Perfect-fluid, generalised Robertson-Walker space-times, and Gray's decomposition

TL;DR

This paper establishes new conditions on the Weyl tensor characterizing perfect-fluid generalized Robertson-Walker space-times, and analyzes the Ricci tensor's form via Gray's decomposition, linking geometric properties to physical equations of state.

Contribution

It provides necessary and sufficient conditions for GRW space-times to be perfect-fluid, and characterizes the Ricci tensor's form using Gray's decomposition, advancing understanding of their geometric and physical structure.

Findings

Weyl tensor conditions for perfect-fluid GRW space-times

Form of Ricci tensor in Gray's decomposition subspaces

Connection between Einstein equations and equations of state in 4D

Abstract

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tensor is Einstein or has the form of perfect fluid. We discuss the corresponding equations of state that result from the Einstein equation in dimension 4, where perfect-fluid GRW space-times are Robertson-Walker.

Tables

Table 1. Table 1. GRW space-times in Gray’s decomposition (E=Einstein, pf=perfect fluid, T [ j ​ k ] = T j ​ k − T k ​ j subscript 𝑇 delimited-[] 𝑗 𝑘 subscript 𝑇 𝑗 𝑘 subscript 𝑇 𝑘 𝑗 T_{[jk]}=T_{jk}-T_{kj} , T ( j ​ k ​ l ) = T j ​ k ​ l + T k ​ l ​ j + T l ​ j ​ k subscript 𝑇 𝑗 𝑘 𝑙 subscript 𝑇 𝑗 𝑘 𝑙 subscript 𝑇 𝑘 𝑙 𝑗 subscript 𝑇 𝑙 𝑗 𝑘 T_{(jkl)}=T_{jkl}+T_{klj}+T_{ljk} , c is a constant).

Subsp.	Condition on ${\nabla }_j{R}_{kl}$			$p\left(\mu \right)$, $n=4$
Trivial	${\nabla }_j{R}_{kl}=0$	Ricci symmetric	E	$p=-\mu$
$ℐ$	${\nabla }_j{R}_{kl}=a_j{g}_{kl}+b_k{g}_{lj}+b_l{g}_{jk}$	Sinyukov	pf	$p=-\frac{5}{3}\mu +c$
$𝒜$	${\nabla }_{(j}{R}_{kl)}=0$	Killing	$\varnothing$	–
$ℬ$	${\nabla }_{[j}{R}_{k]l}=0$	Codazzi	pf	$p=\frac{1}{3}\mu +c$
$ℐ\oplus 𝒜$	${\nabla }_{(j}{R}_{kl)}=\frac{2}{n+2}{\nabla }_{(j}R{g}_{kl)}$	Conformal Killing	pf	$p=-\frac{5}{3}\mu +c$
$ℐ\oplus ℬ$	${\nabla }_{[j}{R}_{k]l}=\frac{1}{2\left(n+2\right)}{\nabla }_{[j}R{g}_{k]l}$	${\mathring{\Re }}_{jkl}={\mathring{\Re }}_{kjl}$	pf	unrestricted
${ℐ}^{⟂}$	${\nabla }_{j}R=0$	Const. scalar curv.	–	–

Equations

$$d{s}^2=-d{t}^2+a^2\left(t\right)g_{\mu \nu }^{\ast }\left(\vec{x}\right)d{x}^{\mu }d{x}^{\nu }$$

$${\nabla }_j{u}_k=\phi \left(g_{jk}+u_j{u}_k\right)$$

$${\mathrm{R}}_{kl}=\frac{\mathrm{R}-n\eta }{n-1}{v}_k{v}_l+\frac{\mathrm{R}-\eta }{n-1}{g}_{kl}$$

$${\mathrm{R}}_{kl}=\frac{\mathrm{R}-n\xi }{n-1}{u}_k{u}_l+\frac{\mathrm{R}-\xi }{n-1}{g}_{kl}-\left(n-2\right){\mathrm{C}}_{jklm}{u}^j{u}^m$$

$$\left(\xi -\frac{\mathrm{R}-\eta }{n-1}\right)u_k=\frac{\mathrm{R}-n\eta }{n-1}(u^l{v}_l)v_k$$

$$(u_i{\mathrm{C}}_{jklm}+u_j{\mathrm{C}}_{kilm}+u_k{\mathrm{C}}_{ijlm})u^m=0.$$

$${\nabla }^k{\mathrm{C}}_{kl}=-\frac{n-3}{2(n-1)(n-2)}({\nabla }_l\mathrm{R}+u_l{u}^k{\nabla }_k\mathrm{R})$$

