Perfect Fluid Spacetimes and Gradient Solitons

In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a perfect fluid spacetime admitting concircular vector field, we prove that the velocity vector field annihilates the conformal curvature tensor. In addition, in dimension 4, we show that a perfect fluid spacetime is a generalized Robertson–Walker spacetime with Einstein fibre. It is proved that if a perfect fluid spacetime furnished with concircular vector field admits a second order symmetric parallel tensor P, then either the equation of state of the perfect fluid spacetime is characterized by p=3-nn-1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\frac{3-n}{n-1} \sigma $$\end{document}, or the tensor P is a constant multiple of the metric tensor. Finally, The perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons, and gradient m -quasi Einstein solitons, are characterized.


Introduction
Let M n be a Lorentzian manifold endowed with the Lorentzian metric g of signature (+, +, . . ., + (n−1)times , −).The idea of generalized Robertson-Walker (GRW ) spacetimes was presented by Alias, Romero and Sanchez [1] in 1995.A Lorentzian manifold M n with n ≥ 3 is named as a GRW spacetime if it can be written as a warped product of an open interval I of R (set of real numbers) and a Riemannian manifold M * of dimension (n − 1), that is, M = −I × f 2 M * , where f > 0 is a smooth function, termed as scale factor or warping function.If the dimension of M * is three and is of constant sectional curvature, then the spacetime reduces to Robertson-Walker (RW ) spacetime.Hence, the GRW spacetime is a spontaneous extension of RW spacetime on which the standard cosmology is modeled.It also includes the Einstein-de Sitter spacetime, the static Einstein spacetime, the Friedman cosmological models, the de Sitter spacetime, and have implementations as inhomogeneous spacetimes obeying an isotropic radiation.The geometrical and physical features of GRW spacetimes have been exhaustively presented in ( [6], [21]).
A Lorentzian manifold M n is called a perfect fluid spacetime if its nonvanishing Ricci tensor S obeys where α, β are scalar fields (not simultaneously zero), ρ is a vector field defined by g(X, ρ) = A(X) for all X.Also, ρ is the unit timelike vector field (also named velocity vector field) of the perfect fluid spacetime.Every RW spacetime is a perfect fluid spacetime [23], where as in 4− dimension, the GRW spacetime is a perfect fluid spacetime if and only if it is a RW spacetime.In differential geometry, the Ricci tensor obeying equation (1.1) is termed as a quasi Einstein manifold [4].For more details, we refer ( [3], [24]) and the references therein.
The problem of discovering a canonical metric on a smooth manifold inspires Hamilton [17] to introduce the concept of Ricci flow.If the metric of a (semi-) Riemannian manifold M n is satisfied by an evolution equation [17].The self-similar solutions to the Ricci flow yield the Ricci solitons.A metric of M n is called a Ricci soliton [16] if it obeys for some real scalar λ.Here L W indicates the Lie derivative operator.We indicate (g, W, λ) as a Ricci soliton on M n .If λ is negative, positive or zero, then the Ricci soliton is said to be shrinking, expanding or steady, respectively.In particular, if W is Killing or identically zero, then the Ricci soliton is trivial and M n is Einstein.Also, if the soliton vector W is the gradient of some smooth function −f , that is, W = −Df , then equation Inspired by the Yamabe's conjecture ("metric of a complete Riemannian manifold is conformally related to a metric with constant scalar curvature"), the idea of Yamabe flow on a complete Riemannian manifold M n was presented by Hamilton [17].A semi-Riemannian manifold M n endowed with a semi-Riemannian metric g is called a Yamabe flow if it obeys: If f is constant (or W is Killing) on M , then gradient Yamabe (or Yamabe) soliton becomes trivial.Sharma [24] investigated the Yamabe soliton on 3-Sasakian manifolds.Also, the 3-Kenmotsu manifolds and almost co-Kähler manifolds with Yamabe solitons have been characterized by Wang [27] and Suh and De [25] respectively.Chen et al. [7] studied the properties of Riemannian manifolds with Yamabe solitons.Some interesting results on Yamabe solitons have been investigated in ( [3], [11], [12]) and also by others.
Recently, in [13], the authors have studied Yamabe and gradient Yamabe solitons in perfect fluid spacetimes.
A semi-Riemannian manifold M n equipped with the semi-Riemannian metric g is said to be a gradient m-quasi Einstein metric [2] if there exists a constant λ, a smooth function f : M n → R and obeys where 0 < m ≤ ∞ is an integer and ⊗ indicate the tensor product.In this case f denotes the m-quasi Einstein potential function [2].Here the Bakry-Emery Ricci tensor S + Hessf − 1 m df ⊗ df is proportional to the metric g and λ = constant [28].
If m = ∞, the foregoing equation (1.6) represents a gradient Ricci soliton and the metric represents almost gradient Ricci soliton if it obeys the condition m = ∞ and λ is a smooth function.Some basic classifications of m-quasi Einstein metrics was characterized by He et al. [18] on Einstein product manifold with non-empty base.Also, a few characterization of mquasi Einstein solitons have been presented (in details) in [19].
The above studies motivate us to study the properties of perfect fluid spacetimes if the Lorentzian metrics are Ricci, gradient Ricci, gradient Yamabe and m-quasi Einstein solitons.We lay out the content of our paper as: In Section 2, we produce the preliminaries idea of perfect fluid spacetime with concircular vector field.The properties of second order symmetric parallel tensor in perfect fluid spacetimes with concircular vector field are studied in Section 3. Section 4 concerns with Ricci soliton and gradient Ricci soliton on a perfect fluid spacetime with concircular vector field.We investigate the properties of perfect fluid spacetimes equipped with gradient yamabe soliton and gradient m-quasi Einstein solitons in Section 5 and Section 6, respectively.

