Chapter 1. Theory of Gravitation

Transcription

1 Chapter 1 Theory of Gravtaton

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3 In ths chapter a theory of gravtaton n flat space-te s studed whch was consdered n several artcles by the author. Let us assue a flat space-te etrc. Denote by x the co-ordnates of space and te then the lne-eleent can be wrtten ds dx dx (1.1) Here, s a syetrc etrc tensor. In addton to the etrc tensor a syetrc contra-varant tensor s defned by k k, k k. (1.) Furtherore, we put det. (1.3) In the specal case of a pseudo-eucldean etrc we have 1 3 x x, x, x, ct. (1.4) 1 3 x, x, x are the Cartesan co-ordnates, t s the te and c s the velocty of lght. Then, the etrc tensor has the for dag 1,1,1, 1. (1.5) Ths s the etrc n whch the ost knds of felds and atter are descrbed. 1.1 Gravtatonal Potentals Slar to Maxwell s theory of Electrodynacs we assue that gravtaton s descrbed by a feld n space and te. The electro-agnetc feld can be descrbed wth the ad of a four-vector called the potentals of the feld and produced by an electrc four-current. 3

4 A Theory of Gravtaton n Flat Space-Te Analogously, the syetrc gravtatonal potentals g are produced by the total energy-oentu of atter and gravtatonal feld. Slar to the equatons (1.) let us defne a syetrc tensor g by g g k k, k g g k (1.6) We put G det g. (1.7) In addton to the te t we defne the proper-te by c d g dx dx. (1.8) The relaton (1.8) s slar to the defnton of the lne-eleent (1.1) wth the etrc tensor. Therefore, theores of gravtaton descrbed by g wth the proper-te (1.8) and wth the lne-eleent (1.1) are called b-etrc theores of gravtaton. 1. Lagrangan The theory of gravtaton s derved fro an nvarant Lagrangan whch s quadratc n the frst order co-varant dervatves of the potentals resp. of the contra-varant tensors. The dervatves are relatve to the flat spacete etrc (1.1) and they are denoted wth a bar /. The Lagrangan has the for G k ln 1 kl G kl / / / / L g g g g g g g (1.9) In addton let us ntroduce the nvarant Lagrangan L G 8 (1.10) 4

5 Chapter 1 Theory of Gravtaton Here, s the cosologcal constant. For splcty we consder dust (no pressure) wth the densty. The Lagrangan for atter can be wrtten n the for L M g u u (1.11) where s the four-velocty. It follows by the use of dx u (1.1) d and relaton (1.8) g u u c. (1.13) By the ntroducng of the constant (1.14) the whole Lagrangan has the for: L L L 8 L. (1.15) G M Here, the constant denotes the gravtatonal constant. 1.3 Feld Equatons The dfferental equatons for the gravtatonal potentals g follow fro the varaton - equaton 4 L d x. (1.16) Fro Euler s equatons we get by the forulas for the covarant dervatves (see e.g., [Sop 76], p.189 ff ) 1 L 1 L g g / k / k (1.17) 5

6 A Theory of Gravtaton n Flat Space-Te plyng by the use of (1.15) L L L L g g g G G M 8 / k / k. (1.18) We use the followng forulas G g 1 G g, g g k l g g kl. Equaton (1.18) ples by the use of these relatons and ultplcaton wth l g the followng forula G k 1 kl g gk g / n gkl g / n 17 1G r k nl 1 kl ggkl g g / g / r g / g / r 1 LG L 4 gu u 4 (1.19) These are the feld equatons of gravtaton for dust. 1.4 Equatons of Moton and the Energy-Moentu We wll now prove the equvalence of the conservaton law of energyoentu and the equatons of oton. It follows fro equaton (1.18) by k ultplcaton wth g / l and suaton g L G G / l g / l/ k g / k g / k / k L L L G r s g / l 8 g / l gr gnsu u g. (1.0) The xed energy-oentu tensors of the gravtatonal feld, of vacuu energy (gven by the cosologcal constant ) and of dust are gven by 6

7 Chapter 1 Theory of Gravtaton 1 G r, k nl 1 kl 1 T G ggkl g g / g / r g / g / r LG 8 (1.1a) T 1 L (1.1b) 16 g u u T M and the correspondng syetrc tensors are defned by Put (1.1c) D g g g G k k / n / (1.). (1.3a) Then, the feld equatons of gravtaton (1.19) have the sple for Here, 1 D 4 D T (1.3b). (1.3c) s the whole energy-oentu tensor of gravtatonal feld, of vacuu energy and of atter. The equatons (1.3) can be rewrtten 1 4 D T T. (1.4) It s worth to enton that the equatons (1.3) are generally co-varant. In partcular, the energy-oentu of gravtaton s a tensor n contrast to the correspondng pseudo-tensor n Ensten s general relatvty. The feld equatons of gravtaton (1.3b) and (1.4) are forally slar to the correspondng equatons of general relatvty. Here, D s a dfferental 7

