Conformal Killing Vectors in LRS Bianchi Type V Spacetimes

In this note, we investigate conformal Killing vectors (CKVs) of locally rotationally symmetric (LRS) Bianchi type V spacetimes. Subject to some integrability conditions, CKVs up to implicit functions of (t,x) are obtained. Solving these integrability conditions in some particular cases, the CKVs are completely determined, obtaining a classification of LRS Bianchi type V spacetimes. The inheriting conformal Killing vectors of LRS Bianchi type V spacetimes are also discussed.     

Matter Collineations in Kantowski-Sachs, Bianchi Types I and III Spacetimes

The matter collineation classifications of Kantowski-Sachs, Bianchi types I and III space times are studied according to their degenerate and non-degenerate energy-momentum tensor. When the energy-momentum tensor is degenerate, it is shown that the matter collineations are similar to the Ricci collineations with different constraint equations. Solving the constraint equations we obtain some cosmological models in this case. Interestingly, we have also found the case where the energy-momentum tensor is degenerate but the group of matter collineations is finite dimensional. When the energy-momentum tensor is non-degenerate, the group of matter collineations is finite-dimensional and they admit either four which coincides with isometry group or ten matter collineations in which four ones are isometries and the remaining ones are proper.     

Tilted Bianchi Type-I Cosmological Model in Lyra Geometry

Tilted Bianchi-I cosmological model is investigated in the frame work of Lyra (in Math. Z. 54:52, 1951) Geometry. Exact solutions to the field equations are derived when the metric potentials are functions of cosmic time only. Some physical and geometrical properties of the solutions are also discussed.     

Energy Conditions and Conservation Laws in LRS Bianchi Type I Spacetimes via Noether Symmetries

In this paper, we have completely classified the locally rotationally symmetric (LRS) Bianchi type I spacetimes via Noether symmetries (NS). The usual Lagrangian corresponding to LRS Bianchi type I metric is used to find the set of determining equations. To achieve a complete classification, these determining equations are generally integrated to find the components of NS vector field and the metric coefficients. During this procedure, several cases arise which give different Noether algebras of dimension 5,..., 9, 11, and 17. A comparison is established between the obtained NS and the Killing and homothetic vectors. Corresponding to all NS generators, the conservation laws are stated by using Noether's theorem. The metrics which we have obtained as a result of our classification are shown to be anisotropic or perfect fluids which satisfy certain energy conditions.     

Comment: Note on Matter Collineations in Kantowski-Sachs, Bianchi Types I and III Spacetimes

We show that the classification of Kantowski-Sachs, Bianchi Types I and III spacetimes admitting Matter Collineations (MCs) presented in a recent paper by Camci et al. [Camci, U., and Sharif, M. Matter Collineations in Kantowski-Sachs, Bianchi Types I and III Spacetimes, (2003) Gen. Rel. Grav. 35, 97-109.] is incomplete. Furthermore for these spacetimes and when the Einstein tensor is non-degenerate, we give the complete Lie Algebra of MCs and the algebraic constraints on the spatial components of the Einstein tensor.     

Anisotropic Dark Energy Bianchi Type-II Cosmological Models in Lyra Geometry

The spatially homogeneous and totally anisotropic Bianchi type-II cosmological model has been discussed in general relativity in the presence of a hypothetical anisotropic dark energy fluid with constant deceleration parameter within the frame work of Lyra’s manifold with uniform and time varying displacement field vector. With the help of special law of variation for Hubble’s parameter proposed by Bermann (Nuovo Cimento 74B:182, 1983) a dark energy cosmological model is obtained in this theory. We use the power law relation between average Hubble parameter H and average scale factor R to find the solution. The assumption of constant deceleration parameter leads to two models of universe, i.e. power law model and exponential model. Some physical and kinematical properties of the model are also discussed.     

