lost in the tensors: einsteins struggles with covariance princip

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Carnegie Mellon University

Research Showcase

Department of Philosophy Dietrich College of Humanities and Social Sciences

1-1-1978

Lost in the Tensors: Einstein's Struggles withCovariance Principles, 1912-1916

John EarmanUniversity of Minnesota

Clark GlymourUniversity of Illinois at Chicago

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Recommended CitationEarman, John and Glymour, Clark, "Lost in the Tensors: Einstein's Struggles with Covariance Principles, 1912-1916" (1978).Department of Philosophy. Paper 336.hp://repository.cmu.edu/philosophy/336
http://repository.cmu.edu/?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/philosophy?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/hss?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/philosophy?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://network.bepress.com/hgg/discipline/525?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]://repository.cmu.edu/philosophy/336?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]://repository.cmu.edu/philosophy/336?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://network.bepress.com/hgg/discipline/525?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/philosophy?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/hss?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/philosophy?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/?utm_source=repository.cmu.edu%2Fphilosophy%2F336&utm_medium=PDF&utm_campaign=PDFCoverPages


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J O H N E A R M A N an d C L A R K G L YM O U R

L O S T I N T H E T E N S O R S :E I N S T E I N ' S S T R U G G L E S W I T H

C O V A R I A N C E P R I N C I P L E S 1 9 1 2 - 1 9 1 6 "I n t r o d u c t i o n

IN 1912 Einstein began to devote a ma jor port ion o f his t ime and energy to anattem pt to const ruct a relat ivist ic theory of gra vitat ion . A strong int ima tion ofthe struggle that lay ahead is cont aine d in a let ter to Arn old Somme rfeld datedOcto ber 29, 1912:

At the moment I am working solely on the problem of gravitation and believe 1will be able to overcome all difficulties with the help of a local, friendly mathemat-ician. But one thing is certain, that I have never worked so hard in my life, and thatI have been injected with a great awe of mathematics, which in my naivet~ untilnow I only viewed as a pure luxury in its subtler forms! Compared to this problemthe original theory of relativity is mere child's play.'

Einstei n's letter containe d only a pe rfunctory reply to a query from Sommerfeldabo ut the De by e- Bo rn theory of specific heats. Obviously disappointed, Som-merfe ld wrote to Hilbert: 'M y letter to Einste in was in vain . . . Einste in isevidently so deeply mired in gravi tat ion that he is deaf to everything else?Somm erf eld 's words were more prophetic th an he could possibly have kno wn;the next three years were to see Einstein deeply mired in gravita tion , sometim esseemi ngly hopelessly so.

In large meas ure, Ein ste in' s struggle resulted fro m his use and his misuse,his unde rs tand ing and his misund ers tan ding of the nature and impl icat ions ofcovariance principles. In brief, considerations of general covariance were b oun dup with Ei nstei n's motive for seeking a 'g eneral ized' theory of relat ivity; mis-unders tan dings about the meani ng and i mplem enta t ion of this mot ivat ionthreatened to wreck the search; and in the end, the desire for general covariancehelped to bring Einstein back onto the track which led to what we now recognize

*Present address c/o Department of Philosophy, University of Minnesota, Minneapolis, Minn,U.S.A. and Department of Philosophy, The University of Illinois at Chicago Circle, Chicago, 111.60680, U.S.A. We are grateful to Dr. Otto Nathan, Trustee of the Einstein Estate, for permissionto quote from Einstein's correspondence. We also want to thank the anonymous referee fornumerous improvements in an earlier draft of this paper.'A. Hermann (ed.), A l b e r t E i n s t e i n / A r n o l d S o m m e r f e l d B r i e fw e c h s el (Basel: Schabe and Co.,1968), p. 26. The 'friendly mathematician' was Marcel Grossmann; various aspects of the Einstein-Grossmann collaboration will be discussed below. Our translations of the Einstein-Sommerfeldcorrespondence are taken from Helga and Roger Stuewer's translation of this book. We are grate-ful to them for their permission to use their unpublished translation.21bid. p. 27.Stud. His t . Phi l . Sc i . , Vol. 9 (1978), No. 4. pp. 251 - 278 . 0039-3681/78/1201-0251$02.00/0 Pergamon Press Ltd. Printed in Great Britain.

251


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252 Studies in History and Philosophy of Sciencea s th e g e n e r a l t h e o r y o f re l a t iv i t y . T h e p u r p o s e o f t h is p a p e r i s t o t r a c e t h ei n f lu e n c e w h i c h E i n s t e in ' s c h a n g i n g a t ti tu d e s t o w a r d s c o v a r i a n c e h a d o n t h ee m e r g e n c e o f t h e g e n e r a l th e o r y . T h o u g h t h e t a s k is b a s i c a l ly a n h i s to r i c a l o n e ,i t i s i n e x t r i c a b l y b o u n d u p w i t h a n u m b e r o f i s s u e s i n t h e f o u n d a t i o n s o f p h y s i c sw h i c h a r e st il l s u b j e c t t o l iv e l y d i s c u s s io n . W e m a k e n o a p o l o g y f o r o u r f o r a y si n to t h es e a r e a s , f o r n o d e t o u r a r o u n d t h e m c a n l e a d to a n y r e al a p p r e c i a t i o no f E i n s t e i n ' s s t r u g g l e s .

1 . E i n s t e i n ' s D e s i r e f o r a 'G e n e r a l i z e d ' T h e o r y o f R e l a t i v it yE i n s t e in ' s m o t i v a t i o n f o r th e s p e c ia l t h e o r y o f r e la t iv i ty d e r iv e d m a i n l y f r o m

h i s c o n c e r n w i t h t h e s p e c i a l p r i n c i p l e o f r e l a t i v i t y a n d n o t w i t h t h e a e t h e rh y p o t h e s i s o r th e c o n s t i tu t i o n o f m a t t e r , t h e p r o b l e m s w h i c h o c c u p i e d L o r e n t za n d P o i n c a r 6 . 3 T o a s i g n i fi c a n t d e g r e e , a s i m i l a r c o n c e r n w i t h a n e x t e n d e dv e r s i o n o f th e p r i n c i p l e o f r e la t i v i ty m o t i v a t e d t h e s e a r c h f o r a t h e o r y o f g r a v i -t a t i o n .

T h e r e a s o n s f o r E i n s t e i n ' s d e s i r e t o g e n e r a l i z e r e l a t i v i t y , a n d t h e i m p o r t a n c eo f c o v a r i a n c e t o t h a t p r o j e c t , a r e t o b e f o u n d i n E i n s t e i n ' s e a r li e s t w r i t in g s o ng r a v i t a ti o n . I n 1 90 7 E i n st e in p u b l i s h e d a s u r v e y a n d d i s c u ss i o n o f e x p e r i m e n t a la n d t h e o r e ti c a l w o r k o n r e la t iv i ty . T h e f i f t h a n d f in a l p a r t o f th i s e s sa y c o n t a i n sa d i s c u s s io n o f t h e c o n n e c t i o n b e t w e e n g r a v i t a t i o n a n d t h e p r i n c i p l e o f r e la -t i v i ty , a n d i t b e g i n s w i t h a q u e s t i o n :

We have t hus f a r t aken t he P r inc ip l e o f Re l a t i v it y - - t he p r inc ip l e o f t he i ndepen-dence o f na tu ra l l aws f rom the s t a t e o f mot ion o f r e f e r ence f r ames - - on ly fo rr e f e rence f r ames f r ee o f acce l e ra t ion . Can i t be t hough t t ha t t he P r inc ip le o f Re l a -t ivity a l so holds for sys tems which a re accelera ted wi th respect to one an oth er? 'E i n s t e i n ' s a n s w e r is i n e f f e c t n e g a t i v e . T h e r e m a i n d e r o f th e s e c t i o n d e v e l o p s

t h e p r i n c i p le o f e q u i v a l e n c e f o r t h e f ir s t t i m e , a n d e x p l o r e s s o m e o f i ts i m p l i c a -t i o n s . T h i s p r i n c i p l e w a s t h e r o c k u p o n w h i c h E i n s t e i n b u i l t h i s g r a v i t a t i o nt h e o r y : i n n i n e y e a r s o f w o r k o n g r a v i t a t i o n b e t w e e n 19 07 a n d 1 9 16 , t h e p r i n c i -p l e o f e q u i v a l e n c e w a s , a b o v e a l l, t h e i d e a E i n s te i n r e f u s e d t o g i v e u p , a n dy e t i n hi s v e r y f i r s t s t a t e m e n t o f t h e p r i n c i p l e h e f i n d s i t i n c o n s i s t e n t w i t h t h ec o n s t a n c y o f th e v e l o c i t y o f l ig h t a n d w i th a n y g e n e r a l i z a t i o n o f th e p r i n c i p l eo f r e l a ti v i ty . T h e v e l o c i t y o f l ig h t c a n n o t b e t h e s a m e in a c c e l e r a t e d a n d u n -a c c e l e ra t e d f ra m e s ; th e la w s in a n u n a c c e l e r a t e d f r a m e m a y h a v e to i n c o r p o r a t ea h o m o g e n e o u s g r a v i t a t i o n a l f i e l d w h i c h w o u l d v a n i s h i n a n a c c e l e r a t e d f r a m e .

E i n st e in p u b l i s h e d n o t h i n g m o r e o n g r a v i t a t i o n f o r t h re e y e a r s , a n d w h e n i n

3See T. Hirosige, 'The Ether Problem , the Mechan istic Wo rldview, and the O rigins of the T heoryof Relativity', Historical Studies in the Physical Sciences, Vol. 7 (Princeton: Princeton UniversityPress, 1976), pp. 3-82.4A. Einstein, 'lQber das Relativit~itsprinzip und die aus dem selben gezog enen Folgerungen ',J~hrbuch der Radioaktivit~t und Elekronik 4 (1907), p. 454 .


