Advances in High Energy Physics

Volume 2016 (2016), Article ID 1986387, 9 pages

http://dx.doi.org/10.1155/2016/1986387

## Tryon’s Conjecture and Energy and Momentum of Bianchi Type Universes

^{1}Department of Electrical Engineering, Indira Gandhi Institute of Technology, Sarang, Dhenkanal District, Odisha 759146, India^{2}Department of Computer Science Engineering and Applications, Indira Gandhi Institute of Technology, Sarang, Dhenkanal District, Odisha 759146, India^{3}Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal District, Odisha 759146, India

Received 5 February 2016; Revised 16 April 2016; Accepted 5 May 2016

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2016 Prajyot Kumar Mishra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The energy and momentum of some diagonal anisotropic Bianchi type Universes are obtained using different energy-momentum complexes in the framework of General Relativity. The Møller energy is found to be zero for all the Universes considered in the present work. In all other prescriptions, the energy and momentum vanish when the sum of the metric parameters and vanishes. This result obviously raises a question: why this specific choice? We have explored Tryon’s conjecture that the Universe must have a zero net value for all conserved quantities to get some ideas on this issue.

#### 1. Introduction

The General Relativity (GR) as formulated by Einstein is now a hundred years old, but the problem of energy-momentum localization in GR has not yet been settled. Einstein conceived the idea of covariant conservation of energy and momenta of gravitational fields along with those of matter and nongravitational fields [1]. However, quantities like energy and momentum at any local point of a manifold should always be conserved as per the usual conservation law , where is the energy-momentum tensor and the comma (,) denotes an ordinary differentiation. The covariant formulation requires nontensorial fields. Obviously, the energy-momentum tensor due to the gravitational field turns out to be nontensorial (pseudotensor). The choice of this pseudotensor is not unique and therefore it has led to the formulation of a number of prescriptions for the calculation of energy and momentum [2–9]. The interesting thing about these prescriptions is that they depend on the coordinate systems used. It has been observed earlier that, for quasi-Cartesian coordinates, all the prescriptions can provide some reasonable and meaningful results. However, some coordinates independent energy-momentum complexes have been proposed by Møller [7], Komar [8], and Penrose [9]. But some of these coordinates independent prescriptions are questioned for their limited applicability.

The issue of energy localization has been widely discussed in the literature in the framework of both GR and teleparallel gravity. Misner et al. showed that energy is localized only for spherically symmetric systems [10]. Cooperstock and Sarracino counter commented on the idea of Misner and established that if energy is localized in spherically symmetric systems then it can be localized in any space-time [11]. Bondi perceived that a nonlocalizable form of energy is not admissible in GR, because any form of energy contributes to gravitation and therefore its location can in principle be found [12]. Virbhadra and his collaborators revived the debate and proved that energy-momentum complexes coincide and give reasonable results for some well known and physically significant space-times [13–21]. Virbhadra showed that different prescriptions can provide the same result for Kerr-Schild space-time when Cartesian coordinates are used [13]. Following Virbhadra, many researchers obtained interesting results on this pressing issue of energy localization [22–39]. Contrary to the previous results, Gad explored the failure of these prescriptions to provide similar results in some specific examples of space-time [40–43]. The issue of energy localization has also been considered in the framework of teleparallel gravity [44–49]. It has been concluded in some recent works that the energy-momentum definitions are identical not only in GR but also in teleparallel gravity [48, 50, 51].

