Generalized Quasi-Conformal Curvature Tensor and the Spacetime of General Relativity

: In this paper, a study of generalized quasi-conformal curvature tensor has been made on the four dimensional spacetime of general relativity. Some results related to the application of such spacetime in the general relativity are obtained. Perfect ﬂuid and dust ﬂuid cosmological models have also been studied.


Introduction
In this paper we study some results on general relativity by the co-ordinate free method of differential geometry.In this method we consider a four dimensional semi-Riemannian manifold (M 4 , g) with a Lorentzian metric g with signature (−, +, +, +).The geometry of the Lorentzian manifold begins with the study of the casual character of vectors of the manifold.It is due to this causality that the Lorentzian manifold became a convenient choice for the study of general relativity.
In [12] Einstein equation on the energy-momentum tensor is of vanishing divergence and the requirement is satisfied if the energy-momentum tensor is covariant-constant.In paper [7] M.C.Chaki and Sarbari Roy showed that a general relativistic spacetime with covariant-constant energy-momentum tensor is Ricci symmetric, that is ∇S = 0, where S is the Ricci tensor of the spacetime.
The generalized quasi conformal curvature tensor of 4-dimensional space time (M n , g) is given by (See [6]) where a 1 and b 1 are real constants, we note that for four dimensions, it has the flavour of Riemannian curvature tensor 6 , −1 6 , 0) and S,r are the Ricci tensor,scalar curvature respectively.

Spacetime with Vanishing Generalized Quasi-Conformal Curvature Tensor
Let V 4 be the spacetime of general relativity, then from (1.1) we have where Let Q be the Ricci operator given by g(QY, X) = S(Y, X), then (2.2) becomes Taking a frame field over X and Y , equation (2.3) becomes Hence we state the following theorem.
Theorem 2.1.A generalized quasi-conformally flat spacetime is an Einstein's spacetime, provided Hence we can state the following theorem: Theorem 2.2.A generalized quasi-conformally flat spacetime is a spacetime of constant curvature, provided Now, we consider a spacetime satisfying the Einstein's field equation with cosmological constant is given by where S and r denote the Ricci tensor and scalar curvature, λ is the cosmological constant, κ is the gravitational constant and T (X, Y ) is the energy momentum tensor.
In view of (2.4) and (2.6), we have Taking covariant derivative of (2.7), we get Since generalized quasi-conformally flat spacetime is Einstein, therefore the scalar curvature r is constant.Hence for all Z.
In view of (2.8) and (2.9), we have Hence we state the following theorem: Theorem 2.3.In a generalized quasi-conformally flat spacetime satisfying Einstein's field equation with cosmological constant, the energy momemtum tensor is covrariant constant.
The curvature collineation studied by Katzin et al. [10], in context of the related particle and field conservation laws that may be admitted in the standard form of general relativity.
The geometrical symmetries of a spacetime is where B represents a geometrical/physical quantity, L ξ denotes the Lie derivative with respect to the vector field ξ and ϕ is a scalar.
One of the most simple and widely used example is the metric inheritance symmetry for B = g(X, Y ) in (2.11).And ξ is the killing vector field if ϕ = 0. (2.12) A spacetime M is said to admit a symmetry called a curvature collineation [8,9] provided there exist a vector field ξ such that where R is the Riemannian curvature tensor.Now we shall investigate the role of such symmetry inheritance for the spacetime admitting generalized quasi-conformal curvature tensor with a killing vector field ξ as a curvature collineation.Then we have If M admits a curvature collineation and then (2.13) becomes where S is the Ricci tensor of the manifold.
Taking Lie derivative of (1.1) and using (2.13), (2.14), (2.15), we have Hence we can state the following theorem: Theorem 2.4.If a spacetime M admitting the generalized quasi-conformal curvature tensor with ξ as a Killing vector field is curvature collineation, then the Lie derivative of the generalized quasi-conformal curvature tensor vanishes along the vector field ξ.
Next, the symmetry of the energy momemtum tensor T is the matter collineation defined by where ξ is the vector field generating the symmetry and L ξ is the lie derivative operator along the vector field ξ.
If ξ is Killing vector field on the spacetime with vanishing generalized quasi-conformal curvature tensor, then where L ξ denotes the Lie derivative with respect to ξ.
Taking Lie derivative on equation (2.7), we get which implies that the spacetime admits matter collineation.
Hence we can state the following theorem: Theorem 2.5.If a spacetime obeying Einstein's field equation has vanishing generalized quasi-conformal curvature tensor, then the spacetime admits matter collineation with respect to a vector field ξ if and only if ξ is a Killing vector field.
If ξ is a conformal Killing vector field, then where α is a scalar.
Using (2.20) in (2.19), we get (2.21) Generalized Quasi-Conformal Curvature Tensor and the Spacetime of General Relativity 5 In view of (2.7) and (2.21), we have Therefore from (2.22) we can say that the energy-momentum tensor has Lie inheritance property along ξ.
Conversely, if (2.22) holds, then it follows that (2.20) holds, that is, ξ is a conformal Killing vector field.Thus we can state the following theorem.
Theorem 2.6.If a spacetime obeying Einstein's field equation has vanishing generalized quasi-conformal curvature tensor, then the vector field ξ on the spacetime is a conformal Killing vector field if and only if the energy momemtum tensor has the Lie inheritance property along ξ.

