A study of mixed generalized quasi-Einstein spacetimes with applications in general relativity

: In the present paper we study Ricci pseudo-symmetry, Z-Ricci pseudo-symmetry and concircularly pseudo-symmetry conditions on a mixed generalized quasi-Einstein spacetime MG ( QE ) 4 . Also, it is proven that if d (cid:44) Λ , then MG ( QE ) 4 spacetime does not admit heat flux, where d and Λ are the function and the cosmological constant, respectively. In the end of this paper we construct a non-trivial example of MG ( QE ) 4 to prove its existence


Introduction
A Riemannian (or semi-Riemannian) manifold (M n , g), (n ≥ 3) is named an Einstein manifold if the Ricci tensor Ric( 0) of type (0, 2) satisfies: Ric = r n g, where r represents the scalar curvature of (M n , g).Einstein manifolds form a natural subclass of several classes of (M n , g) determined by a curvature restriction imposed on their Ricci tensor [1].Also, Einstein manifolds play a key role in Riemannian geometry, the general theory of relativity as well as in mathematical physics.
Approximately two decades ago, the idea of quasi-Einstein manifolds was proposed and studied by Chaki and Maity [2].An (M n , g), (n > 2) is said to be a quasi-Einstein manifold (QE) n if its Ric( 0) satisfies where a, b( 0) ∈ R and U is a non-zero 1-form such that for all vector fields ζ 1 and a unit vector field ϱ called the generator of (QE) n .Also, the 1-form U is named the associated 1-form.From (1.1) it is clear that for b = 0, (QE) n reduces to an Einstein manifold.The notion of (QE) n came into existence during the study of exact solutions of Einstein's field equations as well as during considerations of quasi-umbilical hypersurfaces of semi-Euclidean spaces.For example, the Robertson-Walker spacetimes are (QE) 4 .Also, (QE) 4 can be taken as a model of the perfect fluid spacetime in general relativity [3][4][5].
MG(QE) n has wide applications in cosmology and the general theory of relativity and is studied by several authors, such as [10][11][12][13] and many others.
Putting (1.5), where {e j } is an orthonormal basis of the tangent space at each point of MG(QE) n and taking summation over i (1 ≤ j ≤ n), we get where r is the scalar curvature of MG(QE) n .
Let K be the Riemannian curvature tensor of an (M n , g).The k-nullity distribution N(k) of an (M n , g) is defined by [14,15] where k is some smooth function.
In a similar manner, the k-nullity distribution N(k) of a Lorentzian manifold can be defined.In a (QE) n , if the generator ϱ belongs to some k-nullity distribution N(k), then (M n , g) is called an N(k)-(QE) n [16].In 2007, Özgür and Triphati [17] proved that in an N(k)-(QE) n , k is not arbitrary, that is equal to a+b n−1 .A spacetime is a time oriented (M 4 , g) with Lorentz metric of signature (+, +, +, −).A 4dimensional Lorentzian manifold is said to be MG(QE) 4 with the generator ϱ if its Ric( 0) satisfies (1.5).Here U( 0) and V( 0) being1-forms such that σ is the heat flux vector field perpendicular to the velocity vector field ϱ.Therefore, for any ζ 1 , we have (1.8) From (1.5) and (1.8) we have In [18], a generalized (0, 2) type symmetric Z tensor was introduced by Mantica and Molinari and defined as follows where ϕ is an arbitrary scalar function.The properties of the Z tensor in several ways to a different extent have been studied in [19,20].If the Z tensor at each point of the spacetime vanishes, then the spacetime is said to be Z flat.Einstein's field equation (without cosmological constant) is given by where T and Λ represent the energy-momentum tensor and the Einstein gravitational constant, respectively.The idea of perfect fluid spacetime came into existence while discussing the structure of the universe.In general relativity the matter content of the spacetime is described by T .The matter content is supposed to be a fluid having pressure and density and possessing kinematical and dynamical quantities like acceleration, velocity, vorticity, shear and expansion.In a perfect fluid spacetime, the energy-momentum tensor T is given through the relation where ψ and µ stand for the energy density and isotropic pressure, respectively.ϱ is the unit timelike velocity vector field such that g(ζ 1 , ϱ) = U(ζ 1 ) for all ζ 1 .In case of fluid matter distribution, the energy momentum tensor is given by Ellis [21] as σ is the heat conduction vector field perpendicular to the velocity vector field ϱ.Definition 1.1.An (M n , g) is called Ricci-pseudosymmetric [22], if the tensors K • Ric and Q(g, Ric) are linearly dependent, where and ) where r is the scalar curvature of the manifold.
In view of (1.18), it follows that where Here, K is the curvature tensor of type (0, 4) and is the concircular curvature tensor of type (0,4) which satisfies the following properties: (1.20)
By putting ζ 3 = ϱ and ζ 4 = σ in (2.1) and using (1.8), we obtain In view of (1.8), (2.3) and (2.4), we arrive at Putting ζ 2 = ϱ in (2.5), we get which means that the generator ϱ belongs to the (a−b) 3 -nullity distribution.Thus, the manifold turns into N a−b 3 quasi-Einstein spacetime.Therefore, we can state the following result: holds on the set Also, By virtue of (3.4) and (3.5), (3.3) turns to From (1.5), it follows that By virtue of (3.7), (3.6) becomes (3.8) In view of (1.5) and (1.10), (3.8) takes the form which by putting Taking the inner product of the foregoing equation with ϱ, we lead to An (M n , g), (n ≥ 3) is said to be concircularly pseudo-symmetric if and only if the following relation holds on the set G s , where G s = {ζ 1 ∈ M n : Ric r n g at ζ 1 } and L s is a certain function on G s .In view of (1.14)-(1.16),(4.1) turns to By using (1.5) Putting Putting 3) and using (1.8), we can easily find On simplification, we obtain From (1.18) and (4.6), we obtain In view of (1.9) and (4.6), from (4.5) it follows that Putting Again, putting ζ 2 = σ in (4.8), we get If d = 0, then from (4.9) we find L s = −3b−c 12 , as b 0. If d 0, then L s = −b+c 4 .Comparing this with (4.9), it follows that c = d = 0 and thus MG(QE) 4 spacetime reduces to a (QE) 4 spacetime.Therefore, from (4.7) we have which means that the generator ϱ belongs to the (a−b) 3 -nullity distribution.Thus, the manifold turns into N a−b 3 quasi-Einstein spacetime.Therefore, we have the following result: From (1.5) and (5.1), we have Putting