8 A Theory of Gravtaton n Flat Space-Te operator of order two n dvergence for for g whereas n general relatvty there s nstead of that the Rcc tensor. The source of the gravtatonal feld n flat space-te theory of gravtaton s the whole energy-oentu tensor nclusve the one of the gravtatonal feld whch s not a tensor n Ensten s theory and t does not appear as source for the feld. Relaton (1.0) can be rewrtten L g L L g u u G l n / k k G 8 / k g / l / l.e., we get by the use of (1.1a) and (1.1b) 1 / n T G T / g u u. Ths relaton becoes by the substtuton of (1.1c) and the use of (1.3c) T 1 n 1 / T M / / / / g u u T M g T M. Hence, the conservaton of the whole energy-oentu / 0 s equvalent wth the equatons of oton for atter T (1.5a), (1.6) The conservaton law of the whole energy-oentu (1.5a) can be rewrtten The conservaton of ass s gven by n T n 0 /. (1.5b) u 0 /. (1.7) More general energy-oentu tensors for atter can be consdered, e.g. the atter tensors of perfect flud 8

9 Chapter 1 Theory of Gravtaton T M p u u pc g (1.8) where p denotes the pressure of atter. The conservaton law of the whole energy-oentu and the equvalent equatons of oton are also gven by the equatons (1.5) and (1.6). The conservaton law of the whole energy-oentu (1.5), the equatons of oton (1.6), and the conservaton law of ass (1.7) are gven n covarant for. The equatons of oton (1.6) and the conservaton of ass (1.7) can be rewrtten n non-covarant for k g T M T M 1 1 k x 1 x k u k x The equatons (1.9) gve for a test partcle,.e. p 0 d 1 g g k u u u d x k n (1.9a) 0. (1.9b). (1.30) It follows by dfferentaton, the use of (1.11), and soe eleentary calculatons n d x dx dx G (1.31) d d d where denote the Chrstoffel sybols of g. It s worth entonng that the equatons for the gravtatonal feld can be generalzed ncludng electro-agnetc felds, scalar felds, etc., by addton of the correspondng Lagrangans for these felds to (1.15) whch wll not be consdered. 9

10 A Theory of Gravtaton n Flat Space-Te 1.5 Feld Equatons Rewrtten It s soetes useful for the applcatons of the gravtatonal theory to consder nstead of g syetrc tensors defned by f G g (1.3a) and f G g (1.3b) yeldng k 1 k, f f k k f f. (1.33) Then, the equatons for the gravtatonal feld (1.3) can be rewrtten k k / n 4 / f f f T (1.34) where the energy-oentu tensor of gravtaton has the for 1 r k nl 1 kl 1 T G f fkl f f / r f / f / r f / LG 8 (1.35) wth rs k nl 1 kl LG f fkl f f / r f / s f / r f / s. (1.36) The energy-oentu tensor of perfect flud s gven by F k k T M p f u u pc (1.37) where F det f. (1.38) 10

11 Chapter 1 Theory of Gravtaton The relaton (1.13) has the for F n f u u c. (1.39) 1.6 Feld Strength and Feld Equatons The equatons of oton (1.31) of a test partcle n the gravtatonal feld are not generally co-varant. A co-varant dervatve of the four-vector u of a test partcle s Du du n u u. (1.40) D d are the Chrstoffel sybols of the etrc (1.1). The equatons of oton (1.31) can be rewrtten by the substtuton (1.40) where Du n u u (1.41) D G. (1.4) Eleentary calculatons ply that k s a tensor of rank three. Hence, the equatons of oton (1.41) for a test partcle n the gravtatonal feld g are generally co-varant. Slar to the equatons of oton for a test partcle n the electro-agnetc feld where on the rght hand sde stands the Lorentz-force defned by the electro-agnetc feld strength the tensor k n the equatons (1.41) can be nterpreted as gravtatonal feld strength and the rght hand sde of (1.41) s the gravtatonal force. Eleentary calculatons gve 11