$$u^k{\nabla }_k{\mathrm{C}}_{jl}=-2\phi {\mathrm{C}}_{jl}$$

$${\nabla }^m{\mathrm{C}}_{jklm}=0$$

$$u^m{\mathrm{C}}_{jklm}=0$$

$${\mathrm{C}}_{kl}=0$$

$$u^j{\nabla }^m{\mathrm{C}}_{jklm}=0$$

$$u^m{\mathrm{C}}_{jklm}=u_k{\mathrm{C}}_{jl}-u_j{\mathrm{C}}_{kl}$$

$$u^m{\nabla }^j{\mathrm{C}}_{jklm}=-\phi (n-3){\mathrm{C}}_{kl}+u_k{\nabla }^j{\mathrm{C}}_{jl}$$

$${\mathrm{R}}_{kl}=\frac{\mathrm{R}-n\xi }{n-1}{u}_k{u}_l+\frac{\mathrm{R}-\xi }{n-1}{g}_{kl}$$

$${\nabla }_m{\mathrm{C}}_{jkl}{}^m=\frac{n-3}{n-2}\left[{\nabla }_k{\mathrm{R}}_{jl}-{\nabla }_j{\mathrm{R}}_{kl}-\frac{1}{2(n-1)}(g_{jl}{\nabla }_k\mathrm{R}-g_{kl}{\nabla }_j\mathrm{R})\right]$$

$$0=$$

$$=$$

$$-\frac{1}{2(n-1)}(u_l{\nabla }_k\mathrm{R}-g_{kl}{u}^j{\nabla }_j\mathrm{R})$$

$$=$$

$$0$$

$$=-{\nabla }_k\xi -u^j{\nabla }_j(\xi u_k)+{\mathrm{R}}_{kl}{u}^j{\nabla }_j{u}^l+\frac{1}{2(n-1)}({\nabla }_k\mathrm{R}+u_k{u}^j{\nabla }_j\mathrm{R})$$

$$=-({\nabla }_k\xi +u_k{u}^j{\nabla }_j\xi )+\frac{1}{2(n-1)}({\nabla }_k\mathrm{R}+u_k{u}^j{\nabla }_j\mathrm{R})$$

$$0=$$

$$=$$

$$+\frac{u^j{\nabla }_j\mathrm{R}}{2(n-1)}(g_{kl}+u_k{u}_l)$$

$$=$$

$${\nabla }_j{\mathrm{R}}_{kl}={\mathring{\mathfrak{R}}}_{jkl}+a_j{g}_{kl}+b_k{g}_{jl}+b_l{g}_{jk}$$

$$a_j=\frac{n}{\left(n-1\right)\left(n+2\right)}{\nabla }_j\mathrm{R},b_j=\frac{n-2}{2\left(n-1\right)\left(n+2\right)}{\nabla }_j\mathrm{R}$$

$${\mathring{\mathfrak{R}}}_{jkl}=\frac{1}{3}\left({\mathring{\mathfrak{R}}}_{jkl}+{\mathring{\mathfrak{R}}}_{klj}+{\mathring{\mathfrak{R}}}_{ljk}\right)+\frac{1}{3}\left({\mathring{\mathfrak{R}}}_{jkl}-{\mathring{\mathfrak{R}}}_{kjl}\right)+\frac{1}{3}\left({\mathring{\mathfrak{R}}}_{jkl}-{\mathring{\mathfrak{R}}}_{ljk}\right)$$

$${\nabla }_j{\mathrm{R}}_{kl}=a_j{g}_{kl}+b_k{g}_{lj}+b_l{g}_{jk}$$

$${\nabla }_j{\mathrm{R}}_{kl}+{\nabla }_k{\mathrm{R}}_{lj}+{\nabla }_l{\mathrm{R}}_{jk}=0.$$

$${\nabla }_j{\mathrm{R}}_{kl}={\nabla }_k{\mathrm{R}}_{jl}$$

$${\nabla }_j{\mathrm{R}}_{kl}+{\nabla }_k{\mathrm{R}}_{jl}+{\nabla }_l{\mathrm{R}}_{kj}=\frac{2{\nabla }_j\mathrm{R}}{n+2}{g}_{kl}+\frac{2{\nabla }_k\mathrm{R}}{n+2}{g}_{jl}+\frac{2{\nabla }_l\mathrm{R}}{n+2}{g}_{kj},$$

$${\nabla }_j\left[{\mathrm{R}}_{kl}-\frac{\mathrm{R}}{2\left(n-1\right)}{g}_{kl}\right]={\nabla }_k\left[{\mathrm{R}}_{jl}-\frac{\mathrm{R}}{2\left(n-1\right)}{g}_{jl}\right].$$

$${\nabla }_m{\mathrm{C}}_{jkl}{}^m=-\frac{n-3}{n-2}({\mathring{\mathfrak{R}}}_{jkl}-{\mathring{\mathfrak{R}}}_{kjl})$$

