Characterization of perfect fluid spacetimes admitting gradient η-Ricci and gradient Einstein solitons

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Abstract

We set the goal to study the properties of perfect fluid spacetimes endowed with the gradient η-Ricci and gradient Einstein solitons.

Introduction

Let M be an n-dimensional semi-Riemannian manifold endowed with a semi-Riemannian metric g. In 1988, Hamilton [34] introduced the notion of Ricci flow on a Riemannian manifold. It is noticed that the Ricci flow is capable to resolve many long standing problems of science, medical science and technology. For example, it has applications in the string theory, thermodynamics, general relativity, cosmology, quantum field theory (see [24], [25], [26], [27], [51]). Also, the Poincaré conjecture, differentiable sphere conjecture (extension of Hamilton’s sphere theorem), uniformization theorem, geometrization conjecture have been solved with the help of Ricci flow (see, [3], [8], [9], [24], [25], [42], [43]) and their references.

If t is the time and S denotes the Ricci tensor of M, then the evolution equation tg=2S gives the Ricci flow on M. A triplet (g,V,λ) on M defined by LVg+2S+2λg=0 is said to be a Ricci soliton [33], which is a natural extension of Einstein metric, where LV represents the Lie derivative along the vector field V (soliton vector field) and λ is a real constant. A Ricci soliton is said to be expanding, shrinking or steady if λ is positive, negative or zero, respectively. In 2009, Cho and Kimura [22] generalized the notion of Ricci soliton and defined a new notion, called η-Ricci soliton. According to them, an η-Ricci soliton on M is a quadruple (g,V,λ,μ) if it satisfies the relation LVg+2S+2λg+2μηη=0for some real constant μ, where η is a 1-form. For deep results on Ricci and η-Ricci solitons, we cite ([4], [5], [6], [7], [10], [11], [19], [20], [21], [23], [49], [50]) and the references therein. If V is the gradient of a smooth function f, that is V=Df, where D denotes the gradient operator corresponding to the Levi-Civita connection , then Eq. (1.1) assumes the form Hessf+S+λg+μηη=0.Here Hess=2 is the Hessian, f denotes the potential function of the gradient η-Ricci soliton and represents the Levi-Civita connection.

Let g satisfies the relation S12rg+Hessf+λg=0,where r is the scalar curvature of M, then it is said to be a gradient Einstein soliton [12]. We call the smooth function f as Einstein potential function. If f is parallel (or f is constant), then the gradient Einstein soliton is said to be trivial [12]. If λ is positive, negative or zero, then the Einstein soliton is said to be expanding, shrinking or steady, respectively. Some remarkable results of gradient Einstein solitons have been proved in [13], [28].

Although most of the works on solitons have been done in Riemannian setting, the Ricci solitons and gradient Ricci solitons have been considered in the Lorentzian category [4], [6], [10].

A Lorentzian manifold M of dimension n is an n-dimensional semi-Riemannian manifold endowed with a Lorentzian metric g of signature (+,+,,+(n1)times,). The generalized Robertson–Walker (GRW) spacetime, introduced by Alías, Romero and Sánchez [1], [2], is a Lorentzian manifold M of dimension n3 such that M=I×h2M, where I is an open interval of R (real number), M is an (n1)-dimensional Riemannian manifold and h represents a smooth function, named as warping function or scale factor (see [37]). If M (dimM=3) possesses a constant scalar curvature, then the GRW spacetime reduces to the Robertson–Walker (RW) spacetime. Thus, the GRW spacetime is a natural generalization of RW spacetime. It is noticed that the GRW spacetimes include the Friedmann cosmological models, the static Einstein spacetimes, the Einstein–de Sitter spacetimes, the de Sitter spacetimes and have many applications as inhomogeneous spacetimes admitting an isotropic radiation [46]. The properties of GRW spacetimes have been studied by many researchers but few are [15], [16], [18], [29], [32], [37], [39], [44], [45], [46].

In 2014, Chen [18] has given a simple characterization of Lorentzian manifold of dimension n3 endowed with a time-like concircular vector field as:

Theorem 1.1 [18]

A Lorentzian n-manifold with n3 is a generalized Robertson–Walker spacetime if and only if it admits a time-like concircular vector field.