Perfect fluid spacetime
It is well known that in a perfect fluid spacetimes, ρ is the unit timelike vector field (also termed as velocity vector field ), hence where the vector field U ∈ X(M ) (X(M ) indicates the collection of all C ∞ vector fields of M ) and A indicates the 1-form.Executing the covariant derivative of (2.1) yields where ∇ is the Levi-Civita connection.The Einstein's field equations without cosmological constant have the form where κ and T indicates the gravitational constant and the energy momentum tensor, respectively.In case of perfect fluid spacetime, the energy momentum tensor T is defined as where σ indicates the energy density of the perfect fluid and p is the isotropic pressure.
In a perfect fluid spacetime if we consider an orthonormal frame field and taking contraction of the equation (1.1) over U and V , we obtain The necessary and sufficient condition for the constant scalar curvature of a perfect fluid spacetime is that nU (α) = U (β). Combining the equations (1.1), (2.3) and (2.4), we infer that Moreover, p and σ are interconnected by an equation of state of the form p = p(σ) characterizing the particular sort of perfect fluid under consideration.In this instance, the perfect fluid is called isentropic.In addition, if p = σ, then the perfect fluid is named as stiff matter.Many years ago, a stiff matter equation of state was publicized by Zeldovich [30].The stiff matter era preceded the dust matter era with p = 0 , the radiation and it characterizes the early universe with p − σ 3 = 0 and the dark energy era with p + σ = 0 [5].
The idea of concircular vector field was introduced by Failkow [15].On a semi-Riemannian manifold M , a vector field ρ is called concircular if there exists a smooth function µ (termed as potential function of the concircular vector field) such that In [26], we see that the world lines of receding or colliding galaxies in de Sitter's model of general relativity are trajectories of timelike concircular vector fields.Here, ρ is called non-trivial if ρ is non-constant.The vector field ρ becomes concurrent vector field if µ is non-zero constant.In [8], B.
Y. Chen has investigated concircular vector fields and their applications to Ricci solitons.Also, Deshmukh et al. [14] have studied spheres and Euclidean spaces with the Concircular vector fields.For more information, see ( [9], [10]) and references contained in those.
Utilizing the equations (1.1) and (2.1), we find that and conclude that corresponding to the eigenvector ρ, α − β is an eigenvalue of the Ricci tensor.
Agreement: Throughout the paper, in a perfect fluid spacetime, we consider the velocity vector field is of concircular type.
If the velocity vector field ρ of the perfect fluid spacetime is a concircular vector field, then we have Executing contraction over U , the foregoing equation gives (2.9) Combining the equations (2.7) and (2.9), we infer that (2.10) Using (2.10) into equation (2.8), we find where R is the curvature tensor and Ricci operator Q is defined by g(QU, V ) = S(U, V ), we obtain that Theorem 2.1.In a perfect fluid spacetime with concircular vector field, the velocity vector field annihilates the conformal curvature tensor.
We know that [20], in dimension 4, C(U, V )ρ = 0 is equivalent to Hence, the 1-form A is closed.
In [22], Mantica et al proved that in a 4-dimensional perfect fluid spacetime satisfying divC = 0 is a GRW spacetime with Einstein fibre, provided the velocity vector field ρ is irrotational.
Since C(U, V )W = 0, then divC = 0. Hence, from the above discussion, we say that the perfect fluid spacetime with concircular vector field is a GRW spacetime with Einstein fibre.Thus, we can write: Theorem 2.2.A perfect fluid spacetime with concircular vector field is a GRW spacetime with Einstein fibre.