Perfect Fluid Spacetime with Vanishing Generalized Quasi-Conformal Curvature Tensor
We consider a perfect fluid spacetime with vanishing generalized quasi-conformal curvature tensor obeying Einstein's field equation without cosmological constant.
The energy momentum tensor T of a perfect fluid is given by (See [12]) where δ is the energy density, ρ the isotropic pressure and γ is a non-zero 1-form such that g(X, U ) = γ(X), ∀ X. U being the velocity vector field of the flow,that is, g(U, U ) = −1.
Einstein's field equation without cosmological constant is given by where r is the scalar curvature of the manifold and κ = 0.
Substituting (2.4) and (3.1) in (3.2), we have Taking a frame field and after contraction over X and Y , we get Let Q be the Ricci operator given by g(QX, Y ) = S(X, Y ) and S(QX, Y ) = S 2 (X, Y ).Then we have that γ(QX) = g(QX, U ) = S(X, U ).
Hence equation (3.5) becomes 6S. Girish Babu, P. Siva Kota Reddy, G. S. Shivprasanna, G. Somashekhara and Khaled A. A. Alloush Taking a frame field and after contraction over X and Y in (3.6), we get Hence we can state the following theorem.
Theorem 3.1.If a generalized quasi-conformally flat perfect fluid spacetime obeys Einstein's field equation without cosmological constant, then the square of the length of the Ricci operator of the spacetime is In view of (3.4) and (3.7), we get Substituting (3.8) in (3.1), we get Since the scalar curvature r of generalized quasi-conformally flat spacetime is constant.Hence from (3.7) we have δ = constant, then from (3.8) we obtain ρ = constant.Now δ + ρ = 0 means the fluid behaves as a cosmological constant (See [24]).This is also termed as phantom barrier (See [11]).Now in cosmology we know such a choice δ = −ρ leads to rapid expansion of the spacetime which is now termed as inflation (See [2]).Thus we can state the following theorem.
Theorem 3.2.If a perfect fluid spacetime with vanishing generalized quasi-conformal curvature tensor obeying Einstein's equation without cosmological constant, then the spacetime has constant energy density and isotropic pressure and the spacetime represents inflation and also the fluid behaves as a cosmological constant.
In [13] the Ricci tensor S of type (0, 2) of the spacetime satisfies condition S(X, X) > 0, (3.10) for every timelike vector field X, then (3.10) is called the timelike convergence condition.
In view of (3.1) and (3.2), we have Putting X = Y = V in (3.11) and using (3.4), we get where Since the spacetime under consideration satisfies the timelike convergence condition and κ > 0 then we have Einstein's field equation without cosmological constant for a purely electromagnetic distribution takes the form (See [1]) In view of (3.16) and (3.17), we get t = 0. Thus from (3.15) we have r = 0. Hence equation (2.5) becomes R(Y, U, V, X) = 0 which means that the spacetime is flat.Hence we can state the following theorem.
Theorem 3.5.A generalized quasi-conformally flat spacetime satisfying Einstein's equation without cosmological constant for a purely electromagnetic distribution is an Euclidean space.