Full text

1. Introduction

Generalized Robertson-Walker (GRW) space-times are a natural and wide extension of RW spacetimes, where large scale cosmology is staged. They are Lorentzian manifolds of dimension $n$ characterized by the metric

[TABLE]

where $g_{\mu\nu}^{\ast}\left(\vec{x}\right)$ is the metric tensor of a Riemannian submanifold. A GRW space-time is thus the warped product $-I\times_{a}M^{*}$ where $I$ is an interval of the real line, $\left(M^{\ast},g^{\ast}\right)$ is a Riemannian manifold and $a>0$ is a smooth warping, or scale function. They have been deeply studied in the last years by several authors [1]-[8] (see the review [9]). Few years ago, Bang-Yen Chen [10, 11] characterized them by the presence of a time-like concircular vector in the sense of Fialkow [12]:

Theorem 1.1** (Chen, 2014).**

A Lorentzian manifold of dimension $n>3$ is a GRW space-time if and only if it admits a time-like concircular vector $X_{j}$: $X_{j}X^{j}<0$ and $\nabla_{k}X_{j}=\rho g_{kj}$, where $\rho$ a scalar function.

The associated unit time-like vector field $u_{j}=X_{j}/\sqrt{-X^{j}X_{j}}$ turns out to be torse-forming [13]:

[TABLE]

with $\varphi=\rho/\sqrt{-X^{j}X_{j}}$. In other words, $u_{j}$ is a velocity field without shear, vorticity and acceleration. The field $\varphi$, in the comoving frame, coincides with Hubble’s parameter: $\varphi=\dot{a}/a$.

The alternative characterization was obtained:

Theorem 1.2** (Mantica & Molinari, [9]).**

A Lorentzian manifold of dimension $n>3$ is a GRW space-time if and only if it admits a unit time-like torse-forming vector that is also eigenvector of the Ricci tensor.

A further extension are the twisted space-times, where the scale function $a$ may depend on all coordinates. They were introduced by B.-Y. Chen in 1979 [14], and later characterized by the existence of a time-like ‘torqued’ vector [15].

Theorem 1.3** (Mantica & Molinari, [16]).**

A Lorentzian manifold of dimension $n>3$ is a twisted space-time if and only if it admits a unit time-like torse-forming vector.

A Lorentzian manifold whose Ricci tensor has the form $\mathrm{R}_{kl}=Ag_{kl}+Bv_{k}v_{l}$ with scalar fields $A$, $B$, and a time-like ‘velocity field’, $v_{j}v^{j}=-1$, is named ‘perfect fluid’ space-time [14] (a Robertson-Walker space-time is perfect-fluid). In the geometric literature it is known as quasi-Einstein manifold [17, 18, 19] (without restriction on $v$). It is an Einstein space-time if ${\mathrm{R}}_{ij}=({\mathrm{R}}/n)g_{ij}$.

As $v_{j}$ is an eigenvector, the Ricci tensor can be parameterized in terms of the scalar curvature $\mathrm{R}$ and the eigenvalue $\eta$ as follows:

[TABLE]

In section 2, new necessary and sufficient conditions for a GRW space-time to be a perfect fluid, and for a perfect fluid to be a GRW space-time, will be given. They are based on the Weyl tensor, and extend our result in [3]. Theorems where the Weyl tensor is replaced by other curvature tensors are found in [20].

In section 3 we introduce Gray’s decomposition [21] of the tensor $\nabla_{i}{\mathrm{R}}_{jk}$ into $O(n)$ invariant subspaces, and discuss the special forms of the Ricci tensor of GRW space-times in each subspace. In all subspaces but one, the Ricci tensor is Einstein or perfect-fluid, with different restrictions on the scalar curvature and the eigenvalue. They reflect in the equations of state for the cosmological fluid’s pressure and energy density, determined by the Einstein equations, discussed in section 4 for dimension $n=4$. In $n=4$ the perfect fluid GRW space-times coincide with RW space-times.

