Tryon's conjecture and Energy and momentum of Bianchi Type Universes

The energy and momentum of the Bianchi type $III$ universes are obtained using different prescriptions for the energy-momentum complexes in the framework of General Relativity. The energy and momentum of the Bianchi $III$ universe is found to be zero for the M\o{}ller prescription. For all other prescriptions the energy and momentum vanish when the metric parameter $h$ vanishes. In an earlier work, Tripathy et al. \cite{SKT15} have obtained the energy and momentum of Bianchi $VI_h$ metric and found that the energy of the Universe vanish only for $h=-1$. This result raised a question: why this specific choice?. We explored the Tryon's conjecture that 'the Universe must have a zero net value for all conserved quantities' to get some ideas on the specific values of this parameter for Bianchi type Universes.


I. INTRODUCTION
The General Relativity(GR) as formulated by Einstein is now 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 non gravitational 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 an ordinary differentiation of the energy momentum tensor T j i should vanish i.e. T j i,j = 0. The covariant formulation requires nontensorial fields. Obviously, the energy-momentum due to the gravitational field turns out to be non tensorial (pseudo tensor). The choice of this pseudo tensor is not unique and therefore it has led to the formulation of a number of prescriptions for the calculation of energy and momentum [2][3][4][5][6][7][8][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 coordinate independent energy-momentum complexes have been proposed by Møller [7], Komar [8] and Penrose [9]. But some of these coordinate dependent prescriptions are questioned for their limited applicability.
The issue of energy localization has been widely discussed in literature in the frame work of both General Relativity and teleparallel gravity. Misner et al. showed that energy is localized only for spherically symmetric systems [10]. Cooperstock and Sarracino counter commented the idea of Misner and established that if energy is localized in spherically symmetric system then it can be localizable in any space time [11]. Bondi perceived that, a non localizable form of energy is not admissible in General Relativity, 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][14][15]. Virbhadra showed that different prescriptions can provide same results when Kerr-Schild Cartesian coordinates are used [13]. Followed by Virbhadra, many researchers obtained interesting results on this pressing issue of energy localisation [16].
Contrary to the previous results, Gad explored about the failure of the agreement in some specific examples of space times [17][18][19][20]. Amidst the failures to settle the issue in the context of General Relativity, the energy-momentum has also been formulated in the frame work of teleparallel gravity and has attracted a lot of attention in recent times [21][22][23][24][25][26]. It has been concluded in some recent works that, the energy-momentum definitions are identical not only in General Relativity but also in teleparallel gravity [25,27,28].
Tryon anticipated the net energy of the Universe to be zero [29]. Albrow [30] also had a similar assumption on the net energy of the Universe. Rosen from a calculation of the net energy of a closed homogeneous isotropic universe described by a Friedmann-Robertson-Walker (FRW) metric using Einstein energy-momentum complex showed that the total energy of the Universe is zero everywhere [31]. Cooperstock and Israelit also found similar results for closed FRW Universe [32]. Johri et al. also obtained similar results for a closed FRW Universe in Landau-Lifshitz complex [33]. Vargas calculated the energy-momentum of FRW Universe in Landau-Lifshitz and Einstein prescription in the context of teleparallel gravity and obtained zero total energy of the Universe. In a recent work, Tripathy et al. [34] have obtained the energy and momentum of Bianchi type V I h (BV I h ) Universes in the framework of General Relativity in different prescriptions and have shown that the results can only agree for a specific value of the metric parameter h. They have also questioned on the basis of Tryon's conjecture that, why the specific spacetime requires the specific value of the parameter. In the present work, we have tried to investigate upon the question raised by Tripathy et al. by considering anisotropic Bianchi type Universes. It is worth to mention here that, anisotropic spacetimes 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. Banerjee and Sen [35], Xulu [36] and Aydögdu et al. [23] have obtained the total energy density of Bianchi type-I Universes to be zero everywhere using different nergy-momentum complexes either in General Relativity or in teleparallel gravity . Also, they have calculated the energy of LRS Bianchi II Universe to have consistent results [37]. Radinschi calculated the energy distribution of a Bianchi type V I 0 Universe using different prescriptions like Tolman, Bergman and Thomson and Møller to find zero total energy of the Universe [38]. In another work, Radinschi calculated the energy of Bianchi type V I 0 Universe using the Landu-Lifshitz, Papapetrou and Weinberg prescriptions and found zero net energy due to matter and fields [39]. Aygun and Tarhan have obtained the energy and momentum of Bianchi IV Universe in different complexes in the framework of both the General Relativity and teleparallel gravity [40].
The organisation of the paper is as follows: In Section 2, we present the basics of an anisotropic Bianchi type III Universe. In Section 3, the energy and momentum densities for the anisotropic Bianchi III Universe are obtained using some popular prescriptions.
Results of the present work are discussed and analysed basing upon the Tryon's conjecture advocating a null total energy state of the Universe in Section 4. At the end, in Section 5, the summary and conclusion the present work are presented. 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 8πG = c = 1, G and c being the Newtonian Gravitational constant and speed of light in vacuum respectively.

