Application of Mixed Generalized Quasi-Einstein Spacetimes in General Relativity

: In the present article, some geometric and physical properties of MG ( QE ) n were investigated. Moreover, general relativistic viscous ﬂuid MG ( QE ) 4 spacetimes with some physical applications were studied. Finally, through a non-trivial example of MG ( QE ) 4 spacetime, we proved its existence.


Introduction
A Riemannian or a semi-Riemannian manifold (M n , g) of dimension n(> 2) is termed as an Einstein manifold if its (0, 2)-type Ricci tensor Ric( = 0) satisfies Ric = r n , where r stands for the scalar curvature [1].In addition to Riemannian geometry, Einstein manifolds also have a vital contribution to the general theory of relativity (GTR).
Approximately two decades ago, Chaki and Maity introduced and studied quasi-Einstein manifolds [2].An (M n , g), (n > 2) is said to be a quasi-Einstein manifold (QE) n if its Ric ( = 0) realizes the following condition: where a, b ∈ R such that b = 0 and A( = 0) is the 1-form such that for any vector field U 1 , and a unit vector field ρ called the generator of (M n , g).In addition, A is named the associated 1-form.Einstein manifolds form a natural subclass of the class of (QE) n .Under the study of exact solutions of the Einstein field equations, as well as under the consideration of quasi-umbilical hypersurfaces of semi-Euclidean spaces, (QE) n came into existence.For instance, the Robertson-Walker spacetimes are (QE) n .Thus, (QE) n have great importance in GTR.
An (M n , g), (n ≥ 2) is said to be a generalized quasi-Einstein manifold G(QE) n [3] if its Ric( = 0) realizes the following condition: where a, b, c are non-zero scalars and A, B are two non-zero 1-forms such that g(U 1 , ρ) = A(U 1 ), g(U 1 , σ) = B(U 1 ), where ρ and σ are mutually orthogonal unit vector fields, i. e., g(ρ, σ) = 0.The vector fields ρ and σ are called the generators of the manifold.If c = 0, then the manifold reduces to a quasi-Einstein manifold.
In 2007, Bhattacharya, De and Debnath [4] introduced the notion of a mixed generalized quasi-Einstein manifold.A non-flat Riemannian manifold is said to be a mixed generalized quasi-Einstein manifold and is denoted by MG(QE) n , if its Ric( = 0) satisfies the following condition: where a, b, c, d are non-zero scalars and A, B are two non-zero 1-forms such that where ρ and σ are mutually orthogonal unit vector fields and are called the generators of the manifold.Recently, MG(QE) n have been studied by various geometers in several ways to a different extent, such as [5][6][7][8] and many others.
Putting (5), where {e i } is an orthonormal basis of the tangent space at each point of the manifold, and taking summation over i( A Lorentzian four-dimensional manifold is said to be a mixed generalized quasi-Einstein spacetime with the generator ρ as the unit timelike vector field if its Ric( = 0) satisfies (5).Here, A and B are non-zero 1-forms such that σ is the heat flux vector field perpendicular to the velocity vector field ρ.Therefore, for any vector field U 1 , we have Further, we know that if the Riemannian curvature tensor K of type (0, 4) has the form then the manifold is said to be of constant curvature k.The generalization of this manifold is the manifold of quasi-constant curvature and, in this case, the curvature tensor has the following form: where ), K is the curvature tensor of type (1, 3) and f 1 , f 2 are scalars, and ρ is a unit vector field defined by It can be easily seen that, if the curvature tensor K is of the form (10), then the manifold is conformally flat [3].Thus, a Riemannian or semi-Riemannian manifold is said to be of quasi-constant curvature if the curvature tensor K satisfies the relation (10); we denote such a manifold of dimension n by (QC) n .
A non-flat Riemannian or semi-Riemannian manifold (M n , g) (n ≥ 3) is said to be a manifold of generalized quasi-constant curvature if the curvature tensor K of type (0, 4) satisfies the condition [3] where f 1 , f 2 , f 3 are scalars and A, B are two non-zero 1-forms.ρ and σ are orthonormal unit vectors corresponding to A and B such that g(U 1 , ρ) = A(X), g(U 1 , σ) = B(X) and g(ρ, σ) = 0.Such a manifold is denoted by G(QC) n .
In [9], Bhattacharya and De introduced the notion of mixed generalized quasi-constant curvature.A non-flat Riemannian or semi-Riemannian manifold (M n , g) (n ≥ 3) is said to be a manifold of mixed generalized quasi-constant curvature if the curvature tensor K of type (0, 4) satisfies the condition where f 1 , f 2 , f 3 , f 4 are scalars.A, B are two non-zero 1-forms.ρ and σ are orthonormal unit vectors corresponding to A and B such that g(U 1 , ρ) = A(X), g(U 1 , σ) = B(X) and g(ρ, σ) = 0.Such a manifold is denoted by MG(QC) n .The spacetime of general relativity and cosmology is regarded as a connected four-dimensional semi-Riemannian manifold (M 4 , g) with Lorentzian metric g with signature (−, +, +, +).The geometry of the Lorentz manifold begins with the study of a causal character of vectors of the manifold.Due to this causality, the Lorentz manifold becomes a convenient choice for the study of general relativity.Spacetimes have been studied by various authors in several ways, such as [10][11][12][13][14] and many others.

