Abstract
Following Einstein’s definition of Lagrangian density and gravitational field energy density (Einstein in Ann Phys Lpz 49:806, 1916, Einstein in Phys Z 19:115, 1918, Pauli in Theory of Relativity, B.I. Publications, Mumbai, 1963), Tolman derived a general formula for the total matter plus gravitational field energy (P 0) of an arbitrary system (Tolman in Phys Rev 35:875, 1930, Tolman in Relativity, Thermodynamics & Cosmology, Clarendon Press, Oxford, 1962, Xulu in hep-th/0308070, 2003). For a static isolated system, in quasi-Cartesian coordinates, this formula leads to the well known result \({P_0 = \int \sqrt{-g} (T_0^0 - T_1^1 - T_2^2 - T_3^3) d^3 x,}\) where g is the determinant of the metric tensor and \({T^a_b}\) is the energy momentum tensor of the matter. Though in the literature, this is known as “Tolman Mass”, it must be realized that this is essentially “Einstein Mass” because the underlying pseudo-tensor here is due to Einstein. In fact, Landau–Lifshitz obtained the same expression for the “inertial mass” of a static isolated system without using any pseudo-tensor at all and which points to physical significance and correctness of Einstein Mass (Landau, Lifshitz in The Classical Theory of Fields, Pergamon Press, Oxford, 1962)! For the first time we apply this general formula to find an expression for P 0 for the Friedmann–Robertson–Walker (FRW) metric by using the same quasi-Cartesian basis. As we analyze this new result, it transpires that, physically, a spatially flat model having no cosmological constant is preferred. Eventually, it is seen that conservation of P 0 is honoured only in the static limit.
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References
Einstein A.: Ann. Phys. Lpz. 49, 806 (1916)
Einstein A.: Phys. Z 19, 115 (1918)
Pauli, W.: Theory of Relativity. B.I. Publications, Mumbai (1963) (Trans. by G. Field)
Tolman R.C.: Phys. Rev. 35(8), 875 (1930)
Tolman R.C.: Relativity, Thermodynamics & Cosmology. Clarendon Press, Oxford (1962)
Xulu, S.S.: hep-th/0308070 (2003)
Landau L.D., Lifshitz E.M.: The Classical Theory of Fields, 2nd edn. Pergamon Press, Oxford (1962)
Favata M.: Phys. Rev. D 63, 064013 (2001)
Einstein A.: Berlin Ber., 448 (1918)
Rosen N.: Phys. Rev. 110, 291 (1958)
Rosen N., Virbharda K.S.: Gen. Rel. Grav. 25, 429 (1993)
Møller C.: Ann. Phys. (NY) 4, 347 (1958)
Szabados L.B.: Living Rev. Rel. 7, 4 (2004)
Chang A.-C., Nester J.M., Chen C.-M.: Phys. Rev. Lett. 83, 1897 (1999)
Stephani S.: General Relativity. Cambridge University Press, Cambridge (1996)
Maartens R., Maharaj S.D.: Class. Q. Grav. 3, 1005 (1986)
von Freud P.: Ann. Math. 40, 417 (1939)
Herrera L., Di Prisco A., Hernández-Pastora JL, Santos N.O.: Phys. Lett. A 237, 113 (1998)
Herrera L., Barreto W., Di Prisco A., Santos N.O.: Phys. Rev. D 65(10), 104004 (2002)
Bondi H., van der Burg M.G.J., Metzner A.W.K.: Proc. R. Soc. London A 269, 21 (1962)
Geroch R.P., Winicour J.: J. Math. Phys. 22, 803 (1981)
Penrose R.: Proc. Roy. Soc. London A 284, 159 (1965)
Wald R.M.: General Relativity. University Chicago Press, Chicago (1984)
Mitra, A.: Phys. Rev. D74 024010 (2006). arXiv:gr-qc/0605066
Mitra, A.: arXiv:0806.0706 (2008)
Baryshev Yu., Teerikpori P.: Discovery of Cosmic Fractals. World Scientific, Singapore (2002)
Mitra, A.: Practical Cosmology. In: Baryshev, Y.V., Taganov, I.N., Teerikorpi, P. (eds.) Proceedings of the International Conference held at Russian Geographical Society. TIN, St.-Petersburg, vol. 1, pp. 304–313 (2008). ISBN 978-5-902632. arXiv:0907.2532
Mitra, A.: Practical Cosmology. In: Baryshev, Y.V., Taganov, I.N., Teerikorpi, P. (eds.) Proceedings of the International Conference held at Russian Geographical Society. TIN, St.-Petersburg, vol. 2, pp. 42–51 (2008). ISBN 978-5-902632. arXiv:0907.1521
Faraoni V., Cooperstock F.I.: Astrophys. J. 587(2), 483 (2003)
Vishwakarma R.G.: Mon. Not. Roy. Astron. Soc. 345(2), 545 (2003)
Vishwakarma R.G.: Mon. Not. Roy. Astron. Soc. 361(4), 1382 (2005)
Crawford, D.F.: arXiv:0901.4169
Cwarford, D.F.: arXiv:0901.4172
Wand D.: Nat. Phys. 5(2), 89 (2009)
Mitra A.: New Astron. 12(2), 146 (2006)
Taub, A.H.: Ann. Inst. Henri Poincare, IX(2), 153 (1968). http://www.numdam.org/numdam-bin/fitem?id=AIHPA-1968-9-2-153-0
Mansouri, R.: Ann. Inst. Henri Poincare, XXVII(2), 173 (1977). http://www.numdam.org/numdam-bin/fitem?id=AIHPA-1977–27-2-175-0
Milne E.A.: Q. J. Math. 5, 64 (1934)
McCrea W.H., Milne E.A.: Q. J. Math. 5, 73 (1934)
Lima J.A.S., Zamchin V., Brandenberger R.: Mon. Not. Roy. Astron. Soc. 291, L1 (1997)
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Mitra, A. Einstein energy associated with the Friedmann–Robertson–Walker metric. Gen Relativ Gravit 42, 443–469 (2010). https://doi.org/10.1007/s10714-009-0863-1
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DOI: https://doi.org/10.1007/s10714-009-0863-1