II. ANISOTROPIC BIANCHI TYPE III UNIVERSE
The Universe is observed to be mostly isotropic and can be well explained by the usual ΛCDM ( Λ dominated Cold Dark Matter) model. However, certain measurements of cosmic microwave background from Wilkinson Microwave Anisotropic Probe (WMAP) show some anomalous features of ΛCDM model at large scale [41]. These precise measurements suggest an asymmetric expansion of the Universe with one direction expanding in different manner than the other two transverse directions [42][43][44]. The Planck data [45] shows a slight red shifting of the primordial power spectrum of curvature perturbation from exact scale invariance. It can be inferred from the Planck data that usual ΛCDM model can not be a good fit at least at high multipoles. The issue of global anisotropy can be dealt in many ways.
However, a simple way is to modify the FRW model by considering asymmetric expansion along different spatial directions. In these sense, Bianchi type models play important roles.
The Bianchi type models are homogeneous having anisotropic spatial sections and are the exact solutions of Einstein field equations. In the present work, we have considered the Bianchi type III (BIII) model in its generalised form where A, B and C are the directional scale factors considered as functions of cosmic time t only. The present model is considered in such a manner that the exponent h is a constant of time and can assume any real values compatible to the real universe.
Different non vanishing components of the Einstein tensor G ij = R ij − 1 2 Rg ij for the above metric are Here R ij is the Ricci tensor, R is the Ricci scalar, g ij is the metric tensor. Also, G 10 = G 01 = R 10 . In the above equations an overhead dot over a field variable represents a time derivative.

III. ENERGY-MOMENTUM COMPLEXES
In this section, we present the general results of the energy and momentum of generalised  Table-I. The results for energy and momentum densities will be presented in general form of the directional scale factors A, B, C and the exponent h.
From these general results, one can easily obtain the energy and momentum by considering the time dependence of the scale factors. It is worth to mention here that, we restrict the definitions of the well known energy-momentum prescriptions to the frame work of General Relativity. In the following subsections we report the derived non vanishing components of the super potentials and the consequent energy and momentum densities. The derived energy and momentum densities are given in Table-II.

A. Einstein Energy-Momentum Complex
The required non-vanishing components of the H kl i are The components of energy and momentum densities can now be obtained as

B. Landau and Lifshitz Energy-Momentum Complex
For the the generalised BIII model, the non-vanishing components of λ iklm are obtained as Consequently, the energy and momentum densities in the Landau and Lifshitz prescription become

C. Papapetrou Energy-Momentum Complex
The required non-vanishing components of N iklm for the calculation of the energy (Σ 00 ) and momentum density (Σ α0 ) components are The energy density and momentum density components in the Papapetrou prescription are obtained from (7) as

D. Bergmann-Thompson Energy-Momentum Complex
The non-vanishing components of B km l for BIII Universe are Using eq. (9), the energy and momentum density components i.e. B 00 and B α0 , can be obtained as, The non-vanishing components of χ kl i are The energy and momentum density components for Møller energy-momentum complex are obtained as, Tryon's conjecture, we have calculated the energy of Bianchi V I h Universe in the Møller prescription.
The Bianchi V I h Universes are modelled through the metric The non vanishing components of χ kl i for Bianchi V I h Universes are Consequently, the energy and momentum density components for Møller energymomentum complex are The Møller energy for Bianchi V I h Universe is zero. However, the momentum of this model vanish only for h = −1.
In his interesting work, Tryon [29] assumed that the Universe has appeared from nowhere about 10 10 years ago. As per his thought, at the time of creation of the Universe, the con-  [50][51][52].
More or less, it is now an accepted fact that, our Universe is created out of nothing and its net energy is zero. If this conjecture is to be valid then all prescriptions for energy- in In view of the Tryon's conjecture that advocates a zero total energy of the Universe, it is suggested in the present work that, the Bianchi type Universes described through a general metric in (16) require that the sum α + β should vanish. Therefore, in the present work Bianchi type Universes must also be consistent to that.