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Energy and Momentum of Bianchi Type Universes Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 705262, 8 pages http://dx.doi.org/10.1155/2015/705262 Research Article Energy and Momentum of Bianchi Type VIℎ Universes S. K. Tripathy,1,2 B. Mishra,1,3 G. K. Pandey,1,4 A. K. Singh,1,5 T. Kumar,1,5 andS.S.Xulu6 1 National Institute of Technology, Patna, Bihar 800005, India 2DepartmentofPhysics,IndiraGandhiInstituteofTechnology,Sarang,Dhenkanal,Odisha759146,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 Correspondence should be addressed to S. K. Tripathy; tripathy [email protected] 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. We obtain the energy and momentum of the Bianchi type VIℎ universes using different prescriptions for the energy-momentum complexes in the framework of general relativity. The energy and momentum of the Bianchi VIℎ universes are found to be zero for the parameter ℎ=−1of 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 ℎ=−1case so special?” 1. Introduction they are coordinate-dependent. However, with these defini- tions, meaningful and reasonable results can be obtained if The local distribution of energy and momentum has “Cartesian coordinates” (also called quasi-Cartesian or quasi- remained a challenging domain of research in the context Minkowskian for asymptotically Minkowskian space-times) of Einstein’s general relativity. Einstein proposed the first are used. Some coordinate-independent definitions have also energy-momentum complex [1]thatfollowsthecovariant been proposed by Møller [8], Komar [9]andPenrose[10]. The conservation laws by including the energy and momenta of coordinate-independent prescriptions, including the quasi- gravitationalfieldsalongwiththoseofmatterandnongravita- local mass of Penrose [10], were found to have some serious tional fields. The energy-momentum due to the gravitational shortcomings as these are limited to a certain class of field turns out to be a nontensorial object. The choice ofthe symmetries only (see in [7] and also references therein). gravitational field pseudotensor (nontensor) is not unique The issue of energy localization and the coordinate and therefore, following Einstein, many authors prescribed dependence of these definitions gained momentum with different definitions of energy-momentum complexes based renewed interests after the works of Virbhadra and his on the canonical approach (e.g., Tolman [2], Papapetrou collaborators (notably, Nathan Rosen, the most famous col- [3], Landau and Lifshitz (LL) [4], Bergmann and Thompson laborator of Albert Einstein, of the Einstein-Rosen bridge, (BT) [5], and Weinberg [6]). The Tolman definition is the EPR paradox, and the Einstein-Rosen gravitational waves essentially the same as that of Einstein; however, these two fame) who found a striking similarity in the results for definitions differ in form and sometimes it is easier to use different energy-momentum prescriptions. They considered Tolman’s definition. This was explained by Virbhadra7 [ ]. numerous space-times [11–25] and obtained seminal results The main concern in the use of these definitions isthat that rejuvenated this field of research. 2 Advances in High Energy Physics Virbhadra [7] further investigated whether or not these signals a nontrivial topology of the large scale geometry of the energy-momentum complexes lead to the same results for universe (see [65, 66] and references therein). the most general nonstatic spherically symmetric metric and In recent times this pressing issue of the energy and found that they disagree. Virbhadra and his collaborators [13– momentum localization has been studied widely by many 25] observed that if the calculations of energy-momentum authors using different space-times and definitions of energy- aredoneinKerr-SchildCartesiancoordinates,thenthe momentum complexes. The importance of the study of energy-momentum complexes of Papapetrou [3], Landau energy and momentum distribution lies in the fact that it and Lifshitz [4], and Weinberg [6]producethesameresultas helpsusgettinganideaoftheeffectivegravitationalmass in the Einstein definition. However, if the computations are of metrics of certain symmetries and can put deep insight made in Schwarzschild Cartesian coordinates, these energy- into the gravitational lensing phenomena [67–73]. In fact, momentum complexes disagree [7]. Xulu [26]confirmed the energy-momentum distribution in space-time can be this by obtaining the energy and momentum for the most interpreted as effective gravitational mass if the space-time general nonstatic spherically symmetric system using the has certain symmetry. However, negative and positive energy Møller definition and found a different result in general distributions in space-times always indicate divergent and form than those obtained by Einstein’s definition. Xulu and convergent gravitational lensing, respectively. others [26–54] obtained many important results in this field. Using Einstein definition, Banerjee and Sen [74]studied However, till now, there is no completely acceptable definition the energy distribution with Bianchi type I (BI) space-time. for energy and momentum distributions in general relativity Xulu [33] calculated the total energy in BI universes using the even though prescriptions in teleparallel gravity claim to prescriptions of LL, Pappapetrou, and Weinberg. Radinschi provide a satisfactory solution to this problem [55, 56]. [39] calculated the energy of a model of the universe based Gad calculated the energy and momentum densities of stiff on the Bianchi type VI0 metric using the energy-momentum fluid case using the prescriptions of Einstein, Begmann- complexesofLLandofPapapetrou.Shefoundthattheenergy Thompson, and Landau-Liftshitz in both the general rela- due to the matter plus field is equal to zero. Aydogdu˘ and tivity and the teleparallel gravity and found that different Salti [75], using the Møller’s tetrad, investigated the energy prescriptions do not provide the same results in both theories. of the BI universe. In another work, Aydogdu and Salti [76] Also, they have shown that both the general relativity and calculated the energy of the LRS Bianchi type II metric to get the teleparallel gravity are equivalent [56]. Similar results consistent results. have also been obtained by Aygun¨ and coworkers in [57, 58] In this paper, we obtain the energy and momentum for a where the authors have concluded that energy-momentum more general homogeneous and anisotropic Bianchi type VIℎ definitions are identical not only in general relavity but also metric and its transformation by using different prescriptions in teleparallel gravity. for the energy-momentum complex in general relativity. Bianchi type models are spatially homogeneous and The Bianchi type VIℎ model has already been shown to anisotropic universe models. These models are nine in num- provide interesting results in cosmology in connection with ber, but their classification permits splitting them in two thelatetimeacceleratedexpansionoftheuniversewhenthe classes. There are six models in class (I, II, VI1, VII, VIII, contribution to the matter field comes from one-dimensional and IX) and five in class (III, IV, V, VIℎ,andVIIℎ). Spatially cosmic strings and bulk viscosity [77]. In the present work, homogeneous cosmological models play an important role we have used the convention that Latin indices take values in understanding the structure and properties of the space from0to3andGreekindicesrunfrom1to3.Weuse of all cosmological solutions of Einstein field equations. geometrized units where =1and=1. The composition These spatially homogeneous and anisotropic models are of the paper is as follows. In Section 2,wehavewrittenthe the exact solutions of Einstein field equations and are more energy-momentum tensor for the Bianchi type VIℎ space- general than the Friedman models in the sense that they can time. In Section 3, the Einstein energy-momentum complex provide interesting results pertaining to the anisotropy of the is discussed, following which we investigate the energy- universe. Here, it is worth to mention that the issue of global momentum complex of Landau and Lifshitz, Papapetrou, anisotropy has gained a lot of research interest in recent times. andBergmannThomsonfortheassumedmetricinthe The standard cosmological model (ΛCDM) based upon the subsequent subsections. In the last section, we summarize our spatial isotropy and flatness of the universe is consistent with results. the data from precise measurements of the CMB temperature anisotropy [59] from the Wilkinson Microwave Anisotropy 2. Bianchi Type VIℎ Space-Time Probe (WMAP). However, the ΛCDM model suffers from some anomalous features
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