In the paper the Lorentzian manifolds (space-times) have dimension $n>3$, and are smooth. When used, a dot means the directional derivative $u^{k}\nabla_{k}$.

2. Perfect-fluid and GRW space-times

We give new sufficient conditions for a GRW space-time to be perfect-fluid, and for the opposite occurrence. According to Prop.1.2, a GRW space-time is endowed with the special vector $u_{j}$ (2) that is eigenvector of the Ricci tensor. In Ref.[5] the following general structure of the Ricci tensor was obtained:

[TABLE]

where $\mathrm{R}=\mathrm{R}^{k}{}_{k}$ denotes the scalar curvature, $\mathrm{C}_{jklm}$ is the Weyl tensor and $\xi$ is the eigenvalue.

Remark 2.1**.**

Suppose that the GRW space-time is also perfect-fluid, i.e. there is a vector $v_{j}$ such that the Ricci tensor has the form (3). Then the condition ${\mathrm{R}}_{ij}u^{j}=\xi u_{i}$ gives

[TABLE]

Since both $u_{k}$ and $v_{k}$ are time-like, it cannot be $u^{k}v_{k}=0$. Then, unless the space-time is Einstein, it must be $v_{k}=\pm u_{k}$ and $\xi=\eta$.

We now recall the properties of GRW space-times that are necessary for the discussion. They are mainly taken from Ref. [5].

$\bullet$ Unit torse-forming vectors have the property named Weyl compatibility [22, 23]:

[TABLE]

$\bullet$ The eigenvalue is $\xi=(n-1)(\varphi^{2}+u^{k}\nabla_{k}\varphi)$, and $\nabla_{k}\xi=-u_{k}(u^{j}\nabla_{j}\xi)$.

$\bullet$ The contracted Weyl tensor ${\mathrm{C}}_{kl}=u^{j}u^{m}{\mathrm{C}}_{jklm}$ has the properties [5, eqs.14,15]

[TABLE]

The following proposition contains the new statement (11):

Proposition 2.2**.**

In a GRW space-time, the following statements are equivalent:

[TABLE]

Proof.

The equivalence of (8) with (9) is theorem 3.4 in [5]. The equivalence of (9) with (10) follows from the identity

[TABLE]

which is obtained by contracting (5) with $u^{j}$. Now, (10) is equivalent to (8) that implies (11). Let us show that (11) implies (10).

The covariant divergence of (12) $(\nabla^{j}u^{m}){\mathrm{C}}_{jklm}+u^{m}\nabla^{j}{\mathrm{C}}_{jklm}=(\nabla^{j}u_{k})\mathrm{C}_{jl}+u_{k}\nabla^{j}{\mathrm{C}}_{jl}-(\nabla^{j}u_{j})\mathrm{C}_{kl}-u^{j}\nabla_{j}\mathrm{C}_{kl}$ gives: $u^{m}\nabla^{j}{\mathrm{C}}_{jklm}=-\varphi(n-1)\mathrm{C}_{kl}+u_{k}\nabla^{j}{\mathrm{C}}_{jl}-u^{j}\nabla_{j}\mathrm{C}_{kl}$. Next use (7):

[TABLE]

If $u^{m}\nabla^{j}{\mathrm{C}}_{jklm}=0$ then $\varphi(n-3)\mathrm{C}_{kl}=u_{k}\nabla^{j}{\mathrm{C}}_{jl}$. Contraction with $u^{k}$ gives $\nabla^{j}{\mathrm{C}}_{jl}=0$, but then also ${\mathrm{C}}_{kl}=0$. ∎

In consideration of the general form (4) and of the Remark 2.1, we conclude:

Theorem 2.3**.**

A GRW space-time is perfect fluid if and only if ${\mathrm{C}}_{kl}=0$, or any of the equivalent conditions in Prop. 2.2.

Now we investigate the problem of a perfect-fluid space-time to be GRW. Namely, given

[TABLE]

we give conditions for the unit time-like vector $u_{j}$ to be torse-forming. An answer was given with Th 2.1 in [3]. Now we extend the result:

Theorem 2.4**.**

A perfect-fluid space-time is GRW if the vector field $u_{j}$ has the properties: $u^{j}\nabla^{m}{\mathrm{C}}_{jklm}=0$ and $u^{k}\nabla_{k}u_{j}=0$.

Proof.