12 A Theory of Gravtaton n Flat Space-Te Hence, t follows Therefore, we get g x r r nr n G g G g r r / nr n r g g g. n k / k / kn k g g g g g g. (1.43) Wth the ad of (1.43) all the co-varant dervatves of g can be replaced by the gravtatonal feld strength. Eleentary calculatons gve the Lagrangan r. G k l kl r s r s G l kn rs k ln r sn L g g g (1.44) The energy-oentu tensor of the gravtatonal feld has the for 1 G n k l kl r s r s 1 ln k rs kn l rn s G T( G) g g g L 4 16 (1.45) It follows for the equatons of the gravtatonal feld (1.3b) G g g g 4T k l k n l kn kn The feld equatons (1.4) have the for / /. (1.46a) G 1 k l 4 g n g gl kn T T. (1.46b) Suarzng, we have wrtten the theory of gravtaton n flat space-te by the use of the feld strength of gravtaton slar to Maxwell s theory wrtten wth the ad of the electro-agnetc feld strength. 1

13 Chapter 1 Theory of Gravtaton 1.7 Angular-Moentu We wll now derve the conservaton law of the whole angular-oentu. Let us start fro the conservaton law of the whole energy-oentu (1.5b) whch can be rewrtten 1 T T 0 x where we have ntroduced the non-syetrc energy-oentu tensor T T (1.47a). (1.47b) In an nertal frae,.e. the etrc tensor s constant and therefore k 0 the relaton (1.47a) ples a conservaton law of the whole energyoentu. Therefore, we get P 14 3 T d x (=1-4) (1.48) Where s a constant and the ntegraton s taken over the whole space. Equaton (1.47a) gves 1 x T T x T x The feld equatons (1.3) ply 1 G k l 1 kl T T g gkl g / n gkl g / n 4. (1.49) The substtuton of ths relaton nto equaton (1.49) and the subtracton fro the arsng fro the sae equaton where and are exchanged yelds 1 x T x T x / A x x T /. (1.50) 13

14 A Theory of Gravtaton n Flat Space-Te wth the contra-varant tensor k 1 G k s r s r rs / / A g g g g 4. (1.51) It follows fro equaton (1.50) by the use of relatons for the co-varant dervatves of tensors of order three 1 x x T x T A ( ) n n x x T A A. (1.5) These equatons ply n unforly ovng fraes the conservaton law of the angular-oentu,.e , 1,,3,4 M x T x T A d x (1.53) s constant for all tes. The frst two expressons correspond to the usual defnton of the angular oentu. To study the last expresson we use the frst part of the relaton (1.43) and rewrte (1.51) k 1 G kr s n n / r s s A g g g 4 We now defne the canoncal oentu. (1.54) 1 L 16 g G /4 (1.55a) plyng 1 G 4k 1 g g / k gg n g g. (1.55b) 8 The Haltonan s gven by 1 H g /4 LG. (1.56a)

15 Chapter 1 Theory of Gravtaton Eleentary coputatons gve 4 4 H T G, (1.56b).e. H s the energy densty of the gravtatonal feld. It follows fro (1.54) by the use of relaton (1.55b) A 4 k nl n n g kl. (1.57) We defne for, 1,,3,4 the ant-syetrc four-atrces n n (1.58) wth the proper-values 0,. The relaton (1.57) can be rewrtten A 4 k nl g. (1.59) Hence, the last expresson n equaton (1.53) of the angular oentu can be nterpreted as consequence of the spn of the gravtatonal feld. 1.8 Equatons of the Spn Angular Moentu In ths sub-chapter we follow along the lnes of Papapetrou [Pap 51] who uses a ethod of Fock [Foc 39]. The followng detaled calculatons can be found n [Pet 91]. The equatons of oton for atter (1.9a) can be wrtten n the for: lk T M G T M 1 x (1.60) where t s assued that vanshes outsde of a narrow tube whch surrounds the world lne of the test partcle. The test partcle descrbes a world 4 lne X t X t wth X t ct. Let us put n analogy to [Pap 51] 4 3 M u T M d x (1.61a) 15

16 A Theory of Gravtaton n Flat Space-Te k 4 k k 3 M u x X t T M d x (1.61b) M M. (1.61c) 4 u We obtan the equatons of oton 4 M 4 k d G M G M d u x k (1.6a) and of the spn angular oentu d u d u d d u d u d u 4 G 4 G M u u 4 G 4 G M u (1.6b) Furtherore, we have u 4 4 k k k k k 4 M u u u u (1.63a) u M M u u (1.63b) M u (1.63c) 44 u u 44 d M 4 M 4 M 4 4 G M u u d u d M d u 4 4 G M. (1.63d) Soe of the relatons (1.6) are denttes. Therefore, we have eght equatons (four equatons (1.6a), three equatons (1.6b) and one equaton (1.13) for the 16

