Energy and momentum of Bianchi Type VI_h Universes

We obtain the energy and momentum of the Bianchi type VI_h universes using different prescriptions for the energy-momentum complexes in the framework of general relativity. The energy and momentum of the Bianchi VI_h universe are found to be zero for the parameter h = -1 of the metric. The vanishing of these results support the conjecture of Tryon that Universe must have a zero net value for all conserved quantities.This also supports the work of Nathan Rosen with the Robertson-Walker metric. Moreover, it raises an interesting question:"Why h=-1 case is so special?"


Introduction
The local distribution of energy and momentum has remained a challenging domain of research in the context of Einstein's general relativity. Einstein proposed the first energy-momentum complex [1] that follows the covariant conservation laws by including the energy and momenta of gravitational fields along with those of matter and nongravitational fields. The energy-momentum due to the gravitational field turns out to be a nontensorial object. The choice of the gravitational field pseudotensor (nontensor) is not unique and therefore, following Einstein, many authors prescribed different definitions of energy-momentum complexes based on the canonical approach (e.g., Tolman [2], Papapetrou [3], Landau and Lifshitz (LL) [4], Bergmann and Thompson (BT) [5], and Weinberg [6]). The Tolman definition is essentially the same as that of Einstein; however, these two definitions differ in form and sometimes it is easier to use Tolman's definition. This was explained by Virbhadra [7]. The main concern in the use of these definitions is that they are coordinate-dependent. However, with these definitions, meaningful and reasonable results can be obtained if "Cartesian coordinates" (also called quasi-Cartesian or quasi-Minkowskian for asymptotically Minkowskian space-times) are used. Some coordinate-independent definitions have also been proposed by Møller [8], Komar [9] and Penrose [10]. The coordinate-independent prescriptions, including the quasilocal mass of Penrose [10], were found to have some serious shortcomings as these are limited to a certain class of symmetries only (see in [7] and also references therein).
The issue of energy localization and the coordinate dependence of these definitions gained momentum with renewed interests after the works of Virbhadra and his collaborators (notably, Nathan Rosen, the most famous collaborator of Albert Einstein, of the Einstein-Rosen bridge, the EPR paradox, and the Einstein-Rosen gravitational waves fame) who found a striking similarity in the results for different energy-momentum prescriptions. They considered numerous space-times [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and obtained seminal results that rejuvenated this field of research.
2 Advances in High Energy Physics Virbhadra [7] further investigated whether or not these energy-momentum complexes lead to the same results for the most general nonstatic spherically symmetric metric and found that they disagree. Virbhadra and his collaborators [13][14][15][16][17][18][19][20][21][22][23][24][25] observed that if the calculations of energy-momentum are done in Kerr-Schild Cartesian coordinates, then the energy-momentum complexes of Papapetrou [3], Landau and Lifshitz [4], and Weinberg [6] produce the same result as in the Einstein definition. However, if the computations are made in Schwarzschild Cartesian coordinates, these energymomentum complexes disagree [7]. Xulu [26] confirmed this by obtaining the energy and momentum for the most general nonstatic spherically symmetric system using the Møller definition and found a different result in general form than those obtained by Einstein's definition. Xulu and others  obtained many important results in this field. However, till now, there is no completely acceptable definition for energy and momentum distributions in general relativity even though prescriptions in teleparallel gravity claim to provide a satisfactory solution to this problem [55,56]. Gad calculated the energy and momentum densities of stiff fluid case using the prescriptions of Einstein, Begmann-Thompson, and Landau-Liftshitz in both the general relativity and the teleparallel gravity and found that different prescriptions do not provide the same results in both theories. Also, they have shown that both the general relativity and the teleparallel gravity are equivalent [56]. Similar results have also been obtained by Aygün and coworkers in [57,58] where the authors have concluded that energy-momentum definitions are identical not only in general relavity but also in teleparallel gravity.
Bianchi type models are spatially homogeneous and anisotropic universe models. These models are nine in number, but their classification permits splitting them in two classes. There are six models in class (I, II, VI 1 , VII, VIII, and IX) and five in class (III, IV, V, VI ℎ , and VII ℎ ). Spatially homogeneous cosmological models play an important role in understanding the structure and properties of the space of all cosmological solutions of Einstein field equations. These spatially homogeneous and anisotropic models are the exact solutions of Einstein field equations and are more general than the Friedman models in the sense that they can provide interesting results pertaining to the anisotropy of the universe. Here, it is worth to mention that the issue of global anisotropy has gained a lot of research interest in recent times. The standard cosmological model (ΛCDM) based upon the spatial isotropy and flatness of the universe is consistent with the data from precise measurements of the CMB temperature anisotropy [59] from the Wilkinson Microwave Anisotropy Probe (WMAP). However, the ΛCDM model suffers from some anomalous features at large scale and signals a deviation from the usual geometry of the universe. Recently released Planck data [60][61][62][63] show a slight red shift of the primordial power spectrum from the exact scale invariance. It is clear from the Planck data that the ΛCDM model does not fit well to the temperature power spectrum at low multipoles. Also, precise measurements from WMAP predict asymmetric expansion with one direction expanding differently from the other two transverse directions at equatorial plane [64] which signals a nontrivial topology of the large scale geometry of the universe (see [65,66] and references therein).
In recent times this pressing issue of the energy and momentum localization has been studied widely by many authors using different space-times and definitions of energymomentum complexes. The importance of the study of energy and momentum distribution lies in the fact that it helps us getting an idea of the effective gravitational mass of metrics of certain symmetries and can put deep insight into the gravitational lensing phenomena [67][68][69][70][71][72][73]. In fact, the energy-momentum distribution in space-time can be interpreted as effective gravitational mass if the space-time has certain symmetry. However, negative and positive energy distributions in space-times always indicate divergent and convergent gravitational lensing, respectively.
Using Einstein definition, Banerjee and Sen [74] studied the energy distribution with Bianchi type I (BI) space-time. Xulu [33] calculated the total energy in BI universes using the prescriptions of LL, Pappapetrou, and Weinberg. Radinschi [39] calculated the energy of a model of the universe based on the Bianchi type VI 0 metric using the energy-momentum complexes of LL and of Papapetrou. She found that the energy due to the matter plus field is equal to zero. Aydogdu and Salti [75], using the Møller's tetrad, investigated the energy of the BI universe. In another work, Aydogdu and Salti [76] calculated the energy of the LRS Bianchi type II metric to get consistent results.
In this paper, we obtain the energy and momentum for a more general homogeneous and anisotropic Bianchi type VI ℎ metric and its transformation by using different prescriptions for the energy-momentum complex in general relativity. The Bianchi type VI ℎ model has already been shown to provide interesting results in cosmology in connection with the late time accelerated expansion of the universe when the contribution to the matter field comes from one-dimensional cosmic strings and bulk viscosity [77]. In the present work, we have used the convention that Latin indices take values from 0 to 3 and Greek indices run from 1 to 3. We use geometrized units where = 1 and = 1. The composition of the paper is as follows. In Section 2, we have written the energy-momentum tensor for the Bianchi type VI ℎ spacetime. In Section 3, the Einstein energy-momentum complex is discussed, following which we investigate the energymomentum complex of Landau and Lifshitz, Papapetrou, and Bergmann Thomson for the assumed metric in the subsequent subsections. In the last section, we summarize our results.

Bianchi Type VI ℎ Space-Time
We have considered the Bianchi type VI ℎ space-time in the form The metric potentials , , and are functions of cosmic time only. Further, , = 1, 2, 3, 0, respectively, denote the coordinates , , , and . The exponent ℎ is timeindependent and can assume integral values in the range ℎ = −1, 0, 1. It should be mentioned here that these are not vacuum solutions as ̸ = 0 for all and . For metric (1), the determinant of the metric tensor and the contravariant components of the tensor are, respectively, given as In general relativity, the energy-momentum tensor is given by , where is the Ricci tensor and is the Ricci scalar. The nonvanishing components of the energymomentum tensor for the Bianchi type VI ℎ space time are given below (this is not a new result; however, we put it here because we need it for analysing and discussing results): where the overhead dots hereafter denote ordinary time derivatives.

Einstein Energy-Momentum Complex. The Einstein energy-momentum complex is [1]
Θ 0 0 and Θ 0 stand for the energy and momentum density components, respectively. The energy and momentum components are obtained through a volume integration By applying Gauss' theorem, the above equation can also be reduced to where is the outward unit normal vector over the infinitesimal surface element . 0 and stand for the energy and momentum components, respectively. The required nonvanishing components of for line element (1) are given by Using (8), we obtain the components of energy and momentum densities as If the dependence of , , and on the time coordinate were known, one could evaluate the surface integral. It is clear from the above results that, for the specific choices of ℎ, that is, ℎ = 0 and 1, the energy of the VI ℎ universe in the Einstein prescription is not zero. However, it is interesting to note that the energy and momentum densities vanish for ℎ = −1.