The general formula for the divergence of the Weyl tensor is [24]:

[TABLE]

Contraction with $u^{j}$ and use of (14) give:

[TABLE]

Contraction with $u^{l}$ and use of $u^{j}\nabla_{k}u_{j}=0$ and $u^{j}\nabla_{j}u^{l}=0$ give:

[TABLE]

The relation $\nabla_{k}\xi-\tfrac{1}{2(n-1)}\nabla_{k}\mathrm{R}=-u_{k}u^{j}\nabla_{j}(\xi-\tfrac{1}{2(n-1)}\mathrm{R})$ is inserted back:

[TABLE]

Contraction with $g^{kl}$ gives $\tfrac{\mathrm{R}-n\xi}{n-1}(\nabla_{k}u^{k})=-\tfrac{1}{2}u^{k}\nabla_{k}(\mathrm{R}-2\xi)$. Then, if $\mathrm{R}-n\xi\neq 0$, $\nabla_{k}u_{l}=\frac{\nabla_{j}u^{j}}{n-1}(g_{kl}+u_{k}u_{l})$, i.e. the unit time-like vector is torse-forming. ∎

The case $\xi=\mathrm{R}/n$ corresponds to an Einstein space-time: ${\mathrm{R}}_{ij}=(\mathrm{R}/n)g_{ij}$.

The case $u^{k}\nabla_{k}(\mathrm{R}-2\xi)=0$ with $\xi\neq\mathrm{R}/n$, corresponds to $\nabla_{k}u_{l}=0$. The space-time now factors, as the scale factor in (1) is trivial ($a=1$).

3. Gray’s decomposition and GRW space-times

A. Gray [21] (see also [25][26, Ch.16]) found that the gradient of the Ricci tensor $\nabla_{j}\mathrm{R}_{kl}$ can be decomposed into $O\left(n\right)$ invariant terms (see [27, 28]):

[TABLE]

where $\mathfrak{\mathring{R}}_{jkl}=\mathfrak{\mathring{R}}_{jlk}$ is trace-less i.e. $\mathfrak{\mathring{R}}^{j}{}_{jk}=\mathfrak{\mathring{R}}_{kj}{}^{j}=0$ and

[TABLE]

The trace-less tensor can be decomposed as a sum of orthogonal components

[TABLE]

The decomposition (16), (18) provides $O\left(n\right)$ invariant subspaces, characterized by invariant equations that are linear in $\nabla_{j}\mathrm{R}_{kl}$. In Gray’s notation:

$\bullet$ The trivial subspace $\nabla_{j}\mathrm{R}_{kl}=0$.

$\bullet$ The subspace $\mathcal{I}$ where $\mathfrak{\mathring{R}}_{jkl}=0$, i.e.

[TABLE]

Manifolds satisfying this condition are called Sinyukov manifolds [29].

$\bullet$ The orthogonal complement $\mathcal{I}^{\perp}$ where $\nabla_{j}\mathrm{R}_{kl}=\mathfrak{\mathring{R}}_{jkl}$ or, equivalently, $a_{j}g_{kl}+b_{k}g_{lj}+b_{l}g_{jk}=0$. Then $\mathcal{I}^{\perp}$ is only characterized by the equation $\nabla_{j}\mathrm{R}=0$.

The decomposition (18) of $\mathfrak{\mathring{R}}_{jkl}$ specifies orthogonal subspaces, and $\mathcal{I}^{\perp}=\mathcal{A}\oplus\mathcal{B}\oplus\mathcal{B}^{\prime}$, where ${\mathcal{B}}^{\prime}$ is a copy of ${\mathcal{B}}$ with indices exchanged.

In $\mathcal{A}$ it is $\mathfrak{\mathring{R}}_{jkl}+\mathfrak{\mathring{R}}_{klj}+\mathfrak{\mathring{R}}_{ljk}=0$ and $\nabla_{j}\mathrm{R}=0$, i.e. the Ricci tensor is a Killing tensor [30]:

[TABLE]

In $\mathcal{B}$ and $\mathcal{B^{\prime}}$ it is $\mathfrak{\mathring{R}}_{jkl}-\mathfrak{\mathring{R}}_{kjl}=0$ and $\nabla_{j}\mathrm{R}=0$, i.e. the Ricci tensor is a Codazzi tensor:

[TABLE]

In all cases the condition $\nabla_{j}{\mathrm{R}}=0$ is a consequence. Now, we consider two composite subspaces.

