o c a l i z a t i o n of energy i n f i n i t e r e g i o n s due to the b a s i c approx imate n a t u r e of the " a s y m p t o t i c v i e r b e i n r a d i a t i o n gauge" ( 6 . 5 9 ) . T f e^isi& n - d i T e c t p f l i n e r o f descent from E i n s t e i n ' s o r i g i n a l m e t r i c complex , b e i n g r e a l l y the same t h i n g i n a perhaps more v e r s a t i l e f o r m a l i s m . The U l in icour -Tambur ino f o r m a l i s m has great geometr ic e l e g a n c e , p a r t i c u l a r l y i n i t s t reatment of the a s y m p t o t i c symmetry g roup . It seems s l i g h t l y more remote from any a c t i o n p r i n c i p l e . Under c e r t a i n c o n d i t i o n s i t does l o c a l i z e g r a v i t a t i o n a l energy . 145 In p a r t i c u l a r i t s t a t e s t h a t t h e d e n s i t y o f g r a v i t a t i o n a l energy i n t h e e x t e r i o r S c h u / a r z s c h i l d f i e l d v a n i s h e s , w h i c h seems r a t h e r s u r p r i s i n g i n v i e w of t h e a n a l o g y t o t h e Coulomb f i e l d of an e l e c t r i c c h a r g e . No use has been d e m o n s t r a t e d h e r e f o r any of t h e o t h e r v i e r b e i n c u r r e n t s , i n c l u d i n g t h e s p i n c u r r e n t s . However t h e r ? may be l o c a l u s e s f o r some of t h e s e e x p r e s s i o n s u s i n g i n v a r i a n t v i e r b e i n s as d i s c u s s e d i n C h a p t e r 5 . Then t h e t e r m c u r r e n t wou ld f i n d j u s t i f i c a t i o n . The s p i n c u r r e n t components a r e i m m e d i a t e l y o f some u s e , i f o n l y because of t h e i r c o m p l e t e e q u i v a l e n c e t o t h e s t r u c t u r e c o e f f i c i e n t s and r o t a t i o n c o e f f i c i e n t s of a g i v e n v i e r b e i n . T h i s i n v e s t i g a t i o n c o n c l u d e s w i t h t h e remark t h a t t h e p r e s e n t l y c o n s i d e r e d c o r r e c t t r e a t m e n t of g r a v i t a t i o n a l energy d i f f e r s i n no e s s e n t i a l r e s p e c t f rom t h a t of E i n s t e i n f r o m t h e b e g i n n i n g . 146 BIBLIOGRAPHY A r n o i u i t t , R . , S . D e s e r and C .Ul. M i s n e r » ( 1 9 6 0 ) , P h y s . R e v . 1 1 8 , 1100 A r n o i u i t t , R . , S . D e s e r and C W . M i s n e r , ( 1 9 6 1 ) , P h y s . Rev . 1 2 2 , 997 A r n o w i t t , R . , S . D e s e r and C . W . M i s n e r , ( 1 9 6 2 ) , The Dynamics of G e n e r a l R e l a t i v i t y , i n G r a v i t a t i o n , e d . L . U I i t t e n , U l i l e y B a u e r , H. ( 1 9 1 8 ) , P h y s . Z . 1 9 , 163 B e l i n f a n t e , F . J . ( 1 9 3 9 ) , P h y s i c a , 6, 887? 7 , 449 Bergmann, P . G . , ( 1 9 4 9 ) , P h y s . R e v . , 7 5 , 680 Bergmann, P . G . , and R . S c h i l l e r , ( 1 9 5 3 ) , P h y s . R e v . , 8 9 , 4 Bergmann, P . G . , and R.Thomson, ( 1 9 5 3 ) , P h y s . R e v . 