On generalized quasi-Einstein GRW space-times

TL;DR

This paper investigates generalized quasi-Einstein GRW space-times, showing they have constant scalar curvature and reduce to Einstein or perfect fluid space-times, expanding understanding of their geometric and physical properties.

Contribution

It demonstrates that generalized quasi-Einstein GRW space-times necessarily have constant scalar curvature and simplifies to known space-times, clarifying their structure.

Findings

Generalized quasi-Einstein GRW space-times have constant scalar curvature.

Such space-times reduce to Einstein or perfect fluid space-times.

The results extend the classification of GRW space-times.

Abstract

Recently, it is proven that generalized Robertson-Walker space-times in all orthogonal subspaces of Gray's decomposition but one(unrestricted) are perfect fluid space-times. GRW space-times in the unrestricted subspace are identified by having constant scalar curvature. Generalized quasi-Einstein GRW space-times have a constant scalar curvature. It is shown that generalized quasi-Einstein GRW space-times reduce to Einstein space-times or perfect fluid space-times.

Equations

$$\mathrm{Ric}\left(X,Y\right)=ag\left(X,Y\right)+bA\left(X\right)A\left(Y\right)$$

$$\mathrm{Ric}\left(X,Y\right)=\alpha g\left(X,Y\right)+\beta A\left(X\right)A\left(Y\right)+\gamma \left[A\left(X\right)B\left(Y\right)+A\left(Y\right)B\left(X\right)\right],$$

$$R_{ij}=\alpha g_{ij}+\beta A_i{A}_j+\gamma \left(A_i{B}_j+A_j{B}_i\right),$$

$$R=n\alpha +\beta .$$

$$A^i{A}^j{R}_{ij}$$

$$A^i{B}^j{R}_{ij}$$

$$B^i{B}^j{R}_{ij}$$

$$R=\left(n-1\right)B^i{B}^j{R}_{ij}+A^i{A}^j{R}_{ij}.$$

$$A^i{R}_{ij}=\alpha A^i{g}_{ij}+\beta A^i{A}_i{A}_j+\gamma \left(A^i{A}_i{B}_j+A^i{A}_j{B}_i\right)$$

$$B^i{R}_{ij}=\alpha B^i{g}_{ij}+\beta B^i{A}_i{A}_j+\gamma \left(B^i{A}_i{B}_j+B^i{A}_j{B}_i\right)$$

$$\left(\varphi -\alpha \right)B_j=\gamma A_j.$$

$$R_{ij}=\alpha g_{ij}+\beta A_i{A}_j$$

$$A^i{R}_{ij}=\alpha A^i{g}_{ij}+\beta A^i{A}_i{A}_j=\left(\alpha +\beta \right)A_j.$$

$$B^i{R}_{ij}=\alpha B^i{g}_{ij}+\beta B^i{A}_i{A}_j=\alpha A_j.$$

$${\nabla }_k{R}_{ij}$$

$${\nabla }_i{R}_{kj}$$

$${\mathcal{D}}_{kij}$$

$$A^j{\mathcal{D}}_{kij}$$

$$B^j{\mathcal{D}}_{kij}$$

$$0$$

$${\nabla }_j{R}_{ik}=2\left(\beta A_k+\gamma B_k\right){\nabla }_j{A}_i+2\gamma A_k{\nabla }_j{B}_i.$$

$$0$$

$${\nabla }_k{u}_j=\phi \left(g_{kj}+u_k{u}_j\right),$$

$$R_{ij}=\alpha g_{ij}+\beta A_i{A}_j+\gamma \left(A_i{B}_j+A_j{B}_i\right).$$

$$u^i{R}_{ij}=\alpha u^i{g}_{ij}+\beta u^i{A}_i{A}_j+\gamma \left(u^i{A}_i{B}_j+A_j{u}^i{B}_i\right)$$

$$\left(\xi -\alpha \right)u_j=\beta \left(u^i{A}_i\right)A_j+\gamma \left(u^i{A}_i\right)B_j+\gamma A_j\left(u^i{B}_i\right)$$

$$\left(\xi -\alpha \right)u_j=\left[\beta \left(u^i{A}_i\right)+\gamma \left(u^i{B}_i\right)\right]A_j+\gamma \left(u^i{A}_i\right)B_j$$

$$\left(\xi -\alpha -\beta \right)\left(u^i{A}_i\right)-\gamma \left(u^i{B}_i\right)=0$$