The subspace $\mathcal{I}\oplus\mathcal{A}$ contains tensors that satisfy the cyclic condition

[TABLE]

i.e. the Ricci tensor is a conformal Killing tensor [30]. Note that the cyclic sum of (19) gives (22) (the Ricci tensor of a Sinyukov manifold is conformal Killing).

The subspace $\mathcal{I}\oplus\mathcal{B}$ contains tensors that satisfy the Codazzi condition

[TABLE]

Manifolds satisfying conditions (19)-(23) are also called “Einstein-like manifolds” (see [31] and references therein).

It is interesting to find the form of the Ricci tensor of GRW space-times in Gray’s subspaces. The gradient of the Ricci tensor and the divergence of the Weyl tensor are linked by the identity (15), which becomes:

[TABLE]

3.1. Ricci tensor in the trivial subspace

If $\nabla_{j}\mathrm{R}_{kl}=0$ the gradient of $R_{kl}u^{l}=\xi u_{k}$ gives $\mathrm{R}_{kl}=\xi g_{kl}$: the GRW space-time is Einstein.

3.2. Ricci tensor in the subspace $\mathcal{I}$

The Ricci tensor in the subspace $\mathcal{I}$ satisfies the condition $\mathfrak{\mathring{R}}_{jkl}=0$ or (19). Eq.(24) shows that a Sinyukov manifolds is perfect fluid (quasi-Einstein).

Lemma 3.1**.**

If the tensor $\alpha_{j}g_{kl}+\beta_{k}g_{lj}+\gamma_{l}g_{jk}+\delta_{j}v_{k}v_{l}$, with $v^{2}=v^{k}v_{k}\neq 0$, is zero, then the vector coefficients are zero, $\alpha_{i}=\beta_{i}=\gamma_{i}=\delta_{i}=0$.

Proof.

Contraction with $v^{k}$ gives $\alpha_{j}v_{l}+(\beta_{k}v^{k})g_{lj}+\gamma_{l}v_{j}+v^{2}\delta_{j}v_{l}=0$. Contraction with $\tau^{l}\tau^{j}$, with $\tau^{k}v_{k}=0$ and $\tau^{2}\neq 0$ gives $\beta_{k}v^{k}=0$. Then $(\alpha_{j}+\delta_{j}v^{2})v_{l}+\gamma_{l}v_{j}=0$ i.e. both $\gamma_{l}$ and $(\alpha_{l}+\delta_{l}v^{2})$ are parallel to $v_{l}$.

Similarly, contraction with $v^{l}$ and then with $\tau^{j}\tau^{k}$ gives $\gamma_{l}v^{l}=0$ and both $\beta_{l}$ and $(\alpha_{l}+\delta_{l}v^{2})$ parallel to $v_{l}$. Now, $\beta_{l}$ cannot be orthogonal and parallel to $v_{l}$, and the same for $\gamma_{l}$. Then $\beta_{l}=\gamma_{l}=0$. Next, consider $\alpha_{j}g_{kl}+\delta_{j}v_{k}v_{l}=0$. Contraction with $\tau^{k}$ gives $\alpha_{j}=0$, and then $\delta_{j}=0$. ∎

Theorem 3.2**.**

The Ricci tensor of a GRW space-time belongs to $\mathcal{I}$ if and only if the space-time is perfect fluid and $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$. In the comoving frame it is

[TABLE]

where $\alpha$ and $\beta$ are constants and $a(t)$ is the warping function.

Proof.

If the Ricci tensor belongs to $\mathcal{I}$, then (19) and (17) give $\nabla_{m}\mathrm{C}_{jkl}{}^{m}=0$, the Ricci tensor has the perfect fluid form (14), and $\nabla_{j}{\mathrm{R}}=-u_{j}\dot{\mathrm{R}}$. Now, evaluate:

[TABLE]

By subtracting (19) one obtains a null tensor which, by the previous lemma, implies the constraints:

[TABLE]

The system is supplied with the equation resulting from the covariant derivative of ${\mathrm{R}}_{kl}u^{l}=\xi u_{k}$:

[TABLE]

The system is degenerate and has solution

[TABLE]

The equations (27) can be integrated. In the comoving frame it is $\varphi=\dot{a}/a$, then the second equation yields $\mathrm{R}-n\xi=\beta a(t)^{2}$, where $\beta$ is a constant an $a(t)$ is the warping function. The first equation is now used: $\dot{\mathrm{R}}=-2\beta\frac{n+2}{n-2}a\dot{a}$, and the results are obtained.