8 9 , 400 Bergmann, P . G . , ( 1 9 5 8 ) , P h y s . R e v . , 1 1 2 , 287 B o n d i , H . , ( 1 9 5 9 ) , C o l l o q u e i n t e r n a t i o n a l s u r Les T h e o r i e s  R e l a t i v i s t e s de l a G r a v i t a t i o n , Royaumont, CNRS, 1962 B o n d i , H., ITI.G.J. van der Burg and A .UI.K . M e t z n e r , ( 1 9 6 2 ) , P r o c . R o y . S o c . ( L o n d o n ) , A 2 6 9 , 21 B o u l w a r e , D . G . and S . D e s e r , ( 1 9 6 7 ) , J . M a t h . P h y s . , 8, 1468 C a t t a n e o , C , ( 1 9 6 6 ) , A n n . I n s t . H. P o i n c a r e , JH/, 1 D a v i s , W.R. and J . W . Y o r k , ( 1 9 6 9 ) , Nuovo C i m e n t o , 61B , 271 D i r a c , P . A . M . , ( 1 9 5 9 a ) , P h y s . R e v . L e t t e r s , 2, 368 D i r a c , P . A . M . , ( 1 9 5 9 b ) , C o l l o q u e i n t e r n a t i o n a l s u r Les T h e o r i e s R e l a t i v i s t e s de l a G r a v i t a t i o n , Royaumont, CNRS, 1962 E h l e r s , J . , and W. K u n d t , ( 1 9 6 2 ) , E x a c t S o l u t i o n s of t h e G r a v i t a t i o n a l F i e l d E q u a t i o n s , i n G r a v i t a t i o n , e d . L . W i t t e n , W i l e y E i n s t e i n , A . , ( 1 9 1 5 a ) , S i t z b e r . p r e u s s . A k a d . W i s s . , 778 E i n s t e i n , A . , ( 1 9 1 5 b ) , S i t z b e r . p r e u s s . A k a d . W i s s . , 844 E i n s t e i n , A . , ( 1 9 1 6 a ) , A n n . P h y s i k , 4 9 , 769 E i n s t e i n , A . , ( 1 9 1 6 b ) , S i t z b e r . p r e u s s . A k a d . W i s s . , 1111 147 E i n s t e i n , A . , ( 1 9 1 8 a ) , P h y s . Z . , 1 9 , 115 E i n s t e i n , A . , ( 1 9 1 8 b ) , S i t z b e r . p r e u s s . A k a d . W i s s . , 448 E i n s t e i n , A . , ( 1 9 2 8 ) , S i t z b e r , p r e u s s . A k a d . W i s s . , 466 E i s e n h a r t , L . P . , ( 1 9 2 6 ) , R i e m a n n i a n Geometry , P r i n c e t o n F o c k , V . A . , ( 1 9 5 9 ) , S p a c e , Time and G r a v i t a t i o n , Pergamon, London von F r e u d , P . ( 1 9 3 9 ) , A n n . M a t h . , 4 0 , 417 F r o l o v , B . N . , ( 1 9 6 4 ) , V e s t n i k Moscow U n i v e r i s t y ( F i z , A s -t r o n . ) , 2 , 56 F r o l o v , B . N . , ( 1 9 6 5 ) , i n Sovremenni i P r o b l e m ! G r a v i t a t s a i , e d . D . I v a n e n k o , p. 3 1 1 , T b i l i s i , 1967 G o l d b e r g , J . N . , ( 1 9 5 8 ) , P h y s . R e v . , I l l , 315 H i l b e r t , D . , ( 1 9 1 5 ) , G o t t i n g e r N a c h r . , 395 K a e m p f f e r , F . A . , ( 1 9 6 5 ) , C o n c e p t s i n Quantum M e c h a n i c s , Academic P r e s s K a e m p f f e r , F . A . , ( 1 9 6 8 ) , P h y s . R e v . , 1 6 5 , 1420 K i b b l e , T . W . B . , ( 1 9 6 1 ) , J . M a t h . P h y s . , 2 , 212 K l e i n , F . ( 1 9 1 8 ) , G o t t i n g e r N a c h r . , 394 Komar, A . , ( 1 9 5 9 ) , P h y s . R e v . , 1 1 3 , 934 K o m a r , 9 A . , ( 1 9 6 2 ) , P h y s . R e v . , 1 2 7 , 9 5 9 , 1411 Komar, A . , ( 1 9 6 3 ) , P h y s . R e v . , 1 2 9 , 1873 L a n d a u , L . and E . L i f s h i t z , ( 1 9 5 1 ) , The C l a s s i c a l Theory of  F i e l d s , A d d i s o n - W e s l e y , R e a d i n g L o r e n t z , H . A . , ( 1 9 1 6 ) , C o l l e c t e d P a p e r s , V o l . 5 , p . 2 4 6 , N i j h o f f , t h e Hague, T 9 3 " 7 Magnusson , M . , ( 1 9 6 0 ) , M a t . F y s . Medd. Dan . V i d . S e l s s k . , 3 2 , n o . 