The energy of the universe in the Bianchi type-II cosmological model
aa r X i v : . [ g r- q c ] O c t The energy of the universe in the Bianchi type-IIcosmological model
I-Ching Yang Department of Applied Science, National Taitung University,Taitung 95002, Taiwan (R.O.C.)
ABSTRACT
To investigate the energy of Bianchi type-II cosmological model, I used theenergy-momentum complexes of Einstein and Møller and obtained the zerototal energy in these two prescriptions. This result reinforces the viewpointof Albrow and Tryon that the universe must have a zero net value for allconserved quantities and be equivalent to the previous works of Nester et al.and Aydogdu et al.PACS No.:04.20.Cv, 98.80.Jk.Keywords: Bianchi type-II cosmological model, energy-momentum complexesof Einstein and Møller E-mail:[email protected]
1n view of generally covariant theory, the evolution of Einstein’s gravitytheory, general relativity (GR), is underdetermined. One problematic issue isthe energy localization for the gravitational field. In continuum mechanics,the most general conservation form is given by a continuity equation in a“differential form” ∂ µ J µ = 0, and the conserved quantity would be led. Nev-ertheless, the differential conservation law in curved space-time will becomea covariant derivative ∇ µ T µν = 1 √− g ∂∂x µ (cid:0) √− gT µν (cid:1) − ∂g µα ∂x ν T µα = 0 , (1)and does not lead to any conserved quantity. Einstein proposed the energy-momentum pseudotensor from the gravitational field t µν [1], follows from1 √− g ∂∂x µ (cid:0) √− gt µν (cid:1) ≡ − ∂g µα ∂x ν T µα , (2)and led to energy-momentum complexΘ µν = √− g ( T µν + t µν ) , (3)which satisfies the differential conservation form ∂ µ Θ µν = 0. Thus, the con-served quantity was defined as P ν = Z Θ ν d x. (4)In mathematically, antisymmetric U µρν in their two indices µ and ρ would beintroduced by Θ µν ≡ ∂ U µρν ∂x ρ , (5)and be called “ superpotential ”. There are various energy-momentum com-plexes which are pseudotensors, including those of Einstein [2], Tolman [3],Papapetrou [4], Bergmann-Thompson [5], Laudau-Lifshitz [6], Møller [7] andWeinberg [8].Another idea of conserved quantity proposed by Penrose [9] is “ quasilocal ” (i.e. associated with a closed 2-surface) energy-momentum according asthe 4-coariant expression for the gravitational Hamiltonian. Hence the con-served quantity associated with a local spacetime displacement N of a finitespacelike hypersurface Σ is determined by the integral H ( N , Σ) = Z Σ N µ H µ + I ∂ Σ B ( N ) . (6)2or any choice of N this expression defines a conserved quasilocal quanity,and there have recently been many quasilocal proposals [9, 10, 11, 12, 13].From Eq.(4), the energy-momentum within a finite region is P ( N ) = Z Σ N ν √− g [ T µν + t µν ] d Σ µ (7)= Z Σ (cid:20) N ν √− g (cid:18) T µν − κ G µν (cid:19) + ∂ ρ ( N ν U µρν ) (cid:21) d Σ µ , (8)where √− gt µν = − κ √− gG µν + ∂ ρ U µρν (9)and κ is Einstein’s gravitational constant. Note that H ν is the covariant formof the ADM Hamiltonian density, which has a vanishing numerical value.The boundary term 2-surface integral is determined by the superpotential.Consequently, Nester et al. [14] show that a pseudotensor corresponds to aHamiltonian boundary term P ( N ) = Z Σ N ν H ν + I ∂ Σ B ( N ) ≡ H ( N ) . (10)Furthermore, there have been several studies aimed at uncovering the totalenergy of the cosmological model by using quasilocal energy-momentum [15,16, 17, 18] and energy-momentum pseudotensor [19, 20, 21, 22, 23]. In thisarticle I would like to investigate the energy of Bianchi type-II cosmologicalmodel using energy-momentum complex of Einstein and Møller. Throughthe paper I use geometrized units ( G = 1, c = 1), and follow the traditionthat Latin indices run from 1 to 3 and Greek indices run from 0 to 3.To describe the large scale behavior of the Universe spatially homoge-neous and anisotropic cosmological models are used generally. Suppose thatthe four-dimensional spacetime manifold can be foliated by a family of homo-geneous space-like hypersurfaces Σ t labeled by a constant time t . Spatial ho-mogeneity means that hypersurface is invariant under the three-dimensionalLie group G . The study of G led to the Bianchi classification of spatiallyhomogeneous universes [24]. The spacetime orthonormal coframe of cosmo-logical model has the form θ = dt and θ a = h ak ( t ) σ k , and the spatiallyhomogeneous frame will satisfy dσ k = 12 C kij σ i ∧ σ j , (11)3here the C kij are certain constants. There are nine Bianchi type distin-guished by the particular form of the structure constants C kij . They fallinto two special classes: class A (Types I , II , VI , VII , VIII , IX) and class B(type III , IV , V , VI h , VII h ). Here Bianchi type-II cosmological model [25] isconsidered as ds = η µν θ µ ⊗ θ ν , η µν = diag(1 , − , − , −