On the other hand, suppose that the GRW space-time is perfect fluid, and that $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$ holds. The gradient of the Ricci tensor (14) is

[TABLE]

The first term is zero because for a GRW perfect fluid: $\nabla_{j}({\mathrm{R}}-n\xi)=-u_{j}(\dot{\mathrm{R}}-n\dot{\xi})=-2\varphi u_{j}({\mathrm{R}}-n\xi)$ by (26) and $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$. Then:

[TABLE]

which is the Sinyukov condition (19). ∎

3.3. Ricci tensor in the subspace $\mathcal{A}$

The subspace $\mathcal{A}$ is characterized by the condition $\nabla_{k}\mathrm{R}_{ij}+\nabla_{i}\mathrm{R}_{kj}+\nabla_{j}\mathrm{R}_{ki}=0$, giving $\nabla_{k}\mathrm{R}=0$. We now show that (in a GRW space-time) it is $\nabla_{k}R_{jl}=0$. The subspace is then empty, as the case is accounted for by the trivial subspace.

Theorem 3.3**.**

In the subspace $\mathcal{A}$ the Ricci tensor is Einstein.

Proof.

Contraction with $u^{j}$ of (20) gives:

[TABLE]

Next, use $\nabla_{k}\xi=-u_{k}\dot{\xi}$ to obtain:

[TABLE]

Contraction with $u^{k}$ gives: $0=3u_{l}\dot{\xi}$ i.e. $\dot{\xi}=0$. On the other hand, contraction with $g^{kl}$ gives $\dot{\mathrm{R}}+2\dot{\xi}=2\varphi({\mathrm{R}}-n\xi)$. Since $\dot{\mathrm{R}}=0$ and $\dot{\xi}=0$ it is ${\mathrm{R}}=n\xi$. Then:

[TABLE]

If this is inserted in (29), we obtain $u^{p}\nabla_{p}{\mathrm{C}}_{kl}=2\varphi{\mathrm{C}}_{kl}$. This is in contrast with (7), unless ${\mathrm{C}}_{kl}=0$. ∎

3.4. Ricci tensor in the subspace $\mathcal{B}$

In this subspace the Ricci tensor is Codazzi, (21). A contraction with the metric tensor gives $\nabla_{k}\mathrm{R}=0$, and (15) gives $\nabla_{m}\mathrm{C}_{jkl}{}^{m}=0$. Therefore, the GRW space-time is perfect fluid.

The equation $\dot{\xi}=\varphi(R-n\xi)$ can be integrated: in the comoving frame, where $\varphi=\dot{a}/a$, the eigenvalue depends on time through the warping function as $\xi(t)=\alpha a(t)^{-n}+{\mathrm{R}}/n$, with constant $\alpha$.

3.5. Ricci tensor in the subspace ${\mathcal{I}}^{\perp}$

In this case $\nabla_{k}{\mathrm{R}}=0$. The GRW space-time is not in general perfect-fluid, with the Ricci tensor having the form (4). However, the equation $\dot{\xi}=\varphi(R-n\xi)$ can be integrated and $\xi(t)=\alpha a(t)^{-n}+{\mathrm{R}}/n$, with constant $\alpha$.

3.6. Ricci tensor in the subspace $\mathcal{I}\oplus\mathcal{B}$

The Ricci tensor satisfies the Codazzi condition (23), which is necessary and sufficient for the divergence of the Weyl tensor (15) to vanish. Therefore, the Ricci tensor has the perfect fluid form (14).

3.7. Ricci tensor in the subspace $\mathcal{I}\oplus\mathcal{A}$

In this subspace the Ricci tensor is conformal Killing, (22) (see [32]). We now show that the subspaces $\mathcal{I}\oplus\mathcal{A}$ and $\mathcal{I}$ coincide.

Theorem 3.4**.**

The Ricci tensor in a GRW space-time is conformal Killing if and only if it has the perfect fluid form and $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$, i.e. it belongs to $\mathcal{I}$.

Proof.