Bianchi Type I String Dust Magnetized Cosmological Models in Lyra Geometry

Bianchi type I string dust cosmological models in the presence and absence of magnetic field in the frame work of Lyra geometry are investigated. To get the deterministic model of the universe, we assume that the eigenvalue （σ^11） of shear tensor （σ^ii） is proportional to expansion （θ）. This leads to A = （BC）^n, where A, B, C are metric potentials and n is a constant. To discuss the results in terms of cosmic time t, we have considered n = 1. The physical and geometrical aspects＇ of the models and singularities in the models are also discussed.     

Bianchi Type I String Dust Magnetized Cosmological Models in Lyra Geometry

Bianchi type I string dust cosmological models in the presence and absence of magnetic field in the frame work of Lyra geometry are investigated. To get the deterministic model of the universe, we assume that the eigenvalue (σ₁¹ ) of shear tensor (σ_i ^j ) is proportional to expansion (θ ). This leads to A =(BC)ⁿ, where A, B, C are metric potentials and n is a constant. To discuss the results in terms of cosmic time t, we have considered n = 1. The physical and geometrical aspects of the models and singularities in the models are also discussed.     

Bianchi Type V Barotropic Perfect Fluid Cosmological Model in Lyra Geometry

We have investigated Bianchi Type V barotropic perfect fluid cosmological model in Lyra geometry. To get the deterministic model of the universe, we have assumed the barotropic perfect fluid condition p=γ ρ, 0≤γ≤1 and energy conservation equation i.e. T _i;j ^j =0. The physical and geometrical aspects of the model are discussed. The special cases for γ=1 (stiff fluid distribution), γ=0 (dust distribution), γ=1/3 (disordered radiation) are also discussed.     

A New Class of Bianchi Type-I Cosmological Models in Lyra Geometry

This paper deals with Bianchi type-V cosmological models of the universe filled with a bulk viscous cosmic fluid in the framework of general relativity. A new class of exact solutions has been obtained by considering various well established power law relations among scale factor, cosmological and gravitational constants and cosmic time. Some physical and geometrical behaviors of the models have also been discussed. It has been found that all the models are in fair agreement of observational results.     

On Static Shells and the Buchdahl Inequality for the Spherically Symmetric Einstein-Vlasov System

In a previous work [1] matter models such that the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1, were considered in the context of Buchdahl’s inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, [R ₀, R ₁], R ₀ > 0, satisfies R ₁/R ₀ < 1/4. Moreover, given a sequence of solutions such that R ₁/R ₀ → 1, then the limit supremum of 2M/R ₁ was shown to be bounded by ((2Ω + 1)² − 1)/(2Ω + 1)². In this paper we show that the hypothesis that R ₁/R ₀ → 1, can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of 2M/R ₁ is bounded, but that the limit is ((2Ω + 1)² − 1)/(2Ω + 1)² = 8/9, since Ω = 1 for Vlasov matter. Thus, static shells of Vlasov matter can have 2M/R ₁ arbitrary close to 8/9, which is interesting in view of [3], where numerical evidence is presented that 8/9 is an upper bound of 2M/R ₁ of any static solution of the spherically symmetric Einstein-Vlasov system.     

Particle Motion and Perturbed Dynamical System in Warped Product Spacetimes

In this paper we have used the dynamical systems analysis to study the dynamics of a five-dimensional universe in the form of a warped product spacetime with a spacelike dynamic extra dimension. We have decomposed the geodesic equations to get the motion along the extra dimension and have studied the associated dynamical system when the cross-diagonal element of the Einstein tensor vanishes, and also when it is non-vanishing. Introducing the concept of an energy function along the phase path in terms of the extra-dimensional coordinate, we have examined how the energy function depends on the warp factor. The energy function serves as a measure of the amount of perturbation of geodesic paths along the extra dimension in the region close to the brane. Then we studied the geodesic motion under a conventional metric perturbation in the form of homothetic motion and conformal motion and examined the nature of critical points for a Mashhoon-Wesson-type metric, for timelike and null geodesics when the cross-diagonal term of the Einstein tensor vanishes. Finally we investigated the motion for null and timelike geodesics under the condition when the cross-diagonal element of the Einstein tensor is non-vanishing and examined the effects of perturbation on the critical points of the dynamical system.     

