Einstein energy associated with the Friedmann–Robertson–Walker metric

TLDR: In this paper, the authors derived a general formula for the total matter plus gravitational field energy of an arbitrary system in quasi-Cartesian coordinates and applied it to the Friedmann-Robertson-Walker (FRW) metric.

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.