Full text

1. Introduction

The warped product $M=I\times_{f}M^{\ast}$ of an open connected interval $\left(I,-\mathrm{d}t^{2}\right)$ of $\mathbb{R}$ and a Riemannian manifold $M^{\ast}$ with warping function $f:I\rightarrow\mathbb{R}^{+}$ is called a generalized Robertson-Walker space-time(or GRW space-times)[12, 15]. This family of Lorentzian space-times broadly extends the classical Robertson-Walker space-times, Friedmann cosmological models, Einstein-de Sitter space-times and many others[2, 15]. The classical Robertson-Walker spacetime is regarded as cosmological models since it is spatially homogeneous and spatially isotropic whereas GRW space-times serve as inhomogeneous extension of Robertson-Walker space-times that admit an isotropic radiation[2](see also [4, 15]). A Lorentzian manifold is called a perfect fluid space-time if the Ricci tensor $\mathrm{Ric}$ takes the form

[TABLE]

where $a,b$ are scalars and $A$ is a $1-$form metrically equivalent to a unit time-like vector field[13, 14]. Perfect fluid space-times in the language of differential geometry are called quasi-Einstein spaces where $A$ is metrically equivalent to a unit space-like vector field. Recently, in [14], it is proven that a perfect fluid space-time with divergence-free conformal curvature tensor is a GRW space-time with Einstein fibers given that the scalar curvature is constant. Many sufficient conditions on perfect fluid space-times to be a GRW space-time are derived.

Gray presented an invariant orthogonal decomposition of the covariant derivative of the Ricci tensor[5](see also [10]). Recently, Carlo Mantica et al proved that the Ricci tensor of a generalized Robertson-Walker space-time in all classes of Gray’s decomposition but $\mathcal{A\oplus B}$ is either Einstein or takes the form of a perfect fluid whereas $\mathcal{A\oplus B}$ is not restricted[13]. The class $\mathcal{A\oplus B}$ is characterized by $\nabla R=0$ i.e. the scalar curvature is constant. Now, the following question arises.

Does the Ricci tensor of all GRW space-times in $\mathcal{A\oplus B}$ reduce to be Einstein or take the form of a perfect fluid?

In this work, we get a partial positive answer. A (pseudo-)Riemannian manifold $\left(M,g\right)$ is called a generalized quasi-Einstein manifold if its Ricci curvature satisfies

[TABLE]

where $\alpha,\beta$ and $\gamma$ are non-zero constants, $A$ and $B$ are $1-$forms corresponding to two orthonormal vector field[1, 3, 6, 7, 8, 9]. If $\gamma=0$, then $M$ reduces to a quasi-Einstein manifold. It is clear that generalized quasi-Einstein space-times are generally imperfect fluid space-times with constant scalar curvature $R=n\alpha+\beta$. However, we prove that generalized quasi-Einstein GRW space-times are either Einstein or perfect fluid space-times.

Remark 1**.**

It is noted that any vector field orthogonal to a time-like vector field is space-like. Thus the generators couldn’t be time-like. Now, one may assume that one of the generators is time-like and the other is space-like. Generally, the results of this article still hold in this case with minor changes.

2. Notes on generalized quasi-Einstein manifolds

Let $M$ be a generalized quasi-Einstein (pseudo-)Riemannian manifold i.e.

[TABLE]

where $A_{i}A^{i}=B_{j}B^{j}=1$ and $A_{i}B^{i}=0$. The trace of this equations gives

[TABLE]

It is noted that

[TABLE]

and consequently $\beta=\left(A^{i}A^{j}-B^{i}B^{j}\right)R_{ij}$. Hence we can state the following result.

Proposition 1**.**

Let $M$ be a generalized quasi-Einstein manifold with generators $A$ and $B$. Then the scalar curvature is

[TABLE]

Now assume that $A$ is an eigenvector of the Ricci tensor with eigenvalue $\xi$ i.e. $A^{i}R_{ij}=\xi A_{j}$. A contraction of Equation (2.1) with $A^{i}$ yields

[TABLE]

implies $\left(\xi-\alpha-\beta\right)A_{j}=\gamma B_{j}$. Thus $\gamma=0$ and $\xi=\alpha+\beta$. If $B$ is an eigenvector of the Ricci tensor with eigenvalue $\phi$, then

[TABLE]

infers

[TABLE]

Consequently, $\phi=\alpha$ and $\gamma=0$. Conversely, assume that $M$ is a quasi-Einstein manifold with generator $A$. Then

[TABLE]

yields

[TABLE]

Also,

[TABLE]

This leads to the following.

Theorem 1**.**

Let $M$ be a generalized quasi-Einstein manifold. Then, $M$ reduces to a quasi-Einstein manifold if and only if one of the generators is an eigenvector of the Ricci tensor.