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Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916 2531 91 1 h i s n e x t p a p e r ~ o n t h e s u b j e c t a p p e a r e d , i t w e n t l i tt le b e y o n d h i s 1 9 07 d i s -c u s s i o n i n r e c o n c i li n g t h e p r i n c i p l e s o f e q u i v a l e n c e a n d r e la t iv i t y . E i n s t e in d i d ,h o w e v e r , s u g g e s t a s e n s e i n w h i c h h i s i d e a s o n g r a v i t a t i o n g e n e r a l i z e d t h e t h e o r yo f r e la t iv i ty . F o r i f a n u n a c c e l e r a te d f r a m e o f r e f e re n c e s u b j e c t to a h o m o g e n e o u sg r a v i t a t i o n a l f ie l d is p h y s i c a l l y e q u i v a l e n t t o a n a c c e l e r a t e d f r a m e , t h e n , a tl e a s t w h e n t h e r e i s g r a v i t a t i o n , i t s e e m e d t o E i n s t e i n t h a t t h e r e i s n o s u c h t h i n ga s a b s o l u t e a c c e l e r a t i o n . J u s t a s t h e s p e c ia l t h e o r y e l i m i n a te s a b s o l u t e v e l o c it y ,i ts g e n e r a l i z a t i o n t h r o u g h t h e p r i n c ip l e o f e q u i v a l e n c e e l i m i n a t e s a b s o l u t ea c c e l e r a t i o n . T h i s w a y o f v i e w i n g t h e r e l a t i o n b e t w e e n r e l a t i v i t y a n d t h e p r i n c i p l eo f e q u iv a l e n c e w a s b o u n d t o le a d E i n st ei n t o t r y to e m b o d y c e r t a in o f M a c h ' si d e a s w i t h i n g r a v i t a t i o n a l t h e o r y , a n d M a c h i a n t h e m e s w e r e t o e m e r g e e x p l i c i t l yi n E i n s t e i n ' s 1 9 1 3 t h e o r y . B u t i n 1 91 l , t h e p r i n c i p l e o f r e la t i v it y - - m e a n i n g t h ei n v a r i a n c e o f t h e f o r m o f p h y s i c al la w s in al l a p p r o p r i a t e r e f e r e n c e f r a m e s - -a n d t h e p r i n c ip l e o f e q u i v a l e n c e w e r e st ill a t o d d s . I n d e e d , a s E i n s t e in m a d e h isi d e a s o n g r a v i t a t i o n m o r e e x p l i c i t , t h e c o n t r a d i c t i o n s e e m e d m o r e a n d m o r ei r re c o n c i l a b l e . E i n s t e i n ' s 1 9 11 a n d 1 91 2 p a p e r s o n g r a v i t a t i o n d e v e l o p a s c a la rt h e o r y o f s ta t i c g r a v i t a t i o n a l f i e ld s in w h i c h t h e v e l o c i t y o f li g h t p l a y s t h e r o l eo f t h e g r a v it a t io n a l p o t e n t i a l ? A c c o r d i n g l y , t h e t h e o r y c o u l d n o t b e i n v a ri a n tu n d e r L o r e n t z t r a n s f o r m a t i o n s . T h e p r i n c ip l e o f e q u iv a l e n c e w a s in o u t r ig h tc o n t r a d i c t i o n w i t h t h e s p e c i a l t h e o r y o f r e l a t i v i t y , a n d n o g e n e r a l i z a t i o n s e e m e da t h a n d w h i c h w o u l d s a v e t h e p r in c i p l e s o f th e l a t t e r . W h e n i n 1 91 2 E i n s t e inu n d e r t o o k t o m a s t e r th e t e n s o r c a l c u lu s w i t h t h e h e l p o f h is f r ie n d M a r c e lG r o s s m a n n , i t m a y v e r y w e ll h a v e b e e n e x a c tl y b e c a u s e th e n e w m a t h e m a t i c a la p p a r a t u s a p p e a r e d t o o f f e r a m e a n s t o s a v e g r a v i t a t i o n a l t h e o r y , t h e p ri n c ip l eo f e q u i v a l e n c e a n d t h e p r in c i p l e o f r e la t iv i t y a ll a t o n c e . T h e ' A b s o l u t e D i f f e r -e n t ia l C a l c u l u s ' w o u l d p e r m i t t h e r e p r e s e n t a t i o n o f q u a n t i t ie s a s g e o m e t r i c a lo b j e c t s , a n d w o u l d f u r t h e r p e r m i t t h e s t a t e m e n t o f r e l a ti o n s b e t w e e n s u c hq u a n t it ie s i n s u c h a w a y t h a t t h e y w o u l d h o l d i n e v e r y f r a m e o f r e f e r e n c e if th e yh e l d i n a n y . T h e p r i n c i p l e o f c o v a r i a n c e w o u l d t h u s b r i n g w i t h i t a g e n e r a l i z a t i o no f t h e p r in c i p l e o f r e l a t iv i t y , a n d t h e q u e s t i o n E i n s t e i n a s k e d i n 1 90 7 w o u l d b ea n s w e r e d i n t h e a f f i r m a t i v e .

E i n s t e i n ' s b e l i e f t h a t t h e f i n a l f o r m o f h i s g r a v i t a t i o n a l t h e o r y , a s it e m e r g e di n l a t e 1 9 1 5 a n d e a r l y 1 9 1 6 , e m b o d i e d a g e n e r a l p r i n c i p l e o f r e l a t i v i t y i s s t a t e de x p l i c i t l y :

The general laws of nature are to be expressed by equations which hold good fo rall systems o f coordinates, that is, are covariant with respect to any substitutionswhatever (generally covarianO.I t is clear that a physica l theory which satisfies this po stula te will also be sui tablefor a general pos tula te of re la t ivi ty . For the sum of all subst i tut ions in any case

SA. Einstein, 'tib er den Einfluss der Schw erkraft au f die Ausb reitung des Lichtes', Annalen derPhysik 35 (1911 ), 898-9 08.6Ref. 5 and 'Lichtgeschw indigkeit und Statik des Grav itationsfeldes', Annalen der Physik 38(1912), 355-369 , and 'Zu r Theorie des statischen Gravitationsfelds', ibid. 433-458.


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254 S t u d i e s in H i s t o r y a n d P h i l o s o p h y o f S c i e n cei n c lu d e s t h o s e w h i c h c o r r e s p o n d t o a l l r e la t iv e m o t i o n s o f t h r e e d i m e n s i o n a l s y s te m so f c o o r d i n a t e s . . , t hi s r e qu i re m e n t o f ge n er al c o v a r i a n c e . . , t ak e s a w a y f ro m s pa c ea n d t i m e th e l a s t r e m n a n t o f p h y s i c a l o b j e c t i v i t y . . . 7

T h i s p a s s a g e d i s p l a y s a t o n c e E i n s t e i n ' s t e n d e n c y d u r i n g t h i s p e r i o d t o b l u re a c h o f t h r e e in t e r r e l a t e d d i s t i n c t i o n s : r e f e r e n c e f r a m e s v s c o o r d i n a t e s y s t e m s ;p o i n t t r a n s f o r m a t i o n s v s 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 n d r e l a t i v i t y o ri n v a r i a n c e p r i n c i p le s v s c o v a r i a n c e p r i n c i p l e s . T h e q u o t e d p a s s a g e s h o w s t h a tE i n s t e i n b e l i e v e d t h a t b e c a u s e t h e l a w s o f h i s ' g e n e r a l t h e o r y ' w e r e g e n e r a l l yc o v a r i a n t , t h e y a u t o m a t i c a l l y s a t i s f i e d a g e n e r a l p r i n c i p l e o f r e l a t i v i t y . T h er e a s o n g i v e n f o r th i s b e l i e f s h o w s t h e c o n f u s i o n o f p o i n t a n 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 a n d o f r e f e r e n c e a n d c o o r d i n a t e s y s te m s . S o r t i n g o u t a l l t h e s t r a n d so f th i s t a n g l e w o u l d b e a H e r c u l e a n t a s k , c a l l in g f o r o n e o r m o r e b o o k l e n g t hm o n o g r a p h s , a n d w e d o n o t p r o p o s e t o u n d e r t a k e s u c h a l a b o r h e r e . B u t a f ewp r e l i m i n a r y r e m a r k s a b o u t t h e c o r e o f t h e ta n g l e a r e n e e d e d t o s et u p o u r d i s c u s -s i o n o f E i n s t e i n ' s s t r u g g le s w i t h g r a v i t a t i o n a l t h e o r y .

I f w e u n d e r s t a n d a r e f e r en c e f r a m e t o b e d e f i n e d b y a c o n g r u e n c e o f t i m e l i k ec u r v e s ( e a c h o f w h i c h i s t o b e c o n s i d e r e d t h e w o r l d l i n e o f a r e fe r e n c e p o i n t o ft h e f r a m e ) a n d u n d e r s t a n d a c o o r d i n a t e s y s t e m {x~}, i = 1, 2 , 3, 4, t o b e a d a p t e dt o t h e f r a m e j u s t i n c a s e e a c h c u r v e o f t h e c o n g r u e n c e s a t is f ie s x ~ = c o n s t a n t( a = 1 , 2 , 3 ), t h e n f r a m e s o f r e f e r e n c e a r e r e l a t e d b y p o i n t r a t h e r t h a n b y 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 n d t h e f a c t t h a t l a w s a r e c o v a r i a n t , h o l d i n g i n t h ec o o r d i n a t e s a d a p t e d t o o n e f r a m e o f r e f e r e n c e i f t h e y h o l d i n t h e c o o r d i n a t e sa d a p t e d t o a n o t h e r , s i g n if i e s b y i t s e lf n o t h i n g a b o u t t h e s t a tu s o f t h e r e s p e c t i v ef r a m e s ? T h i s is b u t o n e v e r s i o n o f E i n s t e i n ' s p r a c t i c e i n t h is p e r i o d o f s ee i n gi n th e a b o l i t i o n o f c o o r d i n a t e d e p e n d e n c e f o r p h y s i c a l l a w s a ls o t h e a b o l i t i o n o fs p a c e - t i m e s t r u c t u r e .

I n th e s e c t i o n s b e l o w w e w i ll f o l l o w t h e t h r e a d w h i c h i s m o s t c e n t r a l t o E i n -s t e i n ' s p r o g r e s s , o r l a c k o f it , t o w a r d s a t h e o r y o f g r a v i t a t i o n b e t w e e n 1 91 2a n d 1 91 6; w e w i l l c o n c e n t r a t e f i r s t o n t h e c o n s i d e r a t i o n s w h i c h l e d E i n s t e i n toa b a n d o n t h e r e q u i r e m e n t o f g e n e r a l c o v a r i a n c e i n 1 91 3, a n d w e w i ll t h e n t r a c et h e r e a s o n s w h i c h l e d h i m t o r e a f f i r m i t a t th e e n d o f 1 91 5.

2 . T h e E i n s t e in - G r o s s m a n n T h e o r yT h a t E i n s t e i n s a w in c o v a r i a n c e a s o l u t i o n t o t h e te n s i o n b e t w e e n h i s t w o

~A. Einstein, 'Die Grundlage der allgemeinen Relativit~ttstheorie', A n n a l e n d e r P h y s i k 49 (1916),770-822; t ransla t ion from W. Perret t and G. B. Jeffrey, T h e P r i n c i p l e o f R e l a t i v it y (New York:Dov er , 1923), p . 117. The i ta l ics are E inste in ' s .8Of course , a reference frame can be represented by a maximal c lass of adapted coordinatesystems. A ny two members of such a c lass are re la ted by a coord inate t ransform ation o f the formx '* = x " (x~) , x ' " = x " (x ~ , x ' ) . Thus, coord ina te trans fo rma t ions hav ing th is fo rm can bethough t o f a s g iv ing a recoo rd ina t iza t ion o f the f rame whi le ones no t o f th i s fo rm can be though tof as giving a change o f frame. But such a co ordin ate representation can easily lead to a blurring o fthe crucial d is t inct ions me ntioned above . Fo r more on these m atters , see Sections 3-6 below; seealso J . Earman, 'Covariance, Invariance, and the Equivalence of F ram es ' , F o u n d a t i o n s o f P h y si c s.(1974), 267-289.


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Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916 255c h i e f p r i n c i p le s is e v i d e n c e d i n h i s c o n t r i b u t i o n t o t h e p a p e r h e p u b l i s h e d j o i n t l yw i t h G r o s s m a n n i n 19 13 , ' E n t w u r f e i n e r v e r a l l g e m e i n e r t e n R e l a t iv i t ~ i t s th e o r i e ',t h e f i r s t o f h i s p a p e r s i n w h i c h t h e a p p a r a t u s o f th e t e n s o r c a l c u l u s i s u s e d ?A f t e r e x p l a i n i n g t h a t t h e p r i n c i p l e o f e q u i v a l e n c e r e q u i r e s t h a t t h e v e l o c i t y o fl i g h t n o t b e c o n s t a n t , E i n s t e i n w r o t e :

W i t h t h e i n t r o d u c t i o n o f a s p at ia l v a r i a t i o n i n t h e q u a n t i t y c w e h a v e g o n e b e y o n dw h a t is p r e s e n t l y u n d e r s t o o d b y ' r e l a t i v i t y t h e o r y ' ; f o r t h e e x p r e s s io n o f d s i s n ol o n g e r i n v a r i a n t u n d e r l i n ea r o r t h o g o n 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 . S u p p o s en o w - - w h a t c a n n o t b e in d o u b t - - t h a t t h e p r i n c i p l e o f r e l a ti v i t y is t o b e p r e s e r v e d ;t h e n w e m u s t g e n e r a l i z e r e l a ti v i t y t h e o r y s o th a t t h e s i g n i f i c a n t e l e m e n t s o f t h ef o r e g o i n g t h e o r y o f s t a ti c g r a v i t a t i o n a l f i el d s a r e i n c l u d e d a s a sp e c i al c a se . '

E i n s te i n t h e n i n t r o d u c e d a n a r b i t r a r y c o o r d i n a t e t r a n s f o r m a t i o n a n d r e q u i r e dt h a t t h e m o t i o n o f m a t e r i a l p o i n t s i n a r b i t r a r y c o o r d i n a t e s s a t i s fy