17 Chapter 1 Theory of Gravtaton eleven unknowns quanttes proved n [Pap 51] that 44 M, u 1,,3,4, and, 1,,3 s the coponents of a tensor and the expresson. It s k 4 l M 3 4 G u u kl (1.64) c u s a scalar where equatons (1.6). u g u. We wll now gve a co-varant forulaton of the In analogy to (1.40) we defne the co-varant dervatve D d n n / u u u D d. (1.65) Let us ntroduce the ant-syetrc tensor D n n A u u. D (1.66) Then, we have by (1.6b), (1.63a), (1.4) and (1.65) When we ultply (1.67) wth u 4 u A A A. (1.67) u u u we get 4 u u 4 A A A u c u. (1.68) By the use of the last two relatons we get the co-varant for of (1.6b) 1 A u 0 u A u A. (1.69) c We wll now gve (1.6a) n co-varant for and wrte (1.63d) for 4 wth 4 the ad of (1.63a), (1.63c), M 0, (1.65) and (1.66) 4 4 n 4 u n 4. M G u A M G u (1.70) u 17

18 A Theory of Gravtaton n Flat Space-Te u We get by ultplyng ths relaton wth and the use of (1.64) 4 u u n 4 M G u c A 4 4 u c u Hence, we get fro (1.70) by the use of (1.71) and (1.68) k 4 l 1 k M cu G u u 4 4 kl k A. u u c. (1.71) Now, t follows fro (1.6a) by the use of (1.68), (1.61) and eleentary calculatons d 1 k k l 1 lr cu uka G u kl cu u ra d c c n k l G G G u lk n kl x 0 (1.7) The ntroducton of the co-varant dervatve of a four-vector gves D 1 n 1 nk cu u A u cu u k A D c c 1 n k R k u 0 (1.73) where R k s the curvature tensor of g. Although the equatons (1.6a) and (1.6b) are dentcal wth those of general relatvty the co-varant fors (1.73), (1.69) together wth (1.66) are dfferent fro those of general relatvty [Pap 51]. whch s defned by (1.61c) s not the spn n flat space-te theory of gravtaton. The spn of a partcle ust be defned by 4 3 S x X t T M d x. (1.74) 4 3 x X t T M d x 18

19 Chapter 1 Theory of Gravtaton In Ensten s theory the oton of a spn n free fall can be descrbed accordng to the equatons of parallel transport (see e.g. [We 7]. Ths s not possble by the use of flat space-te theory of gravtaton. 1.9 Transforaton to Co-Movng Frae In the prevous sub-chapter we have seen that there are not enough equatons for the spn coponents. Schff [Sch 80] rearked that one has to transfor the equatons of spn coponents to the co-ovng frae,.e. to the frae of the gyroscope. We use the consderatons of Petry [Pet 86] to transfor fro a preferred frae ' ' dag 1,1,1, 1 to a non-preferred frae wth ovng wth velocty v' v 1 ', v ', v 3 ' 1 3 ' ', ' ', ' ' relatve to the frae ' X t X t X t be the dstance vector of fro '. Then, ' to the correspondng ones n the co- The transforatons of quanttes n ovng frae are gven n [Pet 86] wth. Let d X ' t ' v '. (1.75) dt ' v' x, 1 c v ' x ' x 1 X ' t ', = dt (1.76a) v' c c 1 v' c. (1.76b) It s suffcent to consder (1.76) up to quadratc expressons n the absolute value of the velocty v ',.e. 1 v' v 1 v' x ' x x, X ', dt ' 1 c c c dt. (1.77) 19

20 A Theory of Gravtaton n Flat Space-Te In the frae ' we consder equaton (1.6b), ultpled wth 4 use of (1.63a) and u '/ u ' v '/ c,.e. d / dt ', the d v ' d 4 v ' d 4 v ' 4 ' n ' ' ' G ' G ' v ' dt ' c dt ' c dt ' c v ' 4 n G ' G ' ' v ' c v ' v ' 4 n G ' G ' ' v ' 0 c c (1.78) Furtherore, t follows ' x ' x '. n x x We get after soe calculatons for the spn tensor n 3 k 3 k v ' 4 v ' 4 1 v ' k v ' v ' k v ' ' c c c k1 c c k1 c (1.79a) 3 k 4 1 v' 4 1 v ' k 4 v ' ' 1 c. (1.79b) c k1 c If we substtute (1.79) nto (1.78) and neglect expressons of the for v v t follows by eleentary calculatons c c where 3 d k k k k G 44 v ' v ' dt ' (1.80a) k1 k1 3 k G v ' G c G v ' G v ' G v ' k (1.80b) We wll now apply the result to the spn angular oentu of a test partcle n the gravtatonal feld of a sphercally syetrc body n the preferred frae 0