Example of MG(QE) 4 spacetime
In this section, we construct a non-trivial example to prove the existence of an MG(QE) 4 spacetime.We assume a Lorentzian manifold (M 4 , g) endowed with the Lorentzian metric g given by where i, j = 1, 2, 3, 4 and ω, c are constants.Then the covariant and contravariant components of the metric are respectively given by and 3) The only non-vanishing components of the Christoffel symbols are (6.4) The non-zero derivatives of (6.4) are ∂ ∂r For the Riemannian curvature tensor, The non-zero components of (I) are: and the non-zero components of (II) are: Adding components corresponding to (I) and (II), we have where generators are unit vector fields, then from (1.5), we have ) Ric 33 = ag 33 + bU 3 U 3 + cV 3 V 3 + d(U 3 V 3 + U 3 V 3 ), (6.7) = L.H.S .o f (6.5).(6.9) By a similar argument it can be shown easily that (6.6), (6.7) and (6.8) are also true.Hence, (IR 4 ,g) is an MG(QE) 4 .

Theorem 2 . 1 .
Every Ricci-pseudosymmetric MG(QE) 4 spacetime is a N a−b 3 quasi-Einstein spacetime, for some certain function L s = a−b 3 , where b + c 0. 3. Z-Ricci pseudo-symmetric MG(QE) 4 spacetime An (M n , g), (n ≥ 3) is called Z-Ricci pseudo-symmetric if and only if the following relation