6 f d s = 0 ( 1 )w h e r e d s 2 = g l j d .v d x J. 11 H e p o i n t e d o u t t h a t t h e g o a r e c o m p o n e n t s o f at e n s o r - - t h e s p a c e - t i m e m e t r i c t e n s o r - - a n d r e m a r k e d :

I n th e c a s e o f o r d i n a r y r e l a t iv i t y t h e o r y o n l y l i n e a r o r t h o g o n a l s u b s t i t u t i o n s a r ep e r m i s s i b le . I t w i ll b e s h o w n t h a t i t is p o s s i b le t o o b t a i n e q u a t i o n s f o r t h e e f f e c t s o fg r a v i ta t i o n a l f ie ld s o n m a t e r ia l p h e n o m e n a w h i c h a r e c o v a r i a n t u n d e r a r b i tr a r ys u b s t i t u t i o n s . ~2

E i n s t e i n ' s e q u a t i o n o f m o t i o n f o r m a t e r i a l p a r ti c le s i s s i m p l y t h e r e q u i r e m e n tt h a t t h e i r t r a j e c t o r i e s b e g e o d e s i c s , a n d s o i s g e n e r a l l y c o v a r i a n t . E i n s t e i n a n dG r o s s m a n n w e r e a l s o a b le to g i v e , a s E i n s t e i n p r o m i s e d i n t h e p a s s a g e q u o t e d ,a g e n e r a l l y c o v a r i a n t e q u a t i o n f o r t h e m o t i o n o f i n c o h e r e n t m a t t e r i n a g r a v i t -a t i o n a l f i e ld . B u t a n y f i e l d t h e o r e t i c t r e a t m e n t o f g r a v i t a t i o n r e q u i r e s in a d d i t i o na f i e ld e q u a t i o n w h i c h w i ll d e t e r m i n e t h e r e l a t io n b e t w e e n f i e l d s o u r c e s a n d t h ef i e ld t h e y g e n e r a t e .

9A . E ins te in and M . G ros s m ann , 'En t w u r f e ine r ve ra llgemeiner ten Re la t iv i t a t stheo r ie und e ine rT h e o r i e d e r G r a v i t a t i o n ' , Zeitschriftfiir Mathernatik und Physik 62 (1913) , 225- 261. Eins tein ' sf r i ends h ip w i th G ros s man ex tended back to t l - ? i r s choo l days in Z i i r i ch . I t w as G ros s mann ' sf a the r - in - l aw w ho he lped E ins te in to ge t a job a t the P a ten t O f f ice in Berne , and i t w as G ros s m annw ho he lped to conduc t the nego t ia t ions w h ich b rough t E ins te in back to ZU r ich as a p ro fes s o r o ftheo re t i ca l phys ics a t the ETH . F ur the r , G ros s mann w as an exper t in non-Euc l idean geomet ryand the t ens o r ca lcu lus - - ju s t the mathemat ica l too l s E ins te in s aw he needed fo r h i s w ork ongrav i t a t ion . Thu s , i t w as na tu ra l , i f no t inev i t ab le , tha t E ins te in w ou ld en te r in to a s c ien t i fi cco l l abo ra t ion w i th G ros s mann .~ 'Entwurf ' , p . 228.~ 'F o r s ake o f r eadab i l i ty , w e have modern ized E ins te in ' s no ta t ion . La t in ind ices run f ro m 1 to 4and G reek ind ices f rom 1 to 3 . The E ins te in s umm at ion conven t ion on r epea ted ind ices is in e f f ec t .O rd ina ry de r iva t ives a r e deno ted by a com m a and co var ian t de r iva t ives by a s emi -co lon . In 1913Eins te in d id no t u s e the now s tandard conve n t ion o f upper an d low er ind ices to deno te r es pec t ive lycon t r avar ian t and covar ian t componen ts . Thus , he had to u s e tw o d i f f e r en t s ymbo ls fo r con t r a -va r ian t and c ovar ian t t ens o r s , a p rac t i ce w h ich makes i t ha rd fo r the m odern eye to s can h i s ea r lypaper s on g rav i t a t ion .~2 'Entwurf ' , p . 229.


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256 S t u d i e s in H i s t o r y a n d P h i l o s o p h y o f S cie n ceE i n st e in a n d G r o s s m a n n s e a r ch e d f o r f ie l d e q u a t i o n s o f th e f o r m

A ~ i = K T , . i ( 2 )T h e T ~ a r e th e c o v a r i a n t c o m p o n e n t s o f th e e n e r g y t en s o r f o r ' m a t t e r f i el d s '.I n i t i a l ly , t h r e e r e q u i r e m e n t s w e r e i m p o s e d o n t h e A ~i: ( i) t h e y a r e t o b e c o m -p o n e n t s o f a s e c o n d r a n k t e n s o r s o t h a t ( 2 ) is g e n e ra l l y c o n v a r i a n t , ( ii) t h e y a r e t ob e c o n s t r u c t e d f r o m t h e m e t r i c p o t e n t i a l s g,i a n d t h e i r d e r iv a t i v e s o f f ir s t a n ds e c o n d o r d e r , a n d ( i ii ) t h e y a r e t o b e s u c h t h a t i n t h e N e w t o n i a n l i m i t, ( 2 )r e d u c e s t o a n a n a l o g u e o f t h e P o i s s o n e q u a t i o n

V 2 ~ = c o n s t a n t x o (3)w h e r e q~ is t h e N e w t o n i a n s c a l a r p o t e n t i a l a n d 0 is t h e m a s s d e n s i t y .

A s in d i c a te d b y G r o s s m a n n , a n a t u r a l c a n d i d a t e f o r Av i s t h e R i c c i t e n s o rR i~ . R ~ a u t o m a t i c a l l y s a t i s fi e s ( i) a n d ( ii ). A n d i t s e e m s p r o m i s i n g a s r e g a r d s(iii) . Sin ce th e g,:~ a k e t h e p l a c e o f t h e N e w t o n i a n s c a la r p o t e n t ia l , w h a t is w a n t e dis a n e x p r e s s i o n c o n c o c t e d f r o m t h e g~ w h i c h , i n th e N e w t o n i a n l im i t , r e d u c e st o a s u m o f s e c o n d o r d e r d e r v i a t i v e s o f t h e g,:~. T h i s s u g g e s t s t h a t o n e l o o k f o ra t e n s o r e x p r e s s i o n w h i c h i n v o l v e s a s u m m a t i o n o v e r a r e p e a t e d i n d e x . F o rt e n s o rs , th i s o p e r a t i o n a m o u n t s t o c o n t r a c t i o n , a n d R,~ is t h e o n l y m e a n i n g f u lc o n t r a c t i o n o f t h e R i e m a n n t e n s o r R ~ , . E i n s t e i n a n d G r o s s m a n n h a d a l r e a d yr e c o g n i z e d th a t t h e R i e m a n n t e n s o r p l a y s a f u n d a m e n t a l r o l e in g r a v i ta t i o n a lt h e o r y s in c e i t v a n i sh e s i f a n d o n l y i f t h e m e t r i c is p s e u d o - E u c l i d e a n . G r o s s m a n n ,h o w e v e r , r e j e c t e d t h i s c h o i c e f o r A i r. T h e f a c t t h a t R ~ -R ~i~ i n i t s e l f s h o w e d ,a c c o r d i n g to G r o s s m a n n ' s r e a s o n i n g , t h a t R ~ c a n n o t h a v e t h e ri g h t N e w t o n i a nl im i t . G r o s s m a n ' s i d e a w as t h a t a s t h e g r a v a t i o n a l f i e ld v a n i sh e s , t h e n s o m u s tOR,jk, , s i n ce th i s is a n e c e s s a r y c o n d i t i o n f o r f l a t s p a c e - t i m e a n d s i n c e g r a v i t a -t i o n a l f i e l d i s n u l l o n l y i f s p a c e - t i m e i s f l a t . B u t s i n c e R ,j i s a c o n t r a c t i o n o fR, jk , , i f t h e l a t t e r v a n i s h e s s o m u s t t h e f o r m e r . '3

G r o s s m a f i ' s a r g u m e n t d o e s n o t s h o w w h a t i t w a s in t e n d e d t o s h o w . I t d o e se s t a b l i s h t h a t w h e n t h e g r a v i t a t i o n a l f i e l d v a n i c h e s c o m p l e t e l y , R , : i a l so van i shes ,b u t t h e a r g u m e n t d o e s n o t te ll o n e w h a t h a p p e n s i n s o m e a p p r o p r i a t e l y w e a kb u t n o n - v a n i s h i n g a p p r o x i m a t i o n . G r o s s m a n n c o u l d h a v e a r g u e d v a l i d ly i n t h eo p p o s i t e d i r e c t i o n ; n a m e l y , w h e n T,:~ v a n i s h e s , w e s h o u l d r e t u rn t o t h e s i t u a t i o no f s p e c ia l re l a t iv i t y w i t h i ts f la t s p a c e - t i m e . B u t w i t h t h e c h o i c e o f R ,j f o rAv i n (2 ) , (2 ) d o e s n o t h a v e t h i s c o n s e q u e n c e s in c e in d i m e n s i o n 4 , R ~ = 0 d o e sn o t e n t a i l R i ~ I = 0 . ( T h e e n t a i l m e n t d o e s h o l d f o r lo w e r d i m e n s i o n s . ) W h i l et h i s a rg u m e n t d o e s h a v e t h e v i r t u e o f v a l i d i t y , i t s l e a d i n g i d e a h a s t h e p h y s i c a l l yd i s a st r o u s c o n s e q u e n c e t h a t s p a c e - t i m e c a n n o t b e c u r v e d o u t s i d e m a s s- e n e r g yc o n c e n t r a t i o n s ; t h u s , m a s s iv e b o d i e s w o u l d n o t ' b e n d ' l i g h t, th e su n w o u l d n o t" a t t r a c t ' t h e e a r t h , e tc .

' 3 ' E n t w u r f ' , p . 2 5 7 .


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Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916 257T h e a c t u a l s it u a t i o n a s r e g a r d s t h e N e w t o n i a n l im i t o f

R,j = KT,~ (4)i s m a t h e m a t i c a l l y s t r a i g h t f o r w a r d b u t i s s o m e w h a t c o m p l i c a t e d t o p r e s e n tb e c a u se o f t h e n u m b e r o f a s s um p t i o n s n e e d e d t o g u a r a n t e e t h e c o r r ec t N e w t o n -i a n l i m it . W e d e m a n d f i rs t th a t ( A , ) t h e o n l y s o u r c es o f t h e g r a v i t a t i o n a l f ie lda r e m a s s i v e p a r t i c l e s ( t h i s i s a l r e a d y i m p l i c i t i n t h e N e w t o n i a n e q u a t i o n ( 3 ) ) .F u r t h e r m o r e , w e m a y a s s u m e t h a t (A 2) t h e p a r ti c le s a r e m o v i n g s l o w l y i n c o m -p a r i s o n w i t h t h e sp e e d o f li g h t. A p p l y i n g t h e s e a s s u m p t i o n s t o t h e c o n t r a -v a r i a n t f o r m o f ( 4 ), i t i s e a s il y se e n t h a t t h e o n l y c o m p o n e n t T ~' w h i c h c a n n o tb e n e g l e c t e d t o f i r s t a p p r o x i m a t i o n i s 7" ,, = Q , s o t h e p r o b l e m r e d u c e s t o e x a m -i n in g t h e e q u a t i o n

R"" = K0 (5)I n c o m p u t i n g t h e N e w t o n i a n l im i t o f R " ", w e a r e j u s t i fi e d i n a s s u m i n g t h a t (A 3)t h e s p a c e - t i m e m e t r i c d i f f e r s l it tl e f r o m t h e M i n k o w s k i m e t r i c , i.e., t h a t t h e r eex i s t s a co or d in a t e sys tem in w hich (5) ho lds a n d g~ = A~j + E /3 . w he re A~j i s theM i n k o w s k i m a t r i x a n d t i s a s u f f ic i e n t l y s m a l l p o s i ti v e n u m b e r . I f w e f i n a l lya s s u m e t h a t ( A , ) t h e m e t r i c i s s t a t i o n a r y i n t h e s e n s e t h a t t h e r e i s a c o o r d i n a t es y s t e m i n w h i c h (A 3) h o l d s a n d i n w h i c h O g J O x . = 0 , w e f i n d a f t e r n e g l e c t i n ga l l b u t f i rs t o r d e r i t e m s i n ~ t h a t3

R ,, = 1/2 Y /3"" .,, (6)/ J = lT h u s , u n d e r a s s u m p t i o n s ( A , ) - ( A , ) , t h e c o n t r a v a r i a n t f o r m o f ( 4 ) r e d u c e s t o3 445 -/3 , ,, = c o n s t a n t x 0 (7 )/a= 1w h i c h is a c l o se a n a n a l o g u e o f ( 2) a s c o u l d b e h o p e d f o r .