Second order symmetric parallel tensor in a perfect fluid spacetime
Let P be a symmetric (0,2)-tensor in a perfect fluid spacetime which is parallel with respect to the Levi-civita connection ∇, that is ∇P = 0.Then, by ∇P = 0, we get where U, V, X and Y are arbitrary vectors fields.As P is symmetric , putting = X = Y = ρ in (3.1), we have Utilizing (2.11), we obtain Replacing U by ρ in the foregoing equation and using (2.1), we infer which entails that either α = β, or Since P is parallel, we have Since ∇P = 0, we conclude that P (ρ, ρ) = constant.
From (2.11) we derive Utilizing ∇ U ρ = µU , we infer which implies a Einstein spacetime, that is, trivial Ricci soliton.Therefore, we have Theorem 4.2.In a perfect fluid spacetime endowed with concircular vector field, the Ricci soliton (g, ρ, λ) is trivial and is an Einstein spacetime, provided Next part of this section deals with the investigation of gradient Ricci solitons in perfect fluid spacetimes with concircular vector field.Now, ∀ U, V ∈ X(M ) and from (1.1), we obtain Let us assume that the soliton vector field W of the Ricci soliton (g, W, λ) in a perfect fluid spacetime with concircular vector field is a gradient of some smooth function −f .Then equation (1.2) reduces to for all U ∈ X(M ).The equation (4.5) along with the subsequent relation Executing the covariant derivative of (4.4) and utilizing (4.3), we lead In view of equations (4.7) and (4.8), we infer Taking a set of orthonormal frame field and executing contraction of the equation (4.9), we get Again, from (1.1) we have Suppose that (ρα) = µβ.
Thus, from equations (4.12) we get which shows that either α = β or (ρf ) = 0 on a perfect fluid spacetime with the gradient Ricci soliton.
Case I. We assume that α = β and (ρf ) = 0 and therefore from equation (2.6), we conclude that This gives the equation of state in a perfect fluid spacetime.Also, λ = β −α = 0 and hence the gradient Ricci soliton is steady.
Case II.We consider that (ρf ) = 0 and α = β.Then, we have, f is invariant under the velocity vector field ρ.
By concluding the above facts, we can write our result as:

Gradient Yamabe soliton on perfect fluid spacetimes
From equation (1.5), we find (5.1) Differentiating (5.1) covariantly along the vector field V , we have Interchanging U and V in the foregoing equation and then utilizing the above equation, (5.1) and ( 5.
Considering an orthonormal frame field and contracting the above equation over U , we get From equation (1.1) we have Combining the last two equations, we infer Putting V = ρ in the preceding equation, we get Now, from (5) we infer that Again (2.11) implies that Combining equation (5.5) and (5.6), we have Setting V = ρ in the previous equation gives Utilizing (5.8) in (5.3) we infer that which entails that either β = 0 or (U f ) + (ρf )A(U ) = 0.
If β = 0, then we infer that σ + p = 0.This represents a dark energy.
The covariant derivative of Df = −(ρf )ρ yields where equation (5.1) is used.If f is invariant under the velocity vector field ρ, then we find λ = r. (5.10) From the above we conclude that the scalar curvature of the manifold is constant.Hence from equation (5.8) we have either α = β or Df = 0.
Case I. We suppose that α = β and (Df ) = 0 and therefore from equation (2.6), we conclude that This gives the equation of state in a perfect fluid spacetime.
Case II.We consider that (Df ) = 0 and α = β.The equation Df = 0 states that f is constant and hence the gradient Yamabe soliton is trivial.
Thus, we can write: Next, If f is invariant under the velocity vector field ρ, then equation (5.10) is satisfied.Thus, we conclude that the nature of the flow vary according to the scalar curvature.Thus we can write the subsequent corollaries.
Corollary 5.3.Let a perfect fluid spacetime with concircular vector field admits a gradient Yamabe soliton with β = 0.If f is invariant under the velocity vector field ρ, then the gradient Yamabe soliton is expanding, shrinking or steady according as the scalar curvature is positive, negative or zero, respectively.
Corollary 5.4.If the metric of a perfect fluid spacetime equipped with concircular vector field admits a gradient Yamabe soliton with β = 0, then it possesses the constant scalar curvature, provided f is invariant under the velocity vector field ρ.
Remark 5.5.In [13], the authors have studied gradient Yamabe soliton in perfect fluid spacetimes.But in this current paper, we consider gradient Yamabe soliton in perfect fluid spacetime with an extra condition.Precisely, in a perfect fluid spacetime, we consider the velocity vector field is of concircular type, introduced by Failkow [15] and we obtain some interesting results which are different from the results of the paper [13].