1) (12)and its spacetime orthonormal coframe is θ = dt,θ = Adx,θ = B ( dy − xdz ) ,θ = Cdz. (13)In local coordinate x µ the line element Eq.(12) can be written in the form ds = g µν dx µ ⊗ dx ν , (14)and the matrix representation of metric tensor g µν of Bianchi type-II cosmo-logical model would be expressed as g µν = − A − B xB xB − x B − C , (15)where A, B and C are function of t . Here, the locally rotationally symmetric(LRS) Bianchi type-II cosmological model [26] is a special case of Eq.(14) asselecting C = A .At the beginning the definition of the Einstein energy-momentum com-plex is given as Θ µν = 116 π ∂H µσν ∂x σ , (16)with the Freud’s superpotential H µσν = g νρ √− g ∂∂x α [( − g ) ( g µρ g σα − g σρ g µα )] . (17)4ence four-momentum will be obtained by P ν = Z Θ ν d x. (18)The energy component of Einstein energy-momentum complex is shown as E E = Z Θ d x = 116 π Z ∂H i ∂x i d x. (19)According to Gauss’s theorem, Eq.(18) will become a surface integral E E = 116 π I H i · ˆ n i dS (20)over the surface dS with the outward normal ˆ n i . However, all componentsof H i are equal to zero. From Eq.(19), the energy component of Einsteinenergy-momentum complex is E E = 0 . (21)Afterward according to the definition of the Møller energy-momentum com-plex [7] Θ µν = 18 π ∂χ µσν ∂x σ (22)where the Møller’s superpotential is χ µσν = √− g (cid:18) ∂g να ∂x β − ∂g νβ ∂x α (cid:19) g µβ g σα , (23)the energy component of Møller energy-momentum complex is exhibited by E M = Z Θ d x = 18 π Z ∂χ i ∂x i d x, (24)As well, according to Gauss’s theorem, the energy component can be writtenas E M = 18 π I χ i · ˆ n i dS. (25)Because of no nonvanishing components of χ i , the energy component ofMøller energy-momentum complex is E M = 0 , (26)In conclusion, the zero total energy of Bianchi type-II cosmological mod-els in the Einstein and Møller prescription have been obtained. Albrow [27]and Tryon [28] supposed that the universe may have arisen as a quantum5uctuation of the vacuum and must have a zero net value for all conservedquantities. Thus, the total energy vanishes everywhere means that the en-ergy contributions from the material sources, including dark matter and darkenergy , and gravitational fields inside an arbitrary two-surface boundary ∂ Σof the 3-hypersurface Σ cancel each other. Recently, Nester, So and theircollaborator [16, 17, 18] indicated that the energy vanishes for all regionsin Bianchi type-II cosmological model and my result and theirs are thesame. Furthermore Aydogdu’s result [29, 30] in locally rotationally sym-metric Bianchi type-II cosmological model using Einstein and Møller energy-momentum complex is equivalent to the expression of my result as C = A .The results of this paper also are consistent with those given in the pre-vious works of Rosen [19], Johri et al. [20], Garecki [31] and Vargas [32]for Friedman-Lemaˆıtre-Robertson-Walker universe, and of Banerjee-Sen [21]and Xulu [22] for Bianchi type-I cosmological model. On the other hand,the difference of energy between the Einstein prescription E E and the Møllerprescription E M is defined [33] as∆ E = E E − E M , (27)and its value in this article is ∆ E = 0. For these spacetime structures withtwo event horizons, like as stringy dyonic black hole [34], charged dilaton-axion black hole [35] and Schwarzschild-de Sitter black hole [36], the differ-ence of energy ∆ E will not be equal to zero and is dependent on the heatflow passing through the boundary ∂ Σ. Whatever, Eq.(27) would be inter-esting for studying the thermodynamics of cosmological model, especially theevolution of cosmology.
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