A l t h o u g h G r o s s m a n n ' s o b j e c t i o n t o ( 4) w a s n o t s o u n d , t h e r e a r e re a l p r o b l e m si n v o l v e d . I n a l e t t e r t o B e s s o d a t e d D e c e m b e r 1 0, 1 9 1 5 , E i n s t e i n s a id t h a t h ea n d G r o s s m a n n h a d r e j e c te d (4 ) n o t o n l y b e c au s e o f th e c o n s i d e r a t i o n sm e n t i o n e d a b o v e b u t a l so b e c a u s e t h e y b e l ie v e d t h a t t h e ' c o n s e r v a t i o n l aww o u l d n o t b e s a t i s f ie d ' . '" T o t h e m o d e r n e y e , th e p r o b l e m s e em s o b v i o u s ;n a m e l y , t h e c o n s e r v a ti o n e q u a t i o n

T":j : 0 (8 )d o e s n o t f o l l o w f r o m ( 4) . B u t t hi s is c e r t a i n ly n o t t h e p r o b l e m w h i c h w o r r i e dE i n s t e i n i n 1 9 1 3 , f o r ( 8 ) i s n o t 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 i n a n y o fE i n s t e i n ' s e a r l y t h e o r i e s .

A n o t h e r p r o b l e m i s t h a t ( 8 ) i n c o n j u n c t i o n w i t h ( 4 ) e n t a i l s t h a t t h e e n e r g ys c a l a r T ~ 5 -T ~ is a c o n s t a n t , a s e e m i n g l y i m p l a u s i b l e r e s t r i c t i o n o n m a t t e rf i e ld s . I t is d o u b t f u l , h o w e v e r , t h a t E i n s t e i n w a s a w a r e o f t h is c o n s e q u e n c e i n1 9 1 3 , f o r E i n s t e i n a p p a r e n t l y d i d n o t k n o w t h e k e y i d e n t i t y"P. Speziali (ed.), Albert Einstein/Michele Besso Correspondance(Paris: Her man n, 1972), p. 60.


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258 Studies in History and Philosophy of Science( R i i - V 2 g '~ R ) ; i = 0

(where R -= ZRI ) needed to prove the co nse que nce that T = co nst ant . InNovemb er of 1915, Einstein returned to the equa tion (4), and he did not ment ionthis consequence, although he did go through an eleborate argument to achievethe stronger result T = 0.15 And when he first proposed the final field equa tion s

R , , = K ( T ~ - - V 2 g ~ iT )which are equival ent to

R , i - V z g ( i R = KT, i, (11)the con ser vat ion law (8) was still post ula ted separatel y. '"

Assuming that the recollection Einstein recorded in his le t ter to Besso isaccurate, what probably worried him in 1913,was that, by his lights (8) wasnot a proper co nserva tion law. For Einste in believed that an adequ ate conser-va t ion law would have to inc lude the contr ibu t ion of the ener gy- mom ent umof the gravita t iona l fie ld i tself and that the conse rvat ion law should be express-ed in terms of an ordinary rather than a covariant divergence so that i t couldbe integrated to yield conserved qu anti t ies , lr Evidently , Einste in did n ot see away to achieve a c ons erv ati on law of the desired for m using (4). 1~

The struggle with convariance is evident thro ugh out Einstein and Gro ssm an n'sjoint paper. Based, apparent ly , on Gro ssma n's a rgume nt aga ins t the Riccitensor, Einstein says in his 'Physica l Par t ' of the paper that i t has proved im-possible to f ind a satisfactory ten sor buil t from the metric tensor an d i ts deriv-atives up to secon d order. Eins tei n does allow that it is possible that there m aybe satisfactory third order tensors, but thinks there are no good physical reasonsto expect such an object to be a satisfactory generaliz ation of the Ne wton ianpotent ia l . In his par t of the paper Gr oss man n concludes that

We rot.st therefore leave open the question as to what extent the general theory ofdifferential tensors linked with the gravitational field is related to the problem of

'SA. Einstein, 'Zur RelativitStstheorie (Nachtrag)', S i t zungsbe r i c h t e , de r K Oni g l ic he nP r e u s s i sc h e n A k a d e m i e d e r W i s s e n s c h a f te n z u B e r li n , 1915, 799 - 801.'6A. Einstein, Die Feldgleichungender Gravitation', Sitzungsberichte, de r KO niglichen PreussischenA k a d e m i e d e r W i s s e n s c h a f t en z u B e r li n , 1915, 844-847. For more on this point, see our paper,'Einstein and Hilbert: Two Months in the History of General Relativity', forthcoming inA r c h i v ef o r t h e H i s t o r y o f E x a c t S c ie n ce s ."See Section 3 below for more on Einstein's attitude towards conservation laws.'aE. Zahar has recently claimed that Einstein and Grossmann bad another reason for abandoningthe Ricci tensor. He writes: 'both Einstein and Grossmann thought that, given the appropriateboundary conditions, the ten equations R,~ = 0 would uniquely determine the ten functions g(~.This means that we are not at liberty to choose an arbitrary frame of reference because the functionsg(~ are generally altered by a change in coordinates. Thus it seems that the Relativity Principle isviolated', 'Why Did Einstein's Program Succeed Lorentz's? ' in C. Howson (ed.),M e t h o d a n dAppra i sa l i n t he Phy s i c a l Sc i e nc e s (Cambridge: Cambridge University Press, 1976). This accountappears to be unfounded. The argument ascribed to Einstein and Grossmann is not given in the1913 'Entwurf' paper, nor does it appear in any of Einstein's papers between 1913 and 1916.Zahar puts in evidence a letter from Einstein to Sommerfeld dated November 28, 1915; but thetext of this letter contains nothing of the argument in question, nor anything which could plausiblybe read as implying it (see E i n s t e i n / S o m m e r f e M B r i ef w e ch s el , pp. 32 - 36).


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L o s t i n t h e T e n s o r s : E i n s t e i n ' s S t r u g g l e s w i t h C o v a r i a n c e P r i n c i p l e s 1 9 1 2- 1 91 6 259the equa t i ons o f g rav i t a ti on . Such a con nec t i on m us t ex i st insofa r a s t he equa t i onsa l low a rb i t a ry subs t i tu t i ons [Le . , a re general ly covar iant ] ; b ut in thi s case i t seemstha t d i f f e r en t ia l equa t i ons o f t he second orde r cann ot be ob t a ined . I f , on t he o the rhand , i t cou ld be es t ab l ished t ha t t he equa t i ons o f g rav i t a t ion on ly adm i t a ce r t aingroup of t r ans form at ions , t hen on e cou ld under s t and why the d i f f e ren t i a l t ensor syielded by the general theo ry would n ot b e acceptable. As m entioned in the physicalpar t [ the par t wr i t ten by Eins te in] , we are as yet not in a pos i t ion to di scuss thi sques t i on . ' ~

E i n s t e in ' s o w n a p p r o a c h t o t he p r o b l e m o f g r a v i t a t i o n h a d m a d e h i m r e c ep -t iv e t o d o u b t s a b o u t g e n e r a l c o v a r i a n c e . A s w e h a v e s e en , h is s t r a t e g y w a s t os e e k a r e la t iv i s ti c f ie l d e q u a t i o n (2 ) w h i c h g e n e r a l iz e s t h e N e w t o n i a n e q u a t i o n( 3 ). W i t h i n t h i s s t r a t e g y , a n a t u r a l t a c t i c is t o t r y t o d e r i v e Ao b y f i n d i n g a f o u r -d i m e n s i o n a l c o v a r i a n t a n a l o g u e V~ o f t h e c l a ss ic a l V2 o p e r a t o r a n d t h e n t oa p p l y 7~ t o t h e m e t r i c p o t e n t i a l s g , ~ , w h i c h E i n s t e i n h a d a l r e a d y r e c o g n i z e d a st h e r e la t iv i s ti c g r a v i t a t i o n a l p o t e n t i a l s . A n d t h e m o s t o b v i o u s t r y f o r V ~ i s V~ ( ) =[ g , ,, ( ) ;m ];n . H o w e v e r , i f o n e a s s u m e s , a s E i n s t e in d i d i m p l i c it l y i n th i s p e r i o d ,t h a t t h e a f fi n e c o n n e c t i o n o f s p a c e - t i m e i s c o m p a t i b l e w i t h th e m e t r ic , t h e nV ~(g 'J) = 0 . A f t e r c o n s i d e r i n g s u c h r e s u l t s , E i n s t e i n w a s r e a d y t o e n t e r t a i nn o n - c o v a r i a n t g e n e r a l i z a t io n s o f W . A n d i n f a c t , t h e f ie l d e q u a t i o n s w h i c h h eg a v e in th e ' E n t w u r f ' p a p e r a r e n o t g e n e ra l ly c o v a r i a n t . F o r l a te r r e fe r e n c e ,t h e y a r e

A i j = K T , j (12)A i j = l l x / - - g ( g k , , x / - - g g ' ~ ,, ,) ,k ~ g k " g , r g '" ,k g ~ r,m+ 1 /2 g k~ g 'J g ,r , k g n ~ , m - I A g i J g k ~ g n ., ,g " ~ , m

w he re g = d e t (g~ ).20I n th i s c l im a t e , G r o s s m a n n ' s r e j ec t io n o f ( 4) a n d h i s e x p re s s io n o f d o u b t s

a b o u t t h e p o s s i b i l it y o f f i n d i n g a s u i t a b le t e n s o r e x p r e s s i o n f o r A,~ p r o v i d e d a llt h a t w a s n e e d e d t o m a k e t h e s e ed o f E i n s te i n ' s o w n d o u b t s b l o o m . T h eu n f o r t u n a t e h a r v e s t w a s g i v en i n a n a p p e n d i x a t t a c h e d t o t h e m a i n p a p e r . T h ea p p e n d i x i s s i g n e d w i t h t h e i n i t i a l s ' A . E . ' , a n d n o o n e w h o i s f a m i l i a r w i t hE i n s t e in ' s s ty l e o f a r g u m e n t c a n d o u b t t h a t i t is p u r e E i n s t e in . T w o a r g u m e n t sa r e g i v en f o r a b a n d o n i n g t h e n o w s u s p e c t r e q u i r e m e n t o f g e n e ra l c o v a r i a n c e .B u t b e f o r e tu r n i n g t o a d is c u ss io n o f th e s e a r g u m e n t s , w e s h o u l d m a k e a f e wb r i e f r e m a r k s a b o u t E i n s t e in ' s p e r c e p ti o n o f G r o s s m a n n ' s c o n t r ib u t i o n t o t h e i rj o i n t e f f o r t o f 1 9 1 3.