The covariant derivative of the Ricci tensor of a generalized quasi-Einstein manifold is given by

[TABLE]

and hence

[TABLE]

Thus, the Codazzi deviation tensor $\mathcal{D}$ is

[TABLE]

Now, we have the following cases

[TABLE]

Thus, for a Codazzi Ricci tensor, the generators are both closed. In this case,

[TABLE]

Thus we have the following.

Proposition 2**.**

Let $M$ be a generalized quasi-Einstein manifold. Assume that $M$ is Einstein-like of class $\mathcal{B}$(i.e. the Ricci tensor is a Codazzi tensor). Then $A$ and $B$ are closed. Moreover,

[TABLE]

A contraction of $\mathcal{D}_{kij}$ by $g^{ij}$ and then by the generators $A^{k}$ and $B^{k}$ infers

[TABLE]

Thus $\nabla_{i}A^{i}=\nabla_{i}B^{i}=0$.

Assume that $M$ is Einstein-like of class $\mathcal{P}$( that is, the Ricci tensor is a parallel, $\nabla_{k}R_{ij}=0$). Then,

[TABLE]

Contractions by $A^{k}$ and $B^{k}$ imply

[TABLE]

Assume that $A$ is not parallel, then $\beta=\gamma=0$. Thus we conclude.

Theorem 2**.**

Let $M$ be a Ricci-symmetric generalized quasi-Einstein manifold. Then, $M$ is Einstein if the generator $A$ is not covariantly constant.

3. Generalized quasi-Einstein GRW space-times

A Lorentzian manifold $M$ is a GRW space-time if and only if $M$ has a unit time-like vector field $u_{i}$ such that

[TABLE]

which is also an eigenvector of the Ricci tensor i.e. $R_{ij}u^{i}=\xi u_{j}$ for some scalar functions $\varphi$ and $\xi$[11, 12, 13]. We say that $u$ is a nontrivial torse-forming vector field if $\varphi\neq 0$. This characterization is an alternative of Chen’s theorem in [2]. If $M$ is a generalized quasi-Einstein manifold, then

[TABLE]

A contraction by $u^{i}$ yields

[TABLE]

which implies

[TABLE]

and hence

[TABLE]

Two different contractions by the generators give

[TABLE]

and

[TABLE]

Thus

[TABLE]

and hence

[TABLE]

It is clear that $u_{j}$ is not a linear combination of $A_{j}$ and $B_{j}$ only since $u^{i}$ is time-like whereas $A^{i}$ and $B^{i}$ are orthonormal space-like fields so $\xi=\alpha$. Therefore,

[TABLE]

It is noted that $\gamma=\beta=0$ if $\left(u^{i}A_{i}\right)$ is not zero. Suppose that $\left(u^{i}B_{i}\right)$ does not vanish. Then Equation (3.3) implies that either $\gamma=0$ or $\left(u^{i}A_{i}\right)=0$. The later case with Equation (3.2) yield $\gamma=0$ i.e. $M$ is quasi-Einstein if $\left(u^{i}B_{i}\right)\neq 0$. Now, assume that $u^{i}$ is orthogonal to both the generators i.e. $\left(u^{i}A_{i}\right)=\left(u^{i}B_{i}\right)=0$. The Ricci tensor of a GQE manifold is

[TABLE]

and so

[TABLE]

A contraction by $u^{i}$ implies

[TABLE]

It is noted that $u^{i}$ is an eigenvector of the Ricci tensor ( i.e. $u^{i}R_{ij}=\xi u_{j}$) and $\nabla_{k}u^{i}=\varphi\left(\delta_{k}^{i}+u_{k}u^{i}\right)$. Thus

[TABLE]

So $M$ is Einstein if $u$ is a nontrivial torse-forming vector field.

Theorem 3**.**

Let $M$ be a generalized quasi-Einstein GRW space-time. Then $u^{i}R_{ij}=\alpha u_{j}$ i.e. $\alpha$ is the eigenvalue of the eigenvector $u^{i}$ and

1. (1)

$M$* reduces to be Einstein space-time if $u^{i}$ is orthogonal to both the generators provided $\varphi\neq 0$.* 2. (2)

$M$* reduces to be Einstein space-time if $u^{i}$ is not orthogonal to first generator.* 3. (3)

$M$* reduces to be perfect fluid space-time if $u^{i}$ is not orthogonal to the second generator.*

Corollary 1**.**

Let $M$ be a generalized quasi-Einstein Lorentzian manifold admitting a unit time-like non-trivial torse-forming vector field. Then $M$ reduces to an Einstein GRW space-time or a perfect fluid GRW space-time.