Recently, Mantica and Molinari [37] have also presented a characterization of Lorentzian manifold of dimension n3 equipped with a torse forming vector field as:

Theorem 1.2 [37]

A Lorentzian manifold of dimension n3 is a GRW spacetime if and only if it admits a unit time-like torse forming vector field: kuj=φ(gij+ujui), that is also an eigenvector of the Ricci tensor.

An n-dimensional perfect fluid spacetime is a Lorentzian manifold of dimension n>3 if its non-vanishing Ricci tensor S satisfies the relation S=αg+βηηfor some smooth functions α and β on M. Here ρ is a unit time-like vector field (also called the velocity vector field) of the perfect fluid spacetime, which is metrically equivalent to the 1-form η, that is, g(X,ρ)=η(X) for all X and g(ρ,ρ)=1. Every RW spacetime is a perfect fluid spacetime [40], although in case of n=4, the GRW spacetime is a perfect fluid spacetime if and only if it is a RW spacetime. For more details, we refer [7], [17], [36], [38], [47], [52] and the references therein.

We construct our work as follows:

Section 2 deals with some basic results of perfect fluid spacetimes and definitions. We study the geometrical and physical properties of gradient η-Ricci solitons on the perfect fluid spacetimes with the constant scalar curvature in Section 3. Section 4 is dedicated to study the properties of perfect fluid spacetimes with gradient Einstein solitons.

Section snippets

Perfect fluid spacetimes

In this section, we gather some basic results of the perfect fluid spacetimes and definitions.

From (1.4), we get r=nαβr is constantnX(α)=X(β)and QX=αX+βη(X)ρ,XX(M),where X(M) denotes the collection of all smooth vector fields of M and Q is the Ricci operator such that S(X,Y)=g(QX,Y), X,YX(M).

Lemma 2.1

Every perfect fluid spacetime satisfies (i)(Xη)(ρ)=0,(ii)η(Xρ)=g(Xρ,ρ)=0,XX(M).

Proof

The covariant derivative of g(ρ,ρ)=η(ρ)=1 along the vector field X gives g(Xρ,ρ)=0 for all XX(M). Again, (Yη)(X)

Gradient η-Ricci soliton on perfect fluid spacetimes

Throughout this section, we suppose that the perfect fluid spacetime M possesses the constant scalar curvature.

Proposition 3.1

A perfect fluid spacetime endowed with a gradient η-Ricci soliton satisfies R(X,Y)Df=(YQ)(X)(XQ)(Y)+μ[(Yη)(X)ρ+η(X)Yρ(Xη)(Y)ρη(Y)Xρ] for all X,YX(M).

Proof

From Eq. (1.2), we can write XDf+QX+λX+μη(X)ρ=0for all XX(M). The covariant derivative of Eq. (3.2) along the vector field Y gives YXDf=(YQ)(X)Q(YX)λYXμ[(Yη)(X)ρ+η(YX)ρ+η(X)Yρ]. Interchanging X and Y in Eq. (3.3)

Gradient Einstein solitons on perfect fluid spacetimes

This section is dedicated to study the properties of perfect fluid spacetime M if its metric is a gradient Einstein soliton.

In view of Eq. (1.3), we get XDf=QX+r2XλXfor all XX(M). From Eq. (4.1), it can be easily obtained that R(X,Y)Df=(YQ)(X)(XQ)(Y)+12[X(r)YY(r)X].Considering an orthonormal frame field on M and contracting equation (4.2) along the vector field X, we have S(Y,Df)=n22Y(r).Using Eq. (3.5) in (4.2), we infer R(X,Y)Df=β{(Yη)(X)ρ+η(X)Yρ(Xη)(Y)ρη(Y)Xρ}+Y(α)X+nY(α)η(X)ρ

Acknowledgments

The author expresses his sincere thanks to the Editor and anonymous referees for providing valuable suggestions in the improvement of the paper. He acknowledges authority of the University of Technology and Applied Sciences, Shinas for their continuous support and encouragement to carry out this research work.

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