I n a l e t t e r w h i c h h a s b e e n l o s t , S o m m e r f e l d a p p a r e n t l y a d v i s e d E i n s t e i n t ob e c a u ti o u s i n c re d i ti n g G r o s s m a n n w i th t o o l a rg e a r o le in t h e d e v e l o p m e n t o ft h e t h e o r y o f g r a v i t a t i o n . E i n s t e i n ' s r e p l y , d a t e d J u l y 15 , 1 9 15 , w a s t h a t

lgEntwurf', p. 257.~*For ease o f reference, we will sometimescall these equations the Einstein - Grossm ann equations.In all probab ility, how ever, they are Einstein's invention. The 'Entwu rf' paper consists of two ma insections, a 'Mathem atical Pa rt' by G rossma nn and a 'Physical Pa rt' by Einstein; the field equationsare given n the latter. See also the remarks at the end o f this section.


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260 Studies in Histo ry and Phil osophy o f ScienceGro ssm ann wi ll never c l a im to be cod i scovere r . H e on ly he lped m e or i en t m yse l f inthe ma themat i ca l l i t e r a tu re , bu t con t r i bu t ed no th ing mate r i a l l y t o t he r e su l t s? '

F r o m t h e p o i n t o f v ie w o f h i n d s ig h t , E i n s t e in ' s r e m a r k s s e e m a t o n c e o v e r lyk i n d a n d u n g e n e r o u s . W h i l e G r o s s m a n n d i d h e lp t o o r ie n t E i n st e in in th em a t h e m a t i c a l l i t e r a t u r e , h e a l s o h e l p e d t o d i s o r i e n t h i m i n i t s i n t e r p r e t a t i o n .O n t h e o t h e r h a n d , G r o s s m a n n d i d a l e rt E in s t ei n t o th e p o s s ib l e i m p o r t a n c e o ft h e R i cc i t e n s o r , a n i d e a t h a t c a m e t o E i n s t e i n ' s r e s c u e at th e e n d o f 1 91 5. B u tw r i t i n g i n J u l y , E i n s t e i n w a s a w a r e o f n e i t h e r h i s d i s o r i e n t a t i o n n o r h i s s a l v a t i o n .

3. Einstein's Argum ents Against General CovarianceN o t h i n g is e a s ie r f o r a f ir s t- r a te m i n d t h a n t o f o r m p l a u s ib l e a r g u m e n t s t h a t

w h a t i t c a n n o t d o c a n n o t b e d o n e . I n th e c o u r s e o f w r i ti n g t h e j o i n t p a p e r w i thG r o s s m a n n , E i n s t e i n c o n s t r u c t e d t w o a r g u m e n t s t h a t s e e m e d t o e s t a b l i s h t h a ts a t i s f a c to r y g e n e ra l ly c o v a r i a n t f i e ld e q u a t i o n s a r e i m p o s s i b l e . T h e m a i na r g u m e n t , w h o s e s t y le (t h o u g h n o t i ts i n v a l id i t y ) is a b s o l u t e l y c h a r a c t e r i s t ic o fE i n s t e i n , i s s h o r t , s i m p l e a n d l u c id . I t is a l s o p u r e h o k u m .

E i n s t e i n ' s m a i n a r g u m e n t a g a i n s t g e n e r a l c o v a r i a n c e i s b a s e d o n t h e i d e a t h a tt h e f i e l d e q u a t i o n s s h o u l d s a t i s f y t h e c a u s a l i t y r e q u i r e m e n t t h a t t h e d i s t r i b u t i o no f m a s s - e n e r g y u n iq u e l y d e t e r m i n e s t h e s p a c e - t i m e m e t r i c a n d , h e n c e , t h eg r a v i t a t i o n a l f ie ld . E i n s t e i n ' s a r g u m e n t is s u p p o s e d t o s h o w t h a t c a u s a l i t y int h is f o r m w il l b e v i o l a te d f o r g e n e r a l l y c o v a r i a n t e q u a t i o n s . T o w a r d s t h i s e n d ,w e ar e a sk e d t o c o n s i d e r a d o m a i n D o f s p a c e - t i m e w h i c h is d e v o i d o f m a s s -e n e r g y . F u r t h e r , w e a re a s k e d t o im a g i n e t w o c o o r d i n a t e s y s t e m s {x~} an d {~ }w h i c h a g r e e o u t s i d e D b u t w h i c h d i v e r g e i n si d e D . T h e c o m p o n e n t s o f t h ee n e r g y - m o m e n t u m t e n so r i n t h es e t w o c o o r d i n a t e s y s te m s w ill a g re e e v e r y w h e r es i n c e i n s i d e D w e h a v e T 'j - - T 'j - - 0 a n d o u t s i d e D w e a l s o h a v e T 'j = T 'jb e c a u s e x ~ = x k t h e r e . B u t i n s i d e D , g ,~ ~: g , j . A s s u m i n g g e n e r a l c o v a r i a n c e ,h o w e v e r , t h e gi~ m u s t s a t i s f y t h e f i e ld e q u a t i o n s i f t h e g,~ d o . H e n c e , C a u s a l i t yf a i l s .

A n y s t u d e n t o f m o d e r n d i f f e re n t i a l g e o m e t r y w il l i m m e d i a t e l y p u t h is f in g e ro n t h e f l a w i n E i n s t e i n ' s a r g u m e n t : b y c o n s t r u c t i o n , t h e g~j a n d t h e g~i a r es i m p l y d i f fe r e n t c o o r d i n a t e r e p r e s e n t a t i o n s o f t h e s a m e i n tr i n s ic o b j e c t g , t h em e t r i c te n s o r . T h e r e i s n o t t h e l e a st r e a s o n w e c a n n o t h a v e e q u a t i o n s t h a t r e m a i nt ru e u n d e r a c o o r d i n a t e t r a n s f o r m a t i o n t h a t c h a n g es t h e c o m p o n e n t s o f th em e t r i c te n s o r b u t n o t t h e c o m p o n e n t s o f th e s t re s s - e n e rg y te n s o r . T h u s , t h ea r g u m e n t c a n n o t p o s s i b l y s h o w t h a t g is u n de r d e te r m i n e d .2 2

A s a m a t t e r o f m a t h e m a t i c a l a n d p h y s i c al f a c t , h o w e v e r , t h e r e is a g o o d d e a lo f t r u th i n E i n s t e i n 's c o n c l u s i o n . B u t a p r o p e r u n d e r s t a n d i n g o f t h e b a si s o ft h e k e r n e l o f t r u t h r e v e a l s t h a t g e n e r a l c o v a r i a n c e is i n n o w a y i m p u g n e d . T h e

2' Einstein/Sommerfeld Briefwechsel, p. 30.22See B. Hof fma nn , 'Einstein and Tensors ', Tensor 26 (1972), 157-162.


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Los t in the Tensors : E ins te in ' s S t rugg les w i th Covar iance Pr inc ip les 1912-1916 261m a t h e m a t i c s o f t h e s i tu a t i o n i s s t r a i g h t f o r w a r d , b u t t h e t r e a t m e n t it r e c e iv e s inm o s t s t a n d a r d p h y s i c s t e x t b o o k s i s s t i l l s u f f i c i e n t l y n o n - p e r s p i c u o u s a s t o m a k ea b r i e f d i s c u s s i o n a p p r o p r i a t e h e r e . T h i s d i s c u s s i o n w i ll a l s o f a c i l i ta t e a nu n d e r s t a n d i n g o f a fu r t h e r e v o l u ti o n o f E i n s t e in ' s a r g u m e n t .

L e t u s s u p p o s e t h a t t h e f i e l d e q u a t i o n f o r g r a v i t a t i o n i s l o c a l i n t h e s e n s et h a t i f ( M , g , T ) ( w h e r e M i s a f o u r - d i m e n s i o n a l d i f f e r e n t i a b l e m a n i f o l d , g isL o r e n t z s i g n a t u re m e t ri c f o r M , a n d T is a n e n e r g y m o m e n t u m t e n s o r) is as o l u t i o n a n d D C M i s a n o p e n s u b s e t o f M , t h e n ( D , g l~ ,, T I o ) i s a l s o a s o l u t io n .( M o s t o f th e t h e o r ie s o f g r a v i t a t i o n w h i c h , a t o n e t i m e o r a n o t h e r , h a v e b e e nu n d e r s e r i o u s c o n s i d e r a t i o n d o h a v e t h i s p r o p e r t y . ) T h i s p r o p e r t y a l l o w s o n et o c o n c e n t r a t e o n t h e s o u r c e - f r e e f i e l d e q u a t i o n s w h e n w e a r e i n a n E i n s t e i nd o m a i n D w h e r e T I,, = 0 . T h e s o u r c e f r e e fi el d e q u a t i o n w i ll b e re p r e s e n t e ds y m b o l i c a l l y as

L~r(g,i) = 0 (13)T o e s t a b l i s h E i n s t e i n ' s c o n c l u s i o n , i t m u s t b e s h o w n t h a t i f ( 1 3) is g e n e r a l l yc o v a r i a n t , t h e n t h e r e a r e t w o s o l u t i o n s ( M , g , ) a n d M , g 2) w h e r e g , 4= g 2. T h i s i se a s y t o d o . L e t 0 b y a n y d i f f e o m o r p h i s m o f M o n t o i ts el f. T h e n b y g e n e ra lc o v a r i a n c e , i f ( M , g ) is a s o lu t i o n , s o is ( M 0 * g ) w h e r e 0 * g d e n o t e s t h e' d r a g g i n g a l o n g ' o f g b y 0 . B y c h o o s i n g a m a p w h i c h is n o t a n i s o m e t r y o f g ,i . e . , 0 * g = g , w e h a v e a c a s e i n p o i n t . ( N o t e : g e n e r a l c o v a r i a n c e i s s u f f i c i e n t fo rt h i s r e s u l t ; b u t a s w i ll b e d i s c u s s e d l a t e r , i t is n o t n e c e s s a r y . )I t m u s t b e e m p h a s i z e d , h o w e v e r , t h a t t his k i n d o f f r e e d o m t o p e r f o r m d i f f-e o m o r p h i s m s i s t ri v ia l in t w o s e n se s . F i rs t, a l t h o u g h g a n d 0 * g m a y b e d i f f e r e n tm e t r i c s , t h e y a r e o f t h e s a m e t y p e , e . g . , i f g is f la t o r p s e u d o - E u c l i d e a n , t h e ns o is 0 * g . S e c o n d , t h is f r e e d o m h a s n o t h i n g t o d o w i th g e n e r a l r e l a t iv i t y t h e o r yp e r s e , f o r a s i m i l a r s o r t o f f r e e d o m e x t e n d s t o a n y g e n e r a l l y c o v a r i a n t t h e o r ya s l o n g a s a ll o f t h e q u a n t i t i e s o f th e t h e o r y a r e s i m u l t a n e o u s l y d e t e r m i n e d v i af ie l d e q u a t i o n s . A s a s i m p l e i ll u s t ra t i o n , c o n s i d e r t h e s c a l a r w a v e e q u a t i o n

O 2 / O x a + O 2 / O y 2 + O 2 0 / O z 2 - O 2 / O t 2 = 0(14)

( c - l )e n c o u n t e r e d i n s p e c ia l re l a ti v i ty t h e o r y . T o c o n s t r u c t a g e n e r a l l y c o v a r i a n tv e r s i o n , w e p o s t u l a t e

Ri jk , (g , , , ) = 0 (15)a n d

gm ,~; , , ; - = 0 (16)E q u a t i o n ( 15 ) s a y s t h a t t h e m e t r i c is fl a t, s o t o g e t h e r w i th t h e s id e c o n d i t i o nt h a t t h e m a n i f o l d i s R ' , i t s p e ci fi es t h a t t h e s p a c e - t i m e is a c o p y o f M i n k o w s k is p a c e - t i m e . T h e n i n a n i n e r ti a l c o o r d i n a t e s y s t e m , (1 6 ) r e d u c e s t o t h e m o r e


13/29

262 S t u d i e s in H i s t o r y a n d P h i l o s o p h y o f S c ie n c ef a m i l i a r f o r m g i v e n in ( 14 ). O b v i o u s , i f ( R ' , g , O ) s o l v e s ( 1 5 ) - (1 6 ) , t h e n s od o e s ( R ' , 0 * g , 0 * 0 ) f o r a n y d i f f e o m o r p h i s m 0 o f R 4. N o t e t h a t (1 5) m a k e s t h es p a c e - t i m e m e t r ic a n ' a b s o l u t e o b j e c t ' i n th e s e n se t h a t f o r a n y t w o s o l u t io n s( R ' , g , ) a n d ( R ' , g2 ) o f ( 15 ), t h e r e is a d i f f e o m o r p h i s m 0 o f R ' s u c h t h a t0 * g , = g2. I f o n e t r e a t s th i s a b s o l u t e o b j e c t n o t a s b e i n g d e t e r m i n e d b y t h ef ie l d e q u a t i o n s b u t a s b e i n g g iv e n o n c e a n d f o r a ll b y G o d , t h e n t h e f r e e d o m t op e r f o r m d i f f e o m o r p h i s m s is c u r ta i le d - - t h e n 0 m u s t b e a n i s o m e t r y o f t h e f ix e dm e t r i c - - i n t h is c a se , a P o i n c a r 6 ( p o i n t ) t r a n s f o r m a t i o n . P a r a l le l e x a m p l e s c a na l s o b e c o n s t ru c t e d f o r N e w t o n i a n t h e o r ie s , s o a g a i n n o t h i n g h i n g e s o n f e a t u r e sp e c u l i a r t o r e l a t i v i s ti c p h y s i c s .

I n g e n e r al r e la t i v is ti c th e o r ie s t h e f r e e d o m t o p e r f o r m d i f f e o m o r p h i s m s a l s oo c c u r s i n t h e in i ti a l v a l u e p r o b l e m . F o r t h e s a k e o f c o n c r e t e n e s s , c o n s i d e r t h es o u r c e - f r e e i n it ia l v a l u e p r o b l e m f o r t h e f i e ld e q u a t i o n ( 1 0 -( 1 1) o f t h e u s u a lv e r s i o n o f g e n e r a l r e l a t i v i t y . A n i n i t i a l v a l u e s e t is a t r i p l e ( S, h , X ) w h e r e S is at h r e e m a n i f o l d , a n d h a n d X a r e t e n s o r f ie l ds o n S w h i c h a r e i n t e r p r e t e d r e s p e c -t iv e l y a s t h e f i rs t a n d s e c o n d f u n d a m e n t a l f o r m s o f S ( in t u i ti v e l y , S is a ni n s t a n t a n e o u s t i m e s li ce o f s p a c e - t i m e , h c h a r a c t e r i z e s t h e i n tr i n si c s p a t i a lg e o m e t r y o f S , a n d )~ d e t e r m i n e s h o w S is t o b e i m b e d d e d i n to s p a c e - t i m e ) . B ya s o l u t i o n t o t h e i n i t i a l v a l u e p r o b l e m i s m e a n t a n o b j e c t ( ( M , g ) , ~ ) w h e r e( M , g ) i s a s o l u t i o n t o t h e ( s o u r c e f r e e ) f i e l d e q u a t i o n s a n d u,J : S ~ M i s a ne m b e d d i n g s u c h t h a t ~ ( S ) i s a s p a c e l i k e h y p e r s u r f a c e o f ( M , g ) w h o s e f i rs t a n ds e c o n d f u n d a m e n t a l f o r m s a r e h a n d )~ r e s p e c t i v e l y . A g a i n , i f ( ( M , g ) , qJ) i s as o l u t i o n a n d 0 is a d i f f e o m o r p h i s m o f M , t h e n ( ( M , 0 * g ) , 0 tl ) is a l s o a s o l u t i o n . I fS i s g iv en a p r i o r i o r is s o m e h o w f ix e d o b s e r v a t i o n a l l y , t h e n 0 is c o n s t r a i n e d t op r e s e r v e S p o i n t w i s e , i . e . , t g O ~ - ' = i d ; b u t f o r p o i n t s in M t h a t a r e n o t i n S, 0m a y b e c h o s e n a r b i t r a r i l y . A s i m i l a r r e su l t h o l d s f o r th e i n it ia l v a l u e p r o b l e mw i t h s o u r c e s a n d f o r a n i n i ti a l v a l u e p r o b l e m i n w h i c h t h e i n it ia l d a t a a r e s p ec i -f ie d o n a f in i te ' s a n d w i c h ' o f s p a c e - t i m e r a t h e r t h a n a n i n s t a n t a n e o u s s li ce .T h u s , o n e c a n n o t h o p e t o h a v e c a u s a l i t y in t h e s e n s e t h a t t h e l a w s d e t e r m i n e au n i q u e e x t e n s i o n o f t h e i n it ia l v a l u e s e t; th e b e s t o n e c a n h o p e f o r is u n i q u e n e s su p t o a d i f f e o m o r p h i s m . 23

T h e s i t u a t i o n is n o t q u i t e th e s a m e f o r t h e o r i e s w i t h a n a b s o l u t e s p a c e - t i m es t r u c t u re s i n ce th i s s t ru c t u r e c a n b e u s e d t o ' l i n e - u p ' t h e c o r r e s p o n d i n g s o l u t io n s .I n th e e x a m p l e o f ( 1 5 )- ( 16 ) , w e k n o w t h a t f o r a n y t w o s o l u t io n s ( R ' , g , , 0 , )a n d ( R ' , g 2, O 2), t h e r e is a d i f f e o m o r p h i s m 0 o f R 4 s u c h t h a t 0 * g , = g 2. I f0 * 0 1 = O2 o n a f in i te s a n d w i c h o f s p a c e - t i m e , t h e n 0 * 0 , = O2 e v e r y w h e r e . I nt hi s s en s e, th e s c a f f o l d in g o f t h e a b s o l u t e s p a c e - t i m e b a c k g r o u n d p e r m i t s o n et o c o m e c lo s e r t o a u n i q u e p r o j e c t i o n o f t h e i n it ia l d a t a t h a n is p o s s i b l e in t h ec a s e o f g e n e r a l r e l a t i v i ty .

23This is the geo me tric origin of the 'fo ur u ndeterm ined func tions' in the initial valu e problemof general relativity. Contrary to the im pression sometimes given, these functions do not arisesimply because of general covariance; see the discussion below.


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L o s t i n t h e T e n s o r s : E i n s t e i n ' s S t r u g g l e s w i t h C o v a r i a n c e P r i n c i p l e s 1 9 1 2 - 1 9 1 6 263I t is c l e a r t h e n t h a t i t is n o t g e n e r a l c o v a r i a n c e i t s e lf w h i c h i s r e s p o n s i b l e f o r

t h e f r e e d o m t o p e r f o r m d i f f e o m o r p h i s m s i n t h e i n i t i a l v a l u e p r o b l e m o f g e n e r a lr e l at iv i t y ; r a t h e r , i t is t h e c o m b i n a t i o n o f g e n e r a l c o v a r i a n c e a n d t h e a b s e n c eo f a n y a b s o l u t e o b j e c ts . A n d f u r t h e r , a l t h o u g h t h i s f r e e d o m d o e s e n t a il a c h a n g ei n t h e f o r m u l a t i o n o f t h e c la s si ca l p r i n c ip l e o f L a p l a c i a n c a u s a l it y , i t d o e s n o te x t r a c t t h e t e e t h o f t h e p r i n c ip l e .

T h e d i s c u s s io n s o f a r h as a d m i t t e d l y n o t d o n e c o m p l e t e j u s t ic e t o E i n s t e i n ' si d e a t h a t t h e d i s t r i b u t i o n o f m a s s e n e r g y s h o u l d u n i q u e l y d e t e r m i n e t h e m e t r i c .B u t j u s t ic e i s h a r d t o d o h e r e s i nc e it is d i f fi c u l t to f o r m u l a t e E i n s t e i n ' s i d e a i na c o h e r e n t a n d p r e c is e f a s h i o n . I n s o m e c a se s - - f o r e x a m p l e , w h e n m a t t e re x e r t s a p r e s s u r e - - T 'j w i ll it s e l f e x p l i c i t ly c o n t a i n t h e m e t r i c p o t e n t i a l s . E v e nw h e n T j d o e s n o t e x p l i c i tl y d e p e n d u p o n t h e m e t r i c , i ts i n t e rp r e t a t i o n d o e s . F o re x a m p l e , t h e e n e r g y d e n s it y a s m e a s u r e d b y a n o b s e r v e r w i t h f o u r - v e l o c i t y V isT,~ V i V . B u t t h e a c c u r a c y o f t h is m e a s u r e p r e s u p p o s e s t h a t V is n o r m a l i z e d ,w h i c h i n t u r n a s s u m e s t h e m e t r i c . S t i l l , l e t u s a t t e m p t t o t e s t E i n s t e i n ' s i d e a i nt h e f o l lo w i n g w a y . L e t

L w s ( g l j , T i t) = 0 (17)r e p r e s e n t th e f i e l d e q u a t i o n w i t h s o u rc e s . F o r f i x e d T , s p e c i f ie d s o m e h o w o ro t h e r a s a t e n s o r f i e l d o n a m a n i f o l d M , c a n t h e r e b e m o r e t h a n o n e g w h i c hs o l v es (1 7 ) ? I f ( 1 7 ) is g e n e r a l l y c o v a r i a n t , w e k n o w t h a t w h e n ( M , g , T ) i s as o l u t i o n , s o is ( M , 0 * g , 0 * T ) f o r a n y d i f f e o m o r p h i s m 0 o f M . B u t b y f i x i n g T ,0 is r e q u i r e d t o b e a s y m m e t r y o f T , i . e ., 0 * T = T , a n d i n s p e ci a l c a se s t h e r em a y b e n o n o n - t r i v i a l s o l u t io n s o f t h is c o n d i t i o n . O f c o u r s e , i f (1 7 ) is lo c a l a n di f T v a n i s h e s o n s o m e o p e n s e t D , w e r e t u r n t o t h e s i t u a t i o n d i s c u ss e d a b o v e .T o o b t a i n a n y a d d i t io n a l i n f o r m a t i o n w e m u s t t h e r e f o r e c o n s id e r s p e ci fi ci n s t a n c e s . F o r t h e p a r t i c u l a r c a s e o f t h e f i e l d e q u a t i o n s (1 0 ) - (1 1 ) , t h e f r e e - s p a c es o lu t io n s i n c lu d e o n e s w h i c h a r e m o r e t h a n m e r e d i f f e o m o r p h i c v a r i an t s o fo n e a n o t h e r - - e . g . , t h e y i n c l u d e b o t h f l a t a n d c u r v e d m e t r ic s . A s i m i la r r e su l to b t a i n s w h e n a ' c o sm o l o g i c a l c o n s t a n t ' is a d d e d t o t h e f ie l d e q u a t i o n s o f g e n e ra lr e l a t i v i t y .

I n a d d i t i o n t o h is c a u s a l i ty a r g u m e n t , E i n s t e i n o f f e r e d a s e c o n d r e a s o n f o ra b a n d o n i n g g e n e ra l co v a r i a n c e. H e h a d s h o w n t h a t a s a c o n s e q u e n c e o f e q u a -t i o n (8 ) a n d t h e E i n s t e i n - G r o s s m a n f ie l d e q u a t i o n , a c o n s e r v a t i o n l a w f o r t h ec o m b i n e d m a t t e r a n d g r a v i t a t i o n a l f i e ld s h o l d s i n t h e f o r m

[ x / - - g ( T 4 + t ' / ) l , j = 0 (18)w h e r e t~ is th e o b j e c t E i n s t e i n i n t e r p r e t e d a s c h a r a c t e r i z i n g t h e e n e r g y c o n t e n to f t h e g r a v i t a t i o n a l f i e l d . I n h i s a p p e n d i x , E i n s t e i n s t a t e d t h a t ( 1 8 ) is c o v a r i a n to n l y f o r l i n e ar 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 o n c e a g a i n t h e r e a s o n i n g w a su n s o u n d . I f t , t r a n s f o r m s l i k e a (1 , 1 ) t e n s o r , t h e n i n g e n e ra l ( 1 8) w o u l d b ec o v a r i a n t o n l y u n d e r l i n e a r t r a n s f o r m a t i o n s . B u t i n g e n e r a l t~ t r a n s f o r m s l i k e a


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264 Studies in History and Philosophy of Sciencet e n s o r o n l y f o r li n e a r 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 . S i n ce (1 8) is a c o n s e q u e n c eo f g e n e r a ll y c o v a r i a n t e q u a t i o n ( 8 ) a n d t h e E i n s t e i n - G r o s s m a n n f ie ld e q u a t io n s ,i ts c o v a r i a n c e is j u s t a s w i d e a s t h e c o v a r i a n c e o f t h e f i e ld e q u a t i o n s . T h e p o i n ts h o u l d h a v e b e e n i n t u i t iv e l y e v i d e n t to E i n s t e in . T h e p r i n c i p l e o f e q u i v a l e n c ei m p l i e s t h a t i t is a l w a y s p o s s i b l e t o ' t r a n s f o r m a w a y ' t h e g r a v i t a t i o n a l f ie ld a ta n y p o i n t ; t h u s , w h a t e v e r t h e s t a t e o f th e f i e ld , o n e s h o u l d a l w a y s b e a b l e tof i n d a c o o r d i n a t e s y s t e m s u c h t h a t t = 0 a t t h e c h o s e n p o i n t . 24 T h u s , t{ c a n n o tb e h a v e l ik e a t e n s o r u n d e r n o n - l in e a r t r a n s f o r m a t i o n s .