21 Chapter 1 Theory of Gravtaton ' wth ass M and angular velocty. It holds n approxatons ' up to lnear g km ' 1,, 1,,3 cr km 1, 4 cr kj c r x', 4, 1,,3; 1,,3, 4 (1.81) where J s the oentu of nerta. We get by eleentary coputatons k km r x, x', v',,. k1 c r Put ,, then, we have 3 km 1 kj 1 x', 3 x' v' 3 x' 3. (1.8) c r c r r We defne 3, 31, 1. Relaton (1.81a) s rewrtten n the for d km 3 x ', v '. (1.83a) dt ' cr By the use of the law of Newton dv' x' km 3 dt ' r we get d 1 dv' v', dt ' c dt ' c 1

22 A Theory of Gravtaton n Flat Space-Te 3 1 dv' v' kj x', 3 x' 3 c dt ' c. (1.83b) c r r We consder nstead of the spn. We get n by the use of the standard transforaton forula, consderng only expressons whch are quadratc n the km velocty and lnear n the expresson, the use of (1.81) cr The etrc tensor has the for g km v ' v ' 1,, 1,,3 cr c c v ', 1,,3; 4 c v ', 4; 1,,3 c km 1 4. cr,, 1,,3,4 v', 1,,3; 4 c v', 4; 1,,3 c v' 1, 4. c (1.84) (1.85) We get fro the defnton (1.74), (1.85) and (1.61) for,=1,,3 k 4 3 k4 3 k k S g x T M d x g x T M d x km 1. cr Hence, t holds

23 Chapter 1 Theory of Gravtaton 1 km S. (1.86) cr We have by the substtuton of (1.86) nto the relaton (1.83a) d ' 1 km v S 1 km S dt ' c r c. c r By the use of the conservaton law of energy we get 1 v' km const c cr d S S. (1.87) dt Equaton (1.87) gves the precesson of the spn of a test partcle wth constant angular velocty. It agrees wth the correspondng result of general relatvty [Sch 60]. The angular oentu of a gyroscope processes wthout changng n agntude. The results about the spn angular oentu and the gyroscope agree wth those of general relatvty. All these results of the sub-chapters 1.8 and 1.9 can be found n [Pet 91]. For experental techncal probles copare Wll [Wl 81]. The results of chapter I about the theory of gravtaton n flat space-te can be found n the artcles of Petry [Pet 79, 81a, 8,93b]. It s worth to enton the artcle of Cahll who has studed a theory of gravtaton wth applcaton to cosology by a ethod whch s totally dfferent fro general relatvty and any b-etrc theory Approxate Soluton n Epty Space By the use of general relatvty approxate solutons n epty space are receved by lnearzaton of the non-lnear equatons. Ths can also be consdered by the use of flat space-te theory of gravtaton as wll be seen n sub-chapter.. Therefore, we wll study the lnearzaton of the gravtatonal feld. We start fro the gravtatonal theory n flat space-te (1.3) together 3

24 A Theory of Gravtaton n Flat Space-Te wth the conservaton of the whole energy-oentu (1.5). Forula (1.3b) of the feld equatons ples by the use of covarant dfferentaton, the conservaton law (1.5a) and the use of the pseudo-eucldean geoetry (1.5) Relaton (1.88) gves by the use of lnearzaton,.e. (=1-4). (1.88) the lnearzed expresson Therefore, relaton (1.88) can be wrtten n the for The operator n front of the bracket s the wave operator. Hence we get (=1-4). (1.89) Relaton (1.89) s dentcal wth the result of general relatvty (see e.g. p. 56, whch s used for any applcatons.the dervaton of relaton (1.89) n epty space (no atter) uses the fact that n epty space a gravtatonal feld exsts whch ust be consdered. The qute dfferent study of lnear approxatons of the gravtatonal feld by flat spacete theory of gravtaton and general relatvty follows fro the dfferent sources n the theores. Flat space-te theory of gravtaton has the whole energy-oentu as source whereas general relatvty has only the atter tensor. In general relatvty the energy-oentu s not a tensor whch ples any dffcultes (see the extensve study of Logunov and co-workers (see e.g. ). A coparson of the theory of gravtaton n flat space-te and the theory of general relatvty s gven n [Pet 14a]. 4