Bianchi Type-III Cosmological Models with Anisotropic Dark Energy (DE) in Lyra Geometry

The main purpose of this paper is to explore the solutions of Bianchi type-III cosmological model in Lyra geometry in the background of anisotropic dark energy. The general form of the anisotropy parameter of the expansion for Bianchi type-III space time is obtained in the presence of a single imperfect fluid with a dynamical anisotropic equation of state parameter and a dynamical energy density in Lyra geometry. A special law is assumed for the anisotropy of the fluid with reduces the anisotropy parameter of the expansion to a simple form $\Delta \propto \frac{1}{H^{2}V^{2}}$ . The exact solutions of the field equations, under the assumption on the anisotropy of the fluid, are obtained for exponential and power law volumetric expansion. The isotropy of the fluid, space and expansion are discussed. It is observed that the universe can approach to isotropy monotonically even in the presence of an anisotropic fluid. The anisotropy of the fluid also isotropizes at later times for accelerating models. The expression for the look-back time, proper distance, luminosity distance and angular diameter distance are also derived.     

On the Geometry and Mass of Static,Asymptotically AdS Spacetimes,and the Uniqueness of the AdS Soliton

We prove two theorems, announced in [6], for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with Ricci-flat conformal boundary, the convexity condition is associated with negative mass. The second theorem is a uniqueness theorem for the negative mass AdS soliton spacetime. This result lends support to the new positive mass conjecture due to Horowitz and Myers which states that the unique lowest mass solution which asymptotes to the AdS soliton is the soliton itself. This conjecture was motivated by a nonsupersymmetric version of the AdS/CFT correspondence. Our results add to the growing body of rigorous mathematical results inspired by the AdS/CFT correspondence conjecture. Our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ``ground state' spacetimes such as Minkowski spacetime, namely, the presence of a null line, or complete achronal null geodesic, and the totally geodesic null hypersurface that it determines. En route, we provide an analysis of the boundary data at conformal infinity for the Lorentzian signature static Einstein equations, in the spirit of the Fefferman-Graham analysis for the Riemannian signature case. This leads us to generalize to arbitrary dimension a mass definition for static asymptotically AdS spacetimes given by Chruciel and Simon. We prove equivalence of this mass definition with those of Ashtekar-Magnon and Hawking-Horowitz.     

New Class of Magnetized Inhomogeneous Bianchi Type-I Cosmological Model with Variable Magnetic Permeability in Lyra Geometry

Inhomogeneous Bianchi type-I cosmological model with electro-magnetic field based on Lyra geometry is investigated. Using separated method, the Einstein field equations have been solved analytically with the aid of Mathematica program. A new class of exact solutions have been obtained by considering the potentials of metric and displacement field are functions of coordinates t and x. We have assumed that F ₁₂ is the only non-vanishing component of electro-magnetic field tensor F _ij . The Maxwell’s equations show that F ₁₂ is the function of x alone whereas the magnetic permeability is the function of x and t both. To get the deterministic solution, it has been assumed that the expansion scaler Θ in the model is proportional to the value $\sigma_{1}^{1}$ of the shear tensor $\sigma_{i}^{j}$ . Some physical and geometric properties of the model are also discussed and graphed.     

Bianchi Type-III Cosmological Models with Time Dependent Displacement Vector for Barotropic Fluid Distribution in Lyra Geometry

Bianchi Type-III cosmological models for perfect fluid distribution with time dependent displacement field in the framework of Lyra geometry are investigated. To get the deterministic model of the universe, we have assumed two conditions (i) shear (σ) is proportional to the expansion (θ). This leads to B=C ⁿ where B and C are metric potentials and n is a constant. (ii) Universe is filled with barotropic fluid distribution which leads to p=γ ρ, 0≤γ≤1, p being isotropic pressure and ρ the energy density. The physical and geometrical aspects of the model with a special case and singularities in the models are also discussed.