O n c e E i n s t e i n r e c o g n i z e d h i s m i s t a k e , h e s e t t o w o r k t o f i n d t h e e x t e n t t ow h i c h th e E i n s t e i n - G r o s s m a n n f ie ld e q u a t io n s a r e c o v a r i a n t . T h i s m a t t e r w illb e d i s c u s s e d b e l o w i n S e c . 5 , b u t b e f o r e t u r n i n g t o i t w e w i l l t a k e u p t w o f u r t h e rd e v e l o p m e n t s i n E i n s t e in ' s a r g u m e n t a g a i n s t g e n e r a l c o v a r ia n c e .4 . F u r t h e r D e v e l o p m e n t s i n E i n s t e i n ' s A r g u m e n t s A g a i n s t G e n e r a l C o v a r i a n c e

E i n s te i n h a d s o m e r e a s o n s t o b e p l e a s e d w i th h is n e w t h e o r y o f g r a v i ta t i o n ; i td i d i n c o r p o r a t e a v e rs i o n o f th e p r i n c i p l e o f e q u i v a l e n c e , a n d a l t h o u g h it w a sn o t g e n e r a l l y c o v a r i a n t , h e b e l ie v e d it t o b e c o v a r i a n t u n d e r l i n e a r t r a n s f o r -m a t i o n s a n d t h u s t o s a t is f y a re s t r ic t e d p r i n c i p l e o f r e l a ti v i ty . F u r t h e r m o r e , t h en e w th e o r y a p p e a r e d t o m a k e t h e i n er ti a o f a m a s s p o i n t d e p e n d o n n e i g h b o r in gm a s s e s , a n d E i n s t e i n s a w t h i s r e s u l t a s a n a p p e a l i n g r e a l i z a t i o n o f M a c h ' s i d e a s .B u t t h e t h e o r y w a s n o t e s p e c i a l l y w e l l - r e c e i v e d b y E i n s t e i n ' s c o n t e m p o r a r i e s .T h e d i sc u s si o n o f E i n s te i n 's p a p e r a t th e V i e n n es e N a t u r f o r s c h e r v e r s a m m l u n gw a s d e s u l to r y a t b e s t, a n d d o m i n a t e d b y G u s t a v M i e, w h o c h i d e d E i n st e in f o rf a i li n g to d i s c u ss M i e ' s o w n t h e o r y , a n d t h e n p r o c e e d e d t o d e s c r i b e it a t l e n g t h . 2sE i n s t e i n w r o t e t o B e s s o s h o r t l y a f t e r w a r d s c o m p l a i n i n g t h a t

T h e p h y si ci st s h a v e a n eg a ti v e a t ti tu d e t o w a r d s m y w o r k o n g r a v i t a t i o n . . . A freeand imp ar t ia l v is ion is scarcely pro pe r for the a dul t Ge rm an. (Bl inkers! ) . 2~E v e n a f re e a n d i m p a r t i a l v i si o n , h o w e v e r , m i g h t h a v e h a d v e r y s e ri o u s

r e s e rv a t i o n s a b o u t t h e E i n s t e i n - G r o s s m a n n t h e o r y . I t e x p l a in e d n o k n o w na n o m a l i e s o f N e w t o n i a n g r a v i t a t i o n a l t h e o r y , 27 t h e p r i n c i p le o f e q u i v a l e n c e w a s

2~An explicit demo nst rat ion that this is indeed the case was given by E. Schrodinger , 'DieEnergiekomponenten des Gravitationsfeldes', Phys ikal ische Zei tschr i f t 19 (1918), 4 7.25See A. Einstein, 'Zu m gegenwfirtigen Stande des Grav itat ion spro ble ms', Phys ika l i s cheZeitsehr~ft 14 (1913), 1249-1266, and the discussion following Einstein 's lecture on pp. 1262-1266.The reason given in the body of Einstein's paper for limiting the requirement of covariance to lineartransformations was that the validity of the conservation law in the form (18) is not compatiblewith any wider covariance. But in a footnote (see p. 1257) which was probably added in proof,Einstein states that he has discovered 'in the last days' a demonstration showing that generallycovariant field equat ions canno t be used.~6Einstein/Besso Correspondance, p. 50.~'Besides the ano malo us advance of the perihelion of Mercury, o ther anomalie s had been

established by the end of the 19th century, the most notable one being the movement of the nodesof Venus. See H. Jeffreys, 'The Secular Perturbations of the Four Inner Planets', M o n t h l yNo t ices q / ' the Roya l As t r onomica l Soc ie ty , 77 (1916), pp. 112-118. See also P. Coh en, Relat iv i tyand the Ercess Ad van ces o f Perihelia in Planetary Orbits (University of Pennsylvania M. A. Thesis,1971).


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Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916 265embodied in a strong form without any argument, and, finally, though thesetting of the theory made general covariance almost irresistible, the fieldequations and the conservation laws were not generally covariant . Einstein didnot help matters as he elaborated his views on covariance. In 1914 Mie publisheda critical review article" in which he attacked the Einstein-Grossmann theoryfor, among other things, falling to generalize the principle of relativity to accel-erated reference frames. Einstein replied that he had been misunderstood, andrather than counterattacking, attempted to lay out more clearly the reasoningthat led him to put credence in his theory." He found it unbelievable that thevelocity of light should be entirely unaffected by any other physical phenomena,he wrote. Dropping the requirement that the velocity of light be constantpermits one to use arbitrary coordinates; the equation of motion of a masspoint is the usual Hamiltonian equation (1). Now however, the principle ofequivalence seems to require that the metric components g,j also be the com-ponents of the gravitational field. One can, furthermore, give a generallycovariant equation specifying the effects of the gravi tational field on physicalphenomena. 'It is important for the theory', Einstein wrote, 'that equation [(1)]is invariant under arbitrary transformations'? What more is required is ageneralization of Poisson's equation, that is, a field equation, and that, too,should be generally covariant.

Einstein's entire approach to the theory of gravity, then, demands a generallycovariant theory, for it is exactly the coupling of the admissibility of arbitrarycoordinates with the equivalence principle that leads Einstein to a tensor theory.Yet the field equations of the theory are not generally covariant, Einsteinadmits; still, that is not so disastrous as it seems. Here is Einstein's argument.Suppose a set of equations fail to be generally covariant ; then one of two casesmust obtain. Either there are generally covariant equations which in appropri-ate coordinates reduce to the original equations, or there are not. In the formercase, there is no harm in using the non-covariant equations. In the latter case,however, the non-covariant equations must lack any physical significancewhatsoever and can only be a complex way of specifying coordinate systems.But no one can doubt that the Einstein-Grossmann theory is physicallysignificant.31

The thrust of this argument seems plainly to be that it is of no consequencethat the Einstein-Grossmann field equations are not covariant so long as thereare covariant equations that reduce to them; and, furthermore, that the Einstein-Grossmann equations satisfy a sufficient condition for being special cases of

"G. Mie, 'Bemerkungen zu der Einsteinschen Gravi tations theori e', P h y s i k a l i s c h e Z e i t s c h r i f t14 (1914), 115-122, and 169-176.29A. Einstein, 'PrinzipieUes zur verallgemeinerten Relativit~itstheorie und Gravita tionstheorie' ,P h y s i k a l i s c h e Z e i t s c h r i ft IS (1914), 176-180.3Ibid. p. 177.3'Ibid, p. 177-178,


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266 Studies in History and Philosophy of Sciencec o v a r i a n t r e l a t i o n s . A f t e r w r i t i n g d o w n t h e f i e l d e q u a t i o n s E i n s t e i n c o n t i n u e d :

I t i s i ndubi t ab l e t ha t t hese equa t i ons conform wi th one , even pe rhaps a smal lnumber of general ly covariant equat ions, the statement of which is nei ther of logicalnor o f phys i ca l i n t e r e s t . . . 32E i n s t e i n ' s a r g u m e n t is b o t h p u z z l i n g a n d f a s c i n a t in g , a n d f o r s e v e r a l

r e a s o n s . I n t h e f i rs t p l a c e , t h e a r g u m e n t i s g i v e n in c o n j u n c t i o n w i t h h ise a r l i e r a r g u m e n t s t o t h e c o n c l u s i o n t h a t t h e f i e l d e q u a t i o n s f o r g r a v i t a t i o nc a n n o t e v e n in p r i n c ip l e b e c o v a r i a n t : I f T is t o d e t e r m i n e g u n i q u e l y , t h e n t h ee q u a t i o n s c a n n o t b e g e n e r a l l y c o v a r i a n t ; f u r t h e r , l aw s w h i ch a r e o f th e f o r mo f th e c o n s e r v a t i o n l a w c a n n o t b e g e n e r a l y c o v a r i a n t . I t s e e m s v e r y m u c h a st h o u g h E i n s te i n is s a y i n g b o t h t h a t h i s fi e ld e q u a t i o n s m u s t h a v e a n d d o h a v e ac o v a r i a n t g e n e r a l i z a t io n , a n d f u r t h e r m o r e t h a t th e y c a n n o t h a v e t h e m a n d d on o t h a v e th e m . T h e r e is , p e r h a p s , a c o n s is t e n t w a y to u n d e r s t a n d E i n s t e in ' sp o i n t o f v i e w i n t h i s r e g a r d . I t i s t h a t e v e r y p h y s i c a l l y s i g n i f i c a n t e q u a t i o n w h i c his n o t g e n e r a l l y c o v a r i a n t m u s t b e t h e s p e c i a l iz a t i o n ( t o s p e c ia l c o o r d i n a t e s ) o fs o m e g e n e r a l l y c o v a r i a n t e q u a t i o n , b u t t h a t in sp e c i al c o o r d i n a t e s a n o n -c o v a r i a n t e q u a t i o n m u s t e x p re s s , o r s o m e h o w r e fl e ct , a fe a t u r e o f th e w o r l d( c a u s a li ty o r c o n s e r v a t i o n , f o r e x a m p l e ) t h a t c a n n o t b e c a u g h t b y a n y g e n e r a l lyc o v a r i a n t r e l at io n .