MG(QE) n Admitting the Generators ρ and σ as Recurrent Vector Fields
Let us consider the generators ρ and σ corresponding to the associated recurrent 1-forms A and B.Then, we have where η and ϕ are non-zero 1-forms.
Using ( 5) in (15), we obtain Putting U 2 = U 3 = ρ in ( 16), we obtain By using the fact that A(D U 1 ρ) = 0 and ( 6) in ( 17), we have which can be written as Thus, we have Again, putting U 2 = U 3 = σ in ( 16), we have Again, using the fact that B(D U 1 σ) = 0 and ( 6) in ( 19), we have Thus, we have This implies that either D U 1 ρ ⊥ σ or ρ is a parallel vector field.Hence, we can state the following theorem: Theorem 1.Let a mixed generalized quasi-Einstein manifold MG(QE) n be Ricci-recurrent; then, the following statements are equivalent: (i) ρ and σ are parallel vector fields;

MG(QE) n Admitting the Generators ρ and σ as Concurrent Vector Fields
A vector field π is said to be concurrent if it satisfies the following condition [17,18]: where ξ is constant.
Let us consider the generators ρ and σ corresponding to the associated concurrent 1-forms A and B.Then, we have and where λ and µ are non-zero constants.
In view of ( 27), (26) turns to Thus, by virtue of (28), ( 5) takes the form which is a quasi-Einstein manifold.Thus, we can state the following theorem: Theorem 2. Let MG(QE) n be a mixed generalized quasi-Einstein manifold.If the associated vector fields of MG(QE) n are concurrent and the associated scalars are constants, then the manifold reduces to a quasi-Einstein manifold.

MG(QE) n Admitting Einstein's Field Equations
The Einstein's field equations with and without cosmological constants are given by and respectively; κ is a gravitational constant, λ is a cosmological constant, and T is the energymomentum tensor.Using ( 6) in (31), it follows that Now, taking the covariant derivative of (32) with respect to U 3 , we arrive at Thus, we have a result.
Theorem 3. Let MG(QE) n admit Einstein's field equation without a cosmological constant.If the associated 1-forms A and B are covariantly constant, then the energy-momentum tensor is also covariantly constant."

MG(QE) 4 Spacetime Admitting Space-Matter Tensor
In 1969, Petrov [19] introduced and studied the space-matter tensor P of type (0, 4) and defined by where K is the curvature tensor of type (0, 4), T is the energy-momentum tensor of type (0, 2), κ is the gravitational constant, and ν is the energy density.Furthermore, G and g ∧ T are, respectively, defined by and for all U 1 , U 2 , U 3 , U 4 on M. Using ( 35) and ( 36) in (34), it follows that If P = 0, then (37) gives In view of (5), from (31), it follows that Thus, from ( 38) and (39), we obtain where Thus, we can state the following theorem: Theorem 4. For a vanishing space-matter tensor, MG(QE) 4 spacetime satisfying Einstein's field equation without a cosmological constant is a MG(QC) 4 spacetime.
Next, we investigate the existence of a sufficient condition under which MG(QE) 4 can be a divergence-free space-matter tensor.
From (31) and (37), we obtain By using (divK Let (divP)(U 1 , U 2 , U 3 ) = 0; then, contracting (42) over U 2 and U 3 , we obtain ∂ν(U 1 ) = 0, where (27) is used.Hence, we can state the following theorem: Theorem 5.For a divergence-free space-matter tensor, the energy density in MG(QE) 4 spacetime satisfying Einstein's field equation without a cosmological constant is constant.Now, by using ( 5) in (42), we obtain By assuming that ν, a, b, c, and d are constants and the generator ρ is a parallel vector field, i.e., D U 1 ρ = 0, we obtain In view of (44), we derive Using ( 44) and ( 45), (43) reduces to Thus, we can state the following theorem: Theorem 6.In MG(QE) 4 spacetimes admitting parallel vector field ρ satisfying Einstein's field equation without a cosmological constant, if the energy density and associated scalars constant are constants, then the divergence of the space-matter tensor vanishes.

MG(QE) 4 Spacetime Admitting General Relativistic Viscous Fluid
Ellis [20] defined the energy-momentum tensor for a perfect fluid distribution with heat conduction as where g(U 1 , ρ) = A(U 1 ), g(U 1 , σ) = B(U 1 ), A(ρ) = −1, B(σ) > 0, g(ρ, σ) = 0, and ν, ω are called the isotropic pressure and the energy density, respectively.σ is the heat conduction vector field perpendicular to the velocity vector field ρ.Assuming a mixed generalized quasi-Einstein spacetime satisfying Einstein's field equation without a cosmological con-stant whose matter content is viscous fluid, then, from (31) and ( 46), the Ricci tensor takes the form By comparing ( 5) and (47), we obtain Taking a frame field to contract (48) over U 1 and U 2 , we obtai In view of (49), (47) turns to Now, let R be the Ricci operator given by g(R(U 1 ), Now, contracting (51) over U 1 and U 2 , we obtain For a mixed generalized quasi-Einstein spacetime, from (5), it follows that In view of (48), (49), and (53), we find that By making use of (54), from (52), it follows that Thus, we can state the following theorem: Theorem 7. If MG(QE) 4 spacetime admitting viscous fluid satisfies Einstein's field equation without a cosmological constant, then the square of the length of Ricci operator is κ 2 (ν 3 ω 2 + ν + ω − 3).

Example of MG(QE) 4 Spacetime
In this section, we constructed a non-trivial concrete example to prove the existence of a MG(QE) 4 spacetime.
We assume a Lorentzian manifold (M 4 , g) endowed with the Lorentzian metric g given by ds where u 1 , u 2 , u 3 , u 4 are standard coordinates of M 4 , i, j = 1, 2, 3, 4, and p = e u 1 k −2 , and k is a non-zero constant.Here, the signature of g is (+, +, +, −), which is Lorentzian.Then, the only non-vanishing components of the Christoffel symbols and the curvature tensors are and the components are obtained by the symmetry properties.
The non-vanishing components of the Ricci tensors are Thus, the scalar curvature r is and the 1-forms are defined by where the generators are unit vector fields; then, from (5), we have
Author Contributions: