Advances in High Energy Physics
Volume 2015 (2015), Article ID 705262, 8 pages
http://dx.doi.org/10.1155/2015/705262

Research Article

Energy and Momentum of Bianchi Type Universes

S. K. Tripathy,1,2 B. Mishra,1,3 G. K. Pandey,1,4 A. K. Singh,1,5 T. Kumar,1,5 and S. S. Xulu⁶

¹National Institute of Technology, Patna, Bihar 800005, India
²Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha 759146, India
³Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad Campus, Hyderabad 500078, India
⁴Patna Science College, Patna University, Patna, Bihar 800005, India
⁵Physics Department, Patna University, Patna, Bihar 800005, India
⁶Department of Computer Science, University of Zululand, Kwa-Dlangezwa 3886, South Africa

Received 25 April 2015; Accepted 28 June 2015

Academic Editor: Elias C. Vagenas

Copyright © 2015 S. K. Tripathy 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 SCOAP³.

Abstract.

We obtain the energy and momentum of the Bianchi type universes using different prescriptions for the energy-momentum complexes in the framework of general relativity. The energy and momentum of the Bianchi universes are found to be zero for the parameter of the metric. The vanishing of these results supports the conjecture of Tryon that the 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 is the case so special?”

1. 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 quasi-local 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 [11–25] and obtained seminal results that rejuvenated this field of research.

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 [13–25] 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 energy-momentum 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 [26–54] 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, , , , and ) and five in class (, , , , and ). 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 (CDM) 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 CDM model suffers from some anomalous features at large scale and signals a deviation from the usual geometry of the universe. Recently released Planck data [60–63] show a slight red shift of the primordial power spectrum from the exact scale invariance. It is clear from the Planck data that the CDM 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 energy-momentum 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 [67–73]. 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 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. Aydoğdu 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 metric and its transformation by using different prescriptions for the energy-momentum complex in general relativity. The Bianchi type 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 and . The composition of the paper is as follows. In Section 2, we have written the energy-momentum tensor for the Bianchi type space-time. In Section 3, the Einstein energy-momentum complex is discussed, following which we investigate the energy-momentum 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.

2. Bianchi Type Space-Time

We have considered the Bianchi type space-time in the form

The metric potentials , , and are functions of cosmic time only. Further, , , respectively, denote the coordinates , , , and . The exponent is time-independent and can assume integral values in the range . It should be mentioned here that these are not vacuum solutions as 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 energy-momentum tensor for the Bianchi type 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.

3. Energy-Momentum Complexes

3.1. Einstein Energy-Momentum Complex

The Einstein energy-momentum complex is [1]

where

and 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 . 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 , that is, and 1, the energy of the universe in the Einstein prescription is not zero. However, it is interesting to note that the energy and momentum densities vanish for .

3.2. Landau and Lifshitz Energy-Momentum Complex

The symmetric Landau and Lifshitz energy-momentum complex is [4]

where

Here and 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

The required nonvanishing components of for the present model are obtained as

Using these components in (13), we obtain

where . For the Landau and Lifshitz prescription, the energy of the universe is nonzero for and and it vanishes for .

3.3. Papapetrou Energy-Momentum Complex

The symmetric energy-momentum complex of Papapetrou is given by [3]

where

with

Here and 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 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 and and it becomes zero for . Therefore, the energy and momentum components both are zero for .

3.4. Bergmann-Thompson Energy-Momentum Complex

The Bergmann-Thompson energy-momentum complex is [5]

where

Here and 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

In order to calculate the energy and momentum distribution for Bianchi type space-time using the Bergmann and Thompson energy-momentum complex, we require the following nonvanishing components of :

The energy and momentum density components can be obtained by using (27) in (23) as

It is clear that the energy of the Bianchi universe is zero in Bergmann-Thompson prescription . However, the energy and all momentum components vanish for .

4. Summary

In the present study, we have obtained energy and momentum distributions for the spatially homogeneous and anisotropic Bianchi type metric using Einstein, Landau and Lifshitz, Papapetrou, and Bergmann-Thompson complexes in the framework of general relativity. Bianchi type 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 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 . 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 case. Equation (3) shows that the energy and momentum densities due to matter and nongravitational fields are nonzero even for . 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 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 universe are zero is indeed a very important result. However, one would ask “Why is this true only for the 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 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).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very thankful to their mentor, Dr. K. S. Virbhadra, for his guidance and incredible support throughout the preparation of this project. The authors would like to thank National Institute of Technology (NIT), Patna, India, for inviting them to attend the Workshop on Introduction to Research in Einstein’s General Relativity under TEQIP programme held at NIT, Patna, during which this collaborative research work was initiated. Also, the authors acknowledge the warm hospitality extended by the Director Professor A. De, head of physics department Professor N. R. Lall, and Professor B. K. Sharma of NIT, Patna (India). S. K. Tripathy and B. Mishra like to thank Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune, India, for providing facility and support during an academic visit where a part of this work was carried out. S. S. Xulu thanks the University of Zululand (South Africa) for all support. G. K. Pandey thanks his friends for helping him in checking the calculations.

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