6 M i t s k e v i c h , N . , ( 1 9 5 8 ) , A n n . P h y s i k , 1, 319 M a i l e r , C , ( 1 9 5 8 ) , A n n . of P h y s . , 4 , 347 M p l l e r , C , ( 1 9 6 1 ) , A n n . o f P h y s . , 1 2 , 118 148 M i l l e r , C , ( 1 9 6 1 ) , Mat . F y s . S k r . D a n . U i d . S i l s k . , 1, n o . 10 Mj r f l le r , C , ( 1 9 6 4 ) , Mat . F y s . M e d d . V i d . S e l s k . , 3 4 , n o . 3 M i l l e r , C , £1965) , C o n f e r e n c e G a l i l e o G a l i l e i , F l o r e n c e M i l l e r , C , ( 1 9 6 6 ) , M a t . F y s . M e d d . D a n . V i d . S e l s k . , 3 5 , No . 3 Newman, E. and R . P e n r o s e , ( 1 9 6 2 ) , J . M a t h . P h y s . , 3 , 566? 4 , 998 N o e t h e r , E . , ( 1 9 1 8 ) , G o t t i n g e r N a c h r . , 235 P a p a p e t r o u , A . , ( 1 9 4 8 ) , P r o c . R o y . I r i s h . A c a d . , 42A, 11 P a u l i , Ul. ( 1 9 2 2 ) , Theory of R e l a t i v i t y , Pergamon, 1958 P e l l e g r i n i , C , and J . P l e b a n s k i , ( 1 9 6 2 ) , M a t . F y s . S k r . D a n . V i d . S e l s k . , 2 , 39 P e n r o s e , R . , ( 1 9 6 4 ) , i n R e l a t i v i t y , Groups and T o p o l o g y , e d . B . S . and D.DeWit tT Gordon and B r e a c h P e n r o s e , R . , ( 1 9 6 6 ) , i n P e r s p e c t i v e s i n Geometry and  R e l a t i v i t y , ed B . H o f f m a n , I n d i a n a U n i v e r s i t y P e t r o v , A . Z . , ( 1 9 6 1 ) , C o n f e r e n c e i n t e r n a t i o n a l e s u r l e s t h e o r i e s r e l a t i v i s t e s de l a g r a v i t a t i o n , e d . L . I n f e l d , Warsaw, 1964 P e t r o v , A . Z . , ( 1 9 6 6 ) , i n P e r s p e c t i v e s i n Geometry and  R e l a t i v i t y , e d . B . H o f f m a n , I n d i a n a U n i v e r s i t y P i r a n i , F . A . E . , ( 1 9 5 6 ) , A c t a P h y s . P o l o n . , 1 5 , 389 P i r a n i , F . A . E . , ( 1 9 5 7 ) , P h y s . R e v . , 1 0 5 , 1089 P i r a n i , F . A . E . , ( 1 9 5 9 ) , C o l l o q u e i n t e r n a t i o n a l s u r Les  T h e o r i e s R e l a t i v i s t e s de l a G r a v i t a t i o n , Royaumont , CNRS, 1962 : P l e b a n s k i , J . , ( 1 9 6 2 ) , C o n f e r e n c e i n t e r n a t i o n a l s u r l e s t h e o r i e s r e l a t i v i s t e s de l a g r a v i t a t i o n , ed L . I n f e l d , Warsaw, 1964 P o l i s h c h u k , R . F . , D o k l a d y A k a d . Nauk SSSR, 1 9 4 , 6 2 , (1970) 149 R a y s k i , J . , (1961), A c t a P h y s . P o l o n . , 20, 509 R a y s k i , J . , (1962), A c t a P h y s . P o l o n . , 21, 99 R o d i c h e v , V . I . (1965), I z v e s t i y a VUZ, F i z . , 1, 142 R o d i c h e v , V . I . (1968), i n E i n s h t e i n o v s k i S b o r n i k 1968, 115 I v d a t e l s t V o Nauka, Moscow R o s e n , G., (1959), P h y s . R e v . , U4, 1179 R o s e n , N . , (1940), Phys. Rev., 57, 147 Rosen, N . , (1962), C o n f e r e n c e I n t e r n a t i o n a l e s u r l e s t h e o r i e s r e l a t i v i s t e s de l a g r a v i t a t i o n , ed L . I n f e l d , Warsaw, 1964 R o s e n f e l d , L . , (1930), Ann. P h y s i k , 5, 113 R o s e n f e l d , L., (1940), Acad.Roy.Belg., JL8, n o . 