Tryon anticipated the net energy of the Universe to be zero [52]. Albrow [53] also had a similar assumption on the net energy of the Universe. From the calculation of energy of a closed homogeneous isotropic Friedmann-Robertson-Walker (FRW) Universe, Rosen showed that the total energy of the Universe is zero everywhere [54]. Cooperstock and Israelit [55] and Johri et al. [56] also found similar results for closed FRW Universe. Vargas calculated the energy and momentum of FRW Universe in Landau-Lifshitz and Einstein prescriptions in the context of teleparallel gravity and obtained the total energy of the Universe to be zero. In a recent work, Tripathy et al. [57] have obtained the energy and momentum of Bianchi type () Universes in the framework of GR in different prescriptions and have shown that the results can agree only for a specific value of the metric parameter . They have also raised the question, on the basis of Tryon’s conjecture, of why the space-time requires the specific value of the parameter. In the present work, we have tried to investigate the question raised by Tripathy et al. by considering anisotropic Bianchi type Universes. It is worth mentioning here that anisotropic space-times are more interesting to investigate in the context of recent observations. Many authors have taken interest in the calculation of the energy and momentum of anisotropic Universe in recent times. Using different energy-momentum complexes either in GR or in teleparallel gravity, Banerjee and Sen [58], Xulu [59], and Aydogdu and Salti [46] have obtained the total energy of Bianchi type I () Universes to be zero everywhere. Aydogdu and Salti have also calculated the energy of LRS Bianchi II Universe to get consistent results [60]. Radinschi calculated the energy distribution of a Bianchi type () Universe using Tolman, Bergmann-Thompson, and Møller prescriptions and found the total energy of the Universe to be zero [61]. In another work, Radinschi calculated the energy of Universe using the Landau-Lifshitz, Papapetrou, and Weinberg prescriptions and found similar results [62]. Aygün and Tarhan have obtained the energy and momentum of Bianchi IV Universe in different energy-momentum complexes in the framework of both GR and teleparallel gravity [63].

The organisation of the paper is as follows. In Section 2, we present the basics of diagonal anisotropic Bianchi type (DB) Universes. In Section 3, the energy and momentum densities for these Universes are obtained using some well known prescriptions. Results of the present work are discussed and analysed based upon Tryon’s conjecture advocating a null total energy state of the Universe in Section 4. The summary and conclusion are presented at the end, in Section 5. In the present work, we have used the convention that the Latin indices take values from 0 to 3 and Greek indices run from 1 to 3. Also, we have used the geometrized unit system where , with and being the Newtonian gravitational constant and speed of light in vacuum, respectively.

#### 2. Diagonal Anisotropic Bianchi Type Universe

The Universe is observed to be mostly isotropic and can be well explained by CDM ( dominated Cold Dark Matter) model. However, certain measurements of cosmic microwave background from Wilkinson Microwave Anisotropic Probe (WMAP) show some anomalous features of CDM model at large scale [64]. These precise measurements suggest an asymmetric expansion of the Universe with one direction expanding in different manner than the other two transverse directions [65–67]. The Planck data [68–71] shows slight red shifting of the primordial power spectrum of curvature perturbation from exact scale invariance. It can be inferred from the Planck data that CDM model cannot be a good fit at least at high multipoles. The issue of global anisotropy can be dealt with in many ways. However, a simple way is to modify the FRW model by considering asymmetric expansion along different spatial directions. In this sense, Bianchi type models play important roles. The Bianchi type models are homogeneous space-times having anisotropic spatial sections and are exact solutions of Einstein field equations. In the present work, we have considered diagonal anisotropic Bianchi type Universes modelled through the metricwhere , , and are the directional scale factors and are considered as functions of cosmic time only. The exponents and are constants of time and can assume any real values compatible with the physical Universe. It is worth mentioning here that the models are not vacuum solutions and can have nonvanishing components of the energy-momentum tensor .

The determinant of the metric tensor for the DB space-time in (1) is . The nonvanishing covariant components of the metric tensor are , , , and . The corresponding contravariant components are

#### 3. Energy-Momentum Complexes

We have calculated the energy and momentum of DB Universes described by metric (1) using the Einstein, Landau-Lifshitz, Papapetrou, Bergmann-Thompson, and Møller prescriptions. The definitions of the energy-momentum pseudotensors and corresponding energy-momentum four vectors () for different prescriptions are given in Table 1. We consider the definitions of the well known energy-momentum prescriptions in the framework of General Relativity. In the following subsections, we report the nonvanishing components of the super potentials and the consequent energy and momentum densities. The calculated energy and momentum densities for the DB Universes are given in Table 2. The results for energy and momentum densities are presented in general forms of the directional scale factors , , and and the exponents and . From these general results, the energy and momentum of a given diagonal Bianchi type Universe can be obtained in straightforward manner by incorporating the time dependence of the scale factors and the values of the exponents and .