The Gravitational Energy Problem for Cosmological Models in Teleparallel Gravity

Abstract

We present a method to calculate the gravitational energy when asymptotic boundary conditions for the space-time are not given. This is the situation for most of the cosmological models. The expression for the gravitational energy is obtained in the context of the teleparallel equivalent of general relativity. We apply our method first to the Schwarzschild-de Sitter solution of Einstein's equation, and then to the Robertson-Walker Universe. We show that in the first case our method leads to an average energy density of the vacuum space-time, and in latter case the energy vanishes in the case of null curvature. June 21, 2018 4:18 WSPC/INSTRUCTION FILE GEPforCM 2 Ulhoa et. alThe motivation for finding an expression for the energy of the universe comes from the fact that this subject may help to answer some questions about the origin and evolution of the universe, with consequent improvements for the cosmological models 5 .Several investigations and results indicate that the total energy of the universe is zero. However these results are dubious since most of them are obtained by means of pseudotensors, which are coordinate-dependent expressions whose geometrical meaning is not clear. The results obtained by means of Komar's integrals, on the other hand, depend on the normalization of a Killing vector and in the case of cosmological systems, which are not asymptotically flat, there is no physically preferred choice for such vectors.In the framework of the Teleparallel Equivalent of General Relativity (TEGR) the gravitational energy is well defined for finite volumes of the three dimensional space, and in particular for asymptotically flat space-times. When asymptotic boundary condition are not available, for instance in the case of the Robertson-Walker model and of the de Sitter Universe, a new method has to be used. As remarked by Faddeev 9 , an expression for the gravitational energy must vanish only in the total absence of matter and gravitational field. Therefore in order to avoid the use of boundary conditions, it is possible, and perhaps mandatory, to use regularized expressions 10 , which do not require the establishment of asymptotic conditions. The purpose of this paper is to point out that the use of a regularized expression for the gravitational energy is quite suitable to cosmological models. This paper is organized as follows. In section 2, we introduce the formalism of teleparallel gravity. We show how to define the gravitational energy in this context and present the definition of the regularized expression for the gravitational energy, which can be applied for arbitrary field configurations. In section 3, we propose a method to evaluate the energy based on the regularized expression, since in cosmology the lack of spatial asymptotic conditions prevents the application of ordinary methods. We evaluate the energy for two configurations which have relevance in cosmology, the Schwarzschild-de Sitter's solution and the Robertson-Walker model for the universe. Finally we present some concluding remarks.Not...

The teleparallel equivalent of general relativity

“…We remark that regularized expressions like Eq. (62) are useful in the investigation of cosmological models, when one does not dispose of asymptotic boundary conditions [53]. The evaluation of definition (58) 58) under the local SO(3,1) group reflects the frame dependence of the definition.…”

Cosmic acceleration and phantom crossing in f(T)-gravity

“…It employs the Weitzenböck spacetime which is characterized by the metricity condition and vanishing of the curvature tensor. The approach contains a variety of distinctive manifestation both from physical and geometrical aspects (Hoff da Silva and da Rocha 2010; Hayashi and Shirafuji 1979;Fiorini 2007, 2008;Ulhoa et al 2010;Nashed 2010;Sharif and Taj 2010;Lucas et al 2009;Poplawski 2010, Wu and Yu 2010a, 2010bAo et al 2010;Bengochea andFerraro 2009, 2011;Yang and Zhang 2010).…”

On Gravitational Entropy of de Sitter Universe

“…In addition we have made the identification ≡ (0) . It should be noted that such expression already appeared in [24].…”