Landau and Lifshitz Energy-Momentum Complex. The symmetric Landau and Lifshitz energy-momentum complex is [4]
Here 00 and 0 are the energy and momentum density components.
The energy and momentum can be defined through the volume integral Using Gauss' theorem, the energy and momentum components become Using these components in (13), we obtain where = √ 2 + 2 + 2 . For the Landau and Lifshitz prescription, the energy of the universe is nonzero for ℎ = 0 and 1 and it vanishes for ℎ = −1.
Here Σ 00 and Σ 0 are, respectively, the energy and energy current (momentum) density components. The energy and momentum can be defined as For time-independent metrics, one can have The nonvanishing components of N required to obtain the energy and momentum density components for the space-time described by line element (1) are The Papapertrou energy and energy current density components are obtained by using the above components in (21) as Like the previous cases, it can be concluded from the above components that the energy of the universe in the Papapetrou prescription is nonzero for ℎ = 0 and 1 and it becomes zero for ℎ = −1. Therefore, the energy and momentum components both are zero for ℎ = −1.

Bergmann-Thompson Energy-Momentum Complex. The Bergmann-Thompson energy-momentum complex is [5]
where Here B 00 and B 0 are the energy and momentum densities, respectively. The energy and momentum can be defined as Using Gauss's theorem, the energy and momentum components can be expressed as Advances in High Energy Physics 5 In order to calculate the energy and momentum distribution for Bianchi type VI ℎ space-time using the Bergmann and Thompson energy-momentum complex, we require the following nonvanishing components of B : The energy and momentum density components can be obtained by using (27) in (23) as It is clear that the energy of the Bianchi VI ℎ universe is zero in Bergmann-Thompson prescription ∀ℎ = 0, ±1. However, the energy and all momentum components vanish for ℎ = −1.

Summary
In the present study, we have obtained energy and momentum distributions for the spatially homogeneous and anisotropic Bianchi type VI ℎ metric using Einstein, Landau and Lifshitz, Papapetrou, and Bergmann-Thompson complexes in the framework of general relativity. Bianchi type VI ℎ space-time has an edge over the usual Friedman-Robertson-Walker (FRW) in the sense that it can handle the anisotropic spatial expansion. In the present study, we found that the energy and momentum vanish for the Bianchi VI −1 universe. The only exception to this is the case of Landau and Lifshitz where although energy density components vanish, the momentum density components do not vanish in general for ℎ = −1. Virbhadra (refer to [7] and references in this paper) however pointed out that the Landau and Lifshitz complex does not work as good as the Einstein complex and the latter is the best for energy-momentum calculations. Other energy-momentum complexes agree with that of Einstein's for the ℎ = −1 case. Equation (3) shows that the energy and momentum densities due to matter and nongravitational fields are nonzero even for ℎ = −1. However, as energy-momentum complexes include effects of the gravitational field as well, the total result comes to be zero.
The observation of the cosmic microwave background radiation by Penzias and Wilson [78,79] in the year 1965 strongly supports that some version of the big bang theory is correct and it also suggested a remarkable conjecture regarding the total energy and momentum of the universe. Tryon [80] assumed that our universe appeared from nowhere about 10 10 years ago. He pondered that the conventional laws of physics need not have been violated at the time of creation of the universe. Therefore, he proposed that our universe must have a zero net value for all conserved quantities. Further, he presented arguments suggesting that the net energy and momentum of our universe may be indeed zero. Tryon gave a particular big bang model and according to that our universe is a fluctuation of the vacuum and he predicted a homogeneous, isotropic, and closed universe consisting of equal amount of matter and antimatter. Tryon, in the same paper, also mentioned an excellent topological argument by Bergmann that any closed universe has zero energy. Later, Rosen [81], in the year 1994, considered a closed homogeneous isotropic universe described by the Friedman-Robertson-Walker (FRW) metric and found that the total energy of the universe is zero (sadly, he passed away in the following year (1995)). Excited by these results, Xulu [33] studied energy and momentum in Bianchi type I universes and his results supported the conjecture of Tryon.
In view of the satisfying results mentioned in the above paragraph, the outcome of our research that the energy and momentum of the Bianchi VI −1 universe are zero is indeed a very important result. However, one would ask "Why is this true only for the ℎ = −1 case?" History of science has records that coincidence of results usually points out something very important. Thus, it remains to be investigated: "Why is the ℎ = −1 case so special?" It is likely that the outcome of these investigations would have important implications for general relativity and relativistic astrophysics.

Disclosure
A part of this collaborative research work was done during an International Workshop on Introduction to Research in Einstein's General Relativity at NIT, Patna (India).