Advances n Hgh Energy Physcs Volume 2010, Artcle ID 379473, 6 pages do:10.1155/2010/379473 Research Artcle Møller s Energy n the Kantowsk-Sachs Space-Tme M. Abdel-Meged and Ragab M. Gad Mathematcs Department, Faculty of Scence, Mna Unversty, 61915 El-Mna, Egypt Correspondence should be addressed to Ragab M. Gad, ragab2gad@hotmal.com Receved 17 December 2009; Accepted 24 February 2010 Academc Edtor: Ira Rothsten Copyrght q 2010 M. Abdel-Meged and R. M. Gad. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We have shown that the fourth component of Ensten s complex for the Kantowsk-Sachs spacetme s not dentcally zero. We have calculated the total energy of ths space-tme by usng the energy-momentum defntons of Møller n the theory of general relatvty and the tetrad theory of gravty. 1. Introducton Snce the brth of the theory of general relatvty, ths theory has been accepted as a superb theory of space-tme and gravtaton, as many physcal aspects of nature have been expermentally verfed n ths theory. However, ths theory s stll ncomplete theory; namely, t lacks defnton of energy and momentum. In ths theory many physcsts have ntroduced dfferent types of energy-momentum complexes 1 5, each of them beng a pseudotensor, to solve ths problem. The nontensoral property of these complexes s nherent n the way they have been defned and so much so t s qute dffcult to conceve of a proper defnton of energy and momentum of a gven system. The recent attempt to solve ths problem s to replace the theory of general relatvty by another theory, concentrated on the gauge theores for the translaton group, the so called teleparallel equvalent of general relatvty. We were hopng that the theory of teleparallel gravty would solve ths problem. Unfortunately, the localzaton of energy and momentum n ths theory s stll an open, unsolved, and dsputed problem as n the theory of general relatvty. Møller modfed the theory of general relatvty by constructng a gravtatonal theory based on Wetzenböck space-tme. Ths modfcaton was to overcome the problem of the energy-momentum complex that appears n Remannan space. In a seres of paper 6 8, he was able to obtan a general expresson for a satsfactory energy-momentum complex n
2 Advances n Hgh Energy Physcs the absolute parallelsm space. In ths theory the feld varable are 16 tetrad components h a μ, from whch the Remannan metrc arses as g μν η ab h a μh b v. 1.1 The basc purpose of ths paper s to obtan the total energy of the Kantowsk-Sachs space-tme by usng the energy-momentum defntons of Møller n the theory of general relatvty and the tetrad theory of gravty. The standard representaton of Kantowsk and Sachs space-tmes s gven by 9 ( ds 2 dt 2 A 2 t dr 2 B 2 t dθ 2 sn 2 θdφ 2), 1.2 where the functons A t and B t are functon n t and determned from the feld equatons. For more detaled descrptons of the geometry and physcs of ths space-tme see 9 11. 2. Fourth Component of Ensten s Complex Prasanna has shown that space-tmes wth purely tme-dependent metrc potentals have ther components of total energy and momentum for any fnte volume T 4 t 4 dentcally zero. He had used the Ensten complex for the general Remannan metrc ds 2 g j (x 0) dx dx j 2.1 and concluded the followng: for space-tmes wth metrc potentals g j beng functons of tme varable alone and ndependent of space varable, the components T 4 t 4 vansh dentcally as a consequence of conservaton law. Unfortunately the concluson above s not the soluton to the problem consdered, n the sense that t does not gve the