I n t h e s e c o n d p l a c e , w h a t r e a s o n d i d E i n s te i n h a v e f o r b e l i ev i n g th a t e v e r yp h y s i c a l l y s i g n i f ic a n t e q u a t i o n is e q u i v a l e n t , in t h e s p e ci a l c o o r d i n a t e s u s e d , t oa g e n e r a l l y c o v a r i a n t o n e ? I f w e a r e d e a li n g w i th a d i f f e r e n t i a l e q u a t i o n o r s y s-t e m o f d i f f e r e n t i a l e q u a t i o n s u s i n g g e o m e t r i c a l o b j e c t s o f t h e u s u a l k i n d s , i n as in g l e c o o r d i n a t e s y s t e m , th e n E i n s te i n w a s q u i t e c o r r e c t. I g n o r i n g t e n s o rd e n s i t i e s , o n e c a n a l w a y s f i n d c o r r e s p o n d i n g c o v a r i a n t e q u a t i o n s , f o r e x a m p l e ,s i m p l y b y i n t r o d u c i n g a f la t a f f i n e c o n n e c t i o n w h i c h v a n i s h e s i n th e s p e c ia lc o o r d i n a t e s a n d r e p l a c i n g al l c o o r d i n a t e d e r i v a t i v e s b y c o v a r i a n t d e r i v a t i v e s .B u t i f, as is a c t u a l l y t h e ca s e w i t h t h e E i n s t e i n - G r o s s m a n n t h e o r y , o n e i s g i v e na s et o f e q u a t i o n s w h i c h a r e t o h o l d in n o t o n e b u t a n e n t i r e c o l le c t i o n o fc o o r d i n a t e s y s t e m s , t h e n i t is n o t c l e a r u n d e r w h a t c o n d i t i o n s t h e r e e x i st s ac o v a r i a n t e q u a t i o n w h i c h r e d u c e s t o t h e o r i g i n a l e q u a t i o n s i n a / / o f t h e c o o r d i n -a t e s y s t e m s in t h e c o l le c t io n . T h e t r ic k o f i n t r o d u c i n g a f l at c o n n e c t i o n w ill n o tw o r k i f , f o r e x a m p l e , s o m e o f th e c o o r d i n a t e s y s t e m in th e c o ll e c ti o n a r e n o n -l i n e a rl y re l a t ed . A n a n a l o g o u s t ri c k w ill w o r k p r o v i d e d t h e r e is a n o p e r a t o rw h i c h is a d e r i v a t i o n , a n d w h i c h i s i d e n t i c a l w i t h c o o r d i n a t e d e r i v a t i o n in a ll o ft h e s p e c if ie d c o o r d i n a t e s . W e d o n o t k n o w f o r w h a t g r o u p o f c o o r d i n a t et r a n s f o r m a t i o n s s u c h o p e r a t o r s e x is t.

I n t h e t h i rd p l a c e , t h e a r g u m e n t E i n s te i n g a v e i n r e p l y t o M i e m a r k s a ni m p o r t a n t a n d f u n d a m e n t a l c h a n g e i n h is a t t i t u d e t o w a r d g e n e r a l c o v a r i a n c e .

321bid. p. 179. A similar point is made at the end o f W alter Dallenbach's notes on E instein'slectures at the ETH, presumably delivered during the winter semester, 1913-1914; see EinsteinPapers, Princeton U niversity, microfilm reel I.A.8, no. 4.


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L o s t in th e T e n s o r s : E i n s t e i n ' s S t r u g g l e s w i t h C o v a r i a n c e P r i n c i p l e s 1 9 1 2 -1 9 16 2 6 7H e r e t o f o r e , E i n s t e i n s e e m s t o h a v e s e e n i n g e n e r a l c o v a r i a n c e t h e p e r f e c t a n dc o m p l e t e g e n e r a l i z a t io n o f o n e a s p e c t o f t h e p r i n c i p l e o f r e l a ti v i ty . N o w , i n t h ef a c e o f M i e ' s a t t a c k , E i n s t e in d e n i g r a t e s g e n e r a l c o v a r i a n c e , a n d s ee s t h ed e m a n d f o r i t a s p h y s ic a l ly v a c u o u s ; t h e g r a v i ta t i o n a l t h e o r y h e a n d G r o s s m a n np u t f o r w a r d w a s a n i n t e l li g i b le g e n e r a l i z a t i o n o f s p e c ia l r e l a t i v i t y a n d s a t i sf i e dt h e e q u i v a l e n c e p r i n c ip l e ; th a t i s w h a t m a t t e r e d . A l t h o u g h t h e a r g u m e n t s E i n -s t e i n a d v a n c e d fo r h i s n e w v i e w w e re u n s o u n d o r i n c o m p l e t e , t h e v i e w i t s e l f w a sc l o s er t o t h e t r u t h o f t h e m a t t e r t h a n t h o s e E i n s t e in h a d e a r li e r h el d . T h u sE i n s t e i n ' s a c c o u n t o f c o v a r i a n c e i n r e p l y t o M i e is es s e nt ia l ly t h e o n e g i v e n t w oy e a r s l a t e r b y K r e t s c h m a n n 33 i n c r i ti c i z in g E i n s t e i n ' s 1 91 6 p a p e r o n t h e f o u n -d a t i o n s o f g e n e r a l r e l a t iv i t y . 3 ' Y e t , p a r a d o x i c a l l y , i n f o r m i n g a m o r e a c c u r a t eo p i n i o n o f t h e p h y s i c a l s i g n if i c an c e o f t h e d e m a n d f o r g e n e r a l c o v a r i a n c e ,E i n s t e i n m a y w e l l h a v e m a d e i t m o r e d i f f i c u l t f o r h i m s e l f t o f i n d a s a t i s -f a c t o r y g r a v i t a t i o n a l t h e o r y . C e r t a i n l y it se e m s t h a t i f h e h a d h e l d f a s t tog e n e r a l c o v a r i a n c e a t th i s p o i n t h e m i g h t h a v e b e e n d r i v e n t o r e - e v a l u t eG r o s s m a n n ' s a r g u m e n t s a g a i n s t th e R ic ci t e n s o r .

T h o u g h E i n s t e i n ' s re j e c t i o n o f g e n e r a l c o v a r i a n c e d id n o t w a v e r d u r i n g1 91 4, th e a r g u m e n t a t i o n d i d u n d e r g o a s i g n i fi c a n t s h i ft . I n h is c o m m u n i c a t i o nt o t h e B e r li n A c a d e m y f o r O c t o b e r 1 91 4, 3~ E i n s t e i n ' s a r g u m e n t a g a i n st g e n e r a lc o v a r i a n c e a p p e a r s i n a r ev i s ed f o r m , r e f le c t in g a r e c o g n i t i o n o f t h e m a i n e r r o ro f i n t e r p r e t a t i o n i n t h e o r i g i n a l f o r m . 38 W e a r e a g a i n a s k e d t o c o n s i d e r ad o m a i n D o f s p a c e - t i m e w h i c h i s d e v o i d o f a ll m a t e r ia l p r o c e s se s . T h e n t h et o t a l s t a t e o f D i s c o m p l e t e l y s p e c i f i e d o n c e t h e m e t r i c p o t e n t i a l s g , i ( x k ) asf u n c t i o n s o f s o m e c o o r d i n a t e s y st em { x~} a re g ive n . A ga in l e t {x~} be a secondc o o r d i n a t e s y s t e m w h i c h c o i n c i d e s w i t h t x ~} o u t s i d e D b u t d i v e r g e s i n s i d e .E i n s t e i n n o w c o r r e c t l y s t a te s t h a t t h e m e t r i c p o t e n t i a l s T ,, ( ~ ) a s f u n c t i o n s o ft h e n e w c o o r d i n a t e s d e s c r i b e e x a c t l y t h e s a m e p h y s i c a l s i t u a t i o n a s t h e g , ( x ~ ) .H e r e t h e n w a s a n o p p o r t u n i t y t o e s c a p e f r o m p a s t c o n f u s i o n s a n d t o r e i n s t a t et h e r e q u i r e m e n t o f g e n e r a l c o v a r i a n c e . B u t t h e o p p o r t u n i t y w a s n o t s e iz e d ,a n d E i n s te i n w a s q u i ck l y s e d u c e d b y a n e w f o r m o f t h e s a m e m i s u n d e r s t a n d i n g st h a t c a u s e d t h e o r i g i n a l p r o b l e m .

I f t h e f i e ld e q u a t i o n s a r e g e n e r a l l y c o v a r i a n t , t h e n i t f o l l o w s , E i n s t e ina r g u e s , t h a t i f t h e g l i ( x ~ ) a re s o lu t io ns t he n so a re th e ~, :i(x~ ). Bu t ~ ,~ (x~) andg~i (x k ) d e s c r i b e d i f f e r e n t g r a v i t a t i o n a l f i e ld s i f t h e b a r r e d fu n c t i o n s a r e d i f f e r e n tf r o m t h e u n b a r r e d o n e s , a s c a n a l w a y s b e a r r a n g e d b y t h e c h o i c e o f x ' . T h u s ,

3 3E . K r e t s c h m a n n , 'L ) n e r d e n p h y s i k a l i s c h e n S i n n d e r R e l a t i v i t i t s p o s t u l a t e , A . E i n s t e i n s n e u eu n d s e i n e u r s p r t i n g l i c h e R e l a t i v i t ~ t s t h e o r ie ' , A n n a l e n d e r P h y s i k 5 3 ( 19 1 7 ) , 5 7 5 - 6 1 4 .

~ ' R e f . 7 .3 ~A . E i n s t e i n , ' D i e f o r m a l e G r u n d l a g e d e r a l l g e m e i n e n R e l a t i v i t i t s t h e o r i e ' , Sitzungsberichte, derKO niglichen Preussischen A kadem ie der Wissenschaften zu B erlin, ( 1914), 1030 - 1085.3 eW e h a v e n o t b e e n a b l e t o d e t e r m i n e e i t h e r f r o m E i n s t e i n ' s p u b l i s h e d p a p e r s o r h i s c o r r e s p o n -d e n c e w h o o r w h a t w a s r e s p o n s i b le f o r t h i s c h a n g e . T h i s is j u s t o n e o f t h e m a n y u n s o l v e d m y s t e r ie sa b o u t E i n s t e i n ' s o d y s s e y f r o m t h e s p ec i a l to t h e g e n e r a l t h e o r y o f r e l a ti v i ty .


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26 8 Studies in History and Philosophy of Sciencew e a r r iv e at th e s a m e c o n c l u s i o n : g e n e r a l l y c o v a r i a n t f ie ld e q u a t i o n s v i o l a t eC a u s a l i ty . T h e p r o m i s e s o f E i n s t e in ' s a r g u m e n t a r e l ar g e ly c o r r e c t a n d a m o u n tt o a c o o r d i n a t e r e p r e s e n ta t io n o f th e f r e e d o m t o p e r f o r m d i f f e o m o r p h i s m sd i s c u s s e d i n S e c t i o n 3 , t h o u g h E i n s t e i n ' s c o n s t r u c t i o n p r e s u p p o s e s a n o n -s t a n d a r d n o t i o n o f ' d r a g g i n g a l o n g ' . I n a m o r e s t a n d a r d p r e s e n t a t io n , t h e c o n -s t ru c t io n p r o c e e d s a s fo l lo w s . C o n s i d e r t w o s u b d o m a i n s D , a n d D 2 o f D , a n dl et 0 : D1 ~ D 2 b e a d i f f e o m o r p h i s m . 0 c a n b e g i v e n a c o o r d i n a t e r e p r e s e n t a t i o nb y d e s c r i b in g it a s th e m a p w h i c h t a k e s a p o i n t in D , w i t h c o o r d i n a t e s x k t o ap o i n t i n D 2 w i t h c o o r d i n a t e s f k ( x m ) . W i t h t h i s p o i n t m a p w e c a n a s s o c i a t e ac o o r d i n a t e m a p t o a n e w c o o r d i n a t e s y s t e m {x~} w h e r e x ~ ( O ( p ) ) - x k ( p ) , i . e . ,t h e n e w c o o r d i n a t e s a t t h e n e w p o i n t a r e n u m e r i c a l l y e q u a l t o t h e o l d c o o r d i n -a t e s a t t h e o ld p o i n t . I f F k is t h e i n v e r s e o f f ~ t h e n t h e c o o r d i n a t e t r a n s f o r m a -t i o n i s g i v e n b y ~ ~ = F k ( x m ) . T h e m e t r i c 0 * g d r a g g e d a l o n g b y 0 c a n b ed e f i n e d i n c o o r d

lost in the tensors: einsteins struggles with covariance princip

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