6 Sachs, R ., (1962), Proc.Roy . Soc.(London), A270, 103 Sachs, R., (1962), Phys. Rev., 128, 2851 S a k u r a i , J . J . , ( I 9 6 0 ) , Ann. Phys. N.Y., 11, 1 S a k u r a i , J . J . , (1968), V e c t o r Mesons 1960-1968 i n L e c t u r e s  i n T h e o r e t i c a l P h y s i c s , U n i v e r s i t y of C o l o r a d o , e d . K . M a h a n t h a p p a , W . B r i t t i n and A . B a r u t , Gordon and B r e a c h S b i y t o v , Y, (1965), i n SoveemennE P r o b l e m i G r a v i t a t s i a , e d . D . I v a n e n k o , T b i l i s i , 1967 S c h r o e d i n g e r , E . , (1918), P h y s . Z . , 19, 4 S c h w i n g e r , J . , (1963), P h y s . R e v . , 130, 406, 800, 1253 S c h w i n g e r , J . , (1963), P h y s . R e v . , _132, 1317 S c i a m a , D . W . , (1961), C o n f e r e n c e i n t e r n a t i o n a l e s u r l e s t h e o r i e s r e l a t i v i s t e s de l a g r a v i t a t i o n , e d . L . I n f e l d Warsaw, 1964 Synge, J . L . , (1959), C o l l o q u e i n t e r n a t i o n a l s u r Les T h e o r i e s  R e l a t i v i s t e s de l a G r a v i t a t i o n , Royaumont, CNRS, 1962 Synge, J . L . , (1960), R e l a t i v i t y , t h e G e n e r a l T h e o r y , N o r t h H o l l a n d , Amsterdam T a m b u r i n o , L . and W , H . W i n i c o u r, (1966), P h y s . R e v . , 150,1039 T r a u t m a n , A.,(1957), B u l l . A c a d . P o l o n . S c i . , Cl.111,5, 721 150 T r a u t m a n , A . , ( 1 9 5 9 ) , C o l l o q u e i n t e r n a t i o n a l s u r Les  T h e o r i e s R e l a t i v i s t e s de l a G r a v i t a t i o n , Royaumont , CNRS, 1962 T r a u t m a n , A . , ( 1 9 6 2 ) , C o n s e r v a t i o n Laws i n G e n e r a l R e l a t i v i t y , i n G r a v i t a t i o n , e d . L . W i t t e n , W i l e y T r a u t m a n , A . , ( 1 9 6 4 ) , L e c t u r e s on G e n e r a l R e l a t i v i t y , B r a n d e i s , P r e n t i c e - H a l l , 1965 W a i n w r i g h t , J . , ( 1 9 7 i ) , J . m a t h . P h y s . , 1_2, 828 Weber , J . , ( 1 9 6 9 ) , P h y . R e v . L e t t e r s W e y l , H . , ( 1 9 1 8 ) , R a u m - Z e i t - M a t e r i e , i e d . , p . 184 W e y l , H . , ( 1 9 2 9 ) , Z e i t . P h y s . , 5 6 , 330 W e y l , H . , ( 1 9 5 0 ) , P h y s . R e v . , 7 7 , 699 Winicourtp J . and L . T a m b u r i n o , ( 1 9 6 5 ) , P h y s . R e v . L e t t e r s , 1 5 , 601 W i n i c o u r , J . , ( 1 9 7 0 ) , i n R e l a t i v i t y , e d . M . C a r m e l i , S . F i c k l e r , and L . W i t t e n , Plenum P r e s s Be W i t t , B . S . ( 1 9 6 2 ) , i n G r a v i t a t i o n , ed.. L. W i t t e n , W i l e y De W i t t , B . S . ( 1 9 6 4 ) , R e l a t i v i t y , Groups and T o p o l o g y , Gordon and B r e a c h Yang , C . N . and F t . L . M i l l s , ( 1 9 5 4 ) , P h y s . R e v . , 9 6 , 191 Yano, K . , ( 1 9 5 6 ) , The Theory o f L i e D e r i v a t i v e s and i t s  A p p l i c a t i o n s , N o r t h H o l l a n d , Amsterdam 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084878/manifest

Comment

Related Items