Gradient m-quasi Einstein solitons on perfect fluid spacetimes
Here, we investigate the perfect fluid spacetimes with concircular vector field with m-quasi Einstein metric and at first, we prove the following result Lemma 6.1.Every perfect fluid spacetimes with concircular vector field satisfies the following: for all U, V ∈ X(M ).
Proof.Let us assume that the perfect fluid spacetimes (with concircular vector field) with m-quasi Einstein metric.Then the equation (1.6) may be expressed as After executing covariant derivative of (6.2) along V , we get Exchanging U and V in (6.3), we lead and Utilizing (6.2)-(6.5)and the symmetric property of Levi-Civita connection In view of the equations (4.4), (4.8) and the above Lemma, we infer Taking a set of orthonormal frame field and executing contraction of the equation (6.6), we get

Conclusion
In true sense, solitons are nothing but the waves which is physically propagate with some loss of energy and hold their speed and shape after colliding with one more such wave.In nonlinear partial differential equations describing wave propagation, solitons play an important role in the treatment of initialvalue problems.
In this current investigation, we establish that a perfect fluid spacetime with concircular vector field is a generalized Robertson-Walker spacetime with Einstein fibre.Moreover, we prove that if a perfect fluid spacetime equipped with concircular vector field admits a second order symmetric parallel tensor, then either the state equation of the perfect fluid spacetime is characterized by p = 3−n n−1 σ , or the tensor is a constant multiple of the metric tensor.Also, different metrics like Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons and gradient m-quasi Einstein solitons are studied in the perfect fluid spacetimes with concircular vector field.Specifically, we obtain the condition for which the vector field ρ is steady, expanding and shrinking and observe that the spacetime represents a dark matter era under certain restriction on the vector field ρ.

Robertson-Walker = (RW ).
(1.2) takes the form Hess f − S − λg = 0, (1.3) where Hess and D indicates the Hessian and the gradient operator of g respectively.The metric obeying equation (1.3) is called a gradient Ricci soliton.The smooth function −f is said to be the potential function of the gradient Ricci soliton.

2 L
where t indicates the time and r is the scalar curvature of M .A semi-Riemannian manifold M n equipped with a semi-Riemannian metric g is called a Yamabe soliton if it obeys 1 W g = (r − λ)g (1.4) for real constant λ : M → R.Here L indicates the Lie derivative operator, W is a vector field, termed as the potential vector field and R is the set of real numbers.Yamabe soliton with W = Df reduces to the gradient Yamabe soliton on semi-Riemannian manifold M n .Thus, equation (1.4) takes the form Hessf = (r − λ)g.(1.5)

. 11 )
Applying the above equation and from the subsequent expression of Weyl conformal curvature tensor[29]

Theorem 4 . 3 .
Let the perfect fluid spacetimes with concircular vector field admit a gradient Ricci soliton with (ρα) = µβ.Then either the state equation of the perfect fluid spacetime is governed by p = 3−n n−1 σ and the soliton is steady, or f is invariant under the velocity vector field ρ.

Theorem 5 . 1 .
If the Lorentzian metric of a perfect fluid spacetime equipped with concircular vector field be a gradient Yamabe soliton, then either the spacetime represents a dark energy, or the gradient of Yamabe soliton potential function is pointwise collinear with the velocity vector field of the perfect fluid spacetime.

Corollary 5 . 2 .
Let the Lorentzian metric of a perfect fluid spacetime endowed with concircular vector field admits a gradient Yamabe soliton with β = 0.If f is invariant under the velocity vector field ρ, then either the state equation of the perfect fluid spacetime is governed by p = 3−n n−1 σ or the gradient Yamabe soliton is trivial.

. 7 )Theorem 6 . 2 .
Setting V = ρ in equations (6.7) and(4.11)and then equating the values of S(ρ, Df ), we find( m 1 − n + λ − α)(ρf ) = m{(ρα) − µβ}.(6.8)If f and α are invariant under the velocity vector field ρ, then we find from the foregoing equation that β = 0, since m = 0. Thus, following the proof of Theorem 5.1, we conclude our result as: Let a perfect fluid spacetime endowed with concircular vector field admits a gradient m-quasi Einstein soliton.If f and α are invariant under the velocity vector field ρ, then the spacetime represents a dark energy.