same result, for all metrcs have form of 2.1, usng Ensten complex. If 2.1 s gven n sphercal coordnates, then Prasanna s concluson s correct by usng Møller s complex but not correct for all metrcs by usng Ensten s complex. Because Møller s complex could be utlzed to any coordnate system, Ensten s complex gves meanngful result f t s evaluated n Cartesan coordnates. In the present paper we have found that the total energy for the Kantowsk-Sachs space-tme s dentcally zero by usng Møller s complex, but not zero by usng Ensten s complex. In a recent paper 12, Gad and Fouad have found the energy and momentum dstrbuton of Kantowsk-Scahs space-tme, usng Ensten, Bergmann-Thomson, Landau- Lfshtz, and Papapetrou energy momentum complexes. In ths secton we restrct our attenton to the Ensten s complex whch s defned by 13 θ k T k t k u kj,j, 2.2
Advances n Hgh Energy Physcs 3 wth u kj 1 g n ( [ g g kn g jm g jn g km)] 16π. g,m 2.3 The energy and momentum n the Ensten s prescrpton are gven by P θ 0 dx1 dx 2 dx 3. 2.4 The Ensten energy-momentum complex satsfes the local conservaton law θ k 0. 2.5 xk The energy densty for the space-tme under consderaton, n the Cartesan coordnates, obtaned n 12 s θ 0 0 1 8πAr 4 ( A 2 r 2 B 2), 2.6 and the total energy s E En P 0 1 ( A 2 r 2 B 2). 2Ar 2.7 Followng the approach n 12, we obtan the followng components of θ k 0 : θ 1 0 x [ AA 2 r 2 B ( 2AḂ BA )], 8πA 2 r 4 θ 2 0 y [ AA 2 r 2 B ( 2AḂ BA )], 8πA 2 r 4 θ 3 0 z [ AA 2 r 2 B ( 2AḂ BA )]. 8πA 2 r 4 2.8 The components 2.6 and 2.8 satsfy the conservaton law 2.5. Hence from 2.6 and 2.8, we have θ 0 T 0 t 0 / 0; consequently θ 0 0 T 0 0 t0 0 s not dentcally zero. 3. Energy n the Theory of General Relatvty In the general theory of relatvty, the energy-momentum complex of Møller n a fourdmensonal background s gven as 6 I k 1 8π χkl,l, 3.1
4 Advances n Hgh Energy Physcs where the antsymmetrc superpotental χ kl χ kl χ lk s ( gn g x m g ) m x n g km g nl, 3.2 I 0 0 s the energy densty and I0 α are the momentum densty components. Also, the energymomentum complex I k satsfes the local conservaton laws: I k 0. 3.3 xk The energy and momentum components are gven by P I 0 dx1 dx 2 dx 3 1 χ 0l 8π x l dx1 dx 2 dx 3. 3.4 For the lne element 1.2, the only nonvanshng components of χ kl χ 01 1 t B2 sn θ, A t are χ 02 2 χ 03 3 A t sn θ, A t sn θ. 3.5 Usng these components n 3.1, we get the energy and momentum denstes as follows I 0 0 0, 3.6 I 0 1 I0 3 0, I0 2 A t cos θ. From 3.4 and 3.5 and applyng the Gauss theorem, we obtan the total energy and momentum components n the followng form: P 0 E 0, P α 0. 3.7 4. Energy n the Tetrad Theory of Gravty The superpotental of Møller n the tetrad theory of gravty s gven by see 7, 8, 14 U νβ μ g 2κ P τνβ χρσ[ Φ ρ g σχ g μτ λg τμ γ χρσ 1 2λ g τμ γ σρχ], 4.1
Advances n Hgh Energy Physcs 5 where P τνβ χρσ δ τ χg νβ ρσ δ τ ρg νβ σχ δ τ σg νβ χσ, 4.2 wth g νβ ρσ beng a tensor defned by g νβ ρσ δ ν ρδ β σ δ ν σδ β ρ, 4.3 γ abc s the con-torson tensor gven by γ μνβ h μ h ν;β 4.4 and Φ μ s the basc vector defned by Φ μ γ ρ μρ, 4.5 The energy n ths theory s expressed by the followng surface ntegral: E lm U 0α r 0 n αds, 4.6 r const. where n α s the unt three vector normal to the surface element ds. The tetrad components of the space-tme 1.2,usng 1.1, are as follows h a μ 1,A t,b t,b t sn θ, [ ] h μ a 1,A 1 t,b 1 t, B 1 t. sn θ 4.7 Usng these components n 4.4, we get the nonvanshng components of γ μνβ as follows γ 011 γ 101 A t Ȧ t, γ 022 γ 202 B t Ḃ t, γ 033 γ 303 B t Ḃ t sn 2 θ, 4.8 γ 233 γ 323 B 2 t sn θ cos θ. Consequently, the only nonvanshng components of basc vector feld are { Φ 0 A t 2 A t Φ 2 cot θ B 2 t. } Ḃ t, B t 4.9
6 Advances n Hgh Energy Physcs Usng 4.8 and 4.9 n 4.1 and 4.6, weget E 0. 4.10 5. Summary and Dscusson In ths paper we have shown that the fourth component of Ensten s complex for the Kantowsk-Sachs space-tme s not dentcally zero. Ths gves a counterexample to the result obtaned by Prasanna 15. We calculated the total energy of Kantowsk-Sachs space-tme usng Møller s tetrad theory of gravty. We found that the total energy s zero n ths spacetme. Ths result does not agree wth the prevous results obtaned n both theores of general relatvty 12 and teleparallel gravty 16, usng Ensten, Bergmann-Thomson, and Landau- Lfshtz energy-momentum complexes. In both theores the energy and momentum denstes for ths space-tme are fnte and reasonable. We notce that the results obtaned by usng Ensten, Bergmann-Thomson, and Papapetrou are n conflct wth that gven by Møller s values for the energy and momentum denstes f r tends to nfnty, whle Landau-Lfshtz s values are not n conflct. References 1 R. C. Tolman, Relatvty, Thermodynamcs and Cosmology, Oxford Unversty Press, Oxford, UK, 1934. 2 L. D. Landau and E. M. Lfshtz, The Classcal Theory of Felds, Pergamon Press, Oxford, UK, 1962. 3 A. Papapetrou, Ensten s theory of gravtaton and flat space, Proceedngs of the Royal Irsh Academy. Secton A, vol. 52, pp. 11 23, 1948. 4 P. G. Bergmann and R. Thompson, Spn and Angular Momentum n General Relatvty, Physcal Revew, vol. 89, no. 2, pp. 400 407, 1953. 5 S. Wenberg, Gravtaton and Cosmology: Prncples and Applcatons of General Theory of Relatvty, Wley, New York, NY, USA, 1972. 6 C. Møller, On the localzaton of the energy of a physcal system n the general theory of relatvty, Annals of Physcs, vol. 4, no. 4, pp. 347 371, 1958. 7 C. Møller, Further remarks on the localzaton of the energy n the general theory of relatvty, Annals of Physcs, vol. 12, no. 1, pp. 118 133, 1961. 8 C. Møller, Momentum and energy n general relatvty and gravtatonal radaton, Kongelge Danske Vdenskabernes Selskab, Matematsk-Fysske Meddelelser, vol. 34, no. 3, 1964. 9 R. Kantowsk and R. K. Sachs, Some spatally homogeneous ansotropc relatvstc cosmologcal models, Mathematcal Physcs, vol. 7, no. 3, pp. 443 446, 1966. 10 A. S. Kompaneets and A. S. Chernov, Soluton of the gravtaton equatons for a homogeneous ansotropc model, Sovet Physcs, vol. 20, pp. 1303 1306, 1964. 11 C. B. Collns, Global structure of the Kantowsk Sachs cosmologcal models, Mathematcal Physcs, vol. 18, no. 11, pp. 2116 2124, 1977. 12 R. M. Gad and A. Fouad, Energy and momentum dstrbutons of Kantowsk and Sachs space-tme, Astrophyscs and Space Scence, vol. 310, no. 1-2, pp. 135 140, 2007. 13 A. Ensten, Zur Allgemenen Relatvtaetstheore, Stzungsber, Preussschen Akademe der Wssenschaften, Berln, Germany, 1915. 14 F. I. Mkhal, M. I. Wanas, A. Hndaw, and E. I. Lashn, Energy-momentum complex n Møller s tetrad theory of gravtaton, Internatonal Theoretcal Physcs, vol. 32, no. 9, pp. 1627 1642, 1993. 15 A. R. Prasanna, On certan space-tmes havng the fourth component of energy-momentum complex dentcally zero, Progress of Theoretcal Physcs, vol. 45, pp. 1330 1335, 1971. 16 R. M. Gad, Energy and momentum denstes of cosmologcal models, and equaton of state ρ μ, n general relatvty and teleparallel gravty, Internatonal Theoretcal Physcs, vol. 46, no. 12, pp. 3263 3274, 2007.
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