Suppose that the Ricci tensor is conformal Killing. On multiplying (22) by $u^{j}u^{k}$ we get

[TABLE]

It is $u^{j}u^{k}\nabla_{l}\mathrm{R}_{kj}=u^{j}\nabla_{l}(\xi u_{j})-u^{j}\mathrm{R}_{kj}\nabla_{l}u^{k}=-\nabla_{l}\xi-\xi u_{k}\nabla_{l}u^{k}=-\nabla_{l}\xi=u_{l}\dot{\xi}$. Then:

[TABLE]

Contraction with $u^{l}$ gives $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$. This, when inserted back, gives $\nabla_{k}R=-\dot{\mathrm{R}}u_{k}$ and, because of (6): $\nabla_{k}{\mathrm{C}}^{k}{}_{l}=0$. The general property (26), $\dot{\mathrm{R}}-2\dot{\xi}=-2\varphi(R-n\xi)$ gives:

[TABLE]

Transvect (22) by $u^{j}$ and simplify with $\nabla_{k}\mathrm{R}=-\dot{\mathrm{R}}u_{k}$ and the identity $u^{j}\nabla_{k}{\mathrm{R}}_{jl}=\nabla_{k}(\xi u_{l})-{\mathrm{R}}_{jl}\nabla_{k}u^{j}=-\dot{\xi}u_{k}u_{l}+\varphi\xi g_{kl}-\varphi{\mathrm{R}}_{kl}$

[TABLE]

Now use $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$ and obtain: $u^{j}\nabla_{j}\mathrm{R}_{kl}=2\varphi{\mathrm{R}}_{kl}-2\varphi\xi g_{kl}+\dot{\xi}g_{kl}$. The left-hand side of the equation is now evaluated with (4), with the aid of (7):

[TABLE]

This and the previous equation imply ${\mathrm{C}}_{kl}=0$. Then the GRW space-time is perfect-fluid with $(n+2)\dot{\xi}=2\dot{\mathrm{R}}$.

The proof of the opposite statement runs as for theorem 3.2, and obtains (28). A cyclic summation gives that the Ricci tensor is conformal Killing. ∎

4. Perfect-fluid equations of state

We examine the equations of state that arise from the perfect-fluid solutions in Gray’s subspaces. The perfect-fluid form of the Ricci tensor corresponds, via the Einstein equations ${\mathrm{R}}_{ij}-\tfrac{1}{2}Rg_{ij}=\kappa T_{ij}$ to a perfect fluid energy-momentum tensor $T_{ij}=(p+\mu)u_{i}u_{j}+pg_{ij}$. By assuming the expression (14) for the Ricci tensor, the Einstein equation gives the pressure and the energy-density in terms of $\mathrm{R}$ and $\xi$:

[TABLE]

We recall Proposition 3.1 in [3]: a perfect fluid space-time in dimension $n\geq 4$ with differentiable state equation $p=p\left(\mu\right)$, $p+\mu\neq 0$ and with null divergence of the Weyl tensor $\nabla_{m}\mathrm{C}_{ijk}{}^{m}=0$ is a GRW space-time. Null divergence implies that the Ricci tensor belongs to the subspace $\mathcal{I\oplus B}$. On the other hand, consider a perfect fluid GRW space-time with state equation $p=\frac{\mu}{n+1}+\mathrm{constant}$: it follows that $\nabla_{k}R=0$. This and $\nabla_{m}\mathrm{C}_{ijk}{}^{m}=0$ give that the Ricci tensor is Codazzi, i.e. it belongs to $\mathcal{B}$. Then a perfect fluid GRW space-time with a state equation different from $p=\frac{\mu}{n+1}+\mathrm{constant}$ belongs to $\mathcal{I\oplus B}$ and not to $\mathcal{B}$.

Hereafter, we restrict to dimension $n=4$, where a GRW perfect-fluid space-time is exactly a Robertson-Walker (RW) space-time (this follows from $u^{m}{\mathrm{C}}_{jklm}=0$ which, in $n=4$, is equivalent to $u_{i}{\mathrm{C}}_{jklm}+u_{j}{\mathrm{C}}_{kilm}+u_{k}{\mathrm{C}}_{ijlm}=0$ as shown in Lovelock and Rund, [33] page 128. Contraction with $u^{i}$ gives ${\mathrm{C}}_{jklm}=0$).

The cases are:

$\bullet$ In the trivial subspace the space-time is Einstein $({\mathrm{R}}=4\xi$, $p=-\mu$).

$\bullet$ In $\mathcal{B}$ the Ricci tensor is Codazzi and the RW space-time is a “Yang’s Pure Space” [34]. Since ${\mathrm{R}}$ is constant, eq.(31) gives the equation of state $p=\frac{1}{3}\mu-\frac{1}{3\kappa}{\mathrm{R}}$.

The RW spaces with constant ${\mathrm{R}}$ are described, for example, [35]. In the expanding ones, the time evolution of the eigenvalue $\xi(t)=\alpha a(t)^{-4}+\tfrac{1}{4}{\mathrm{R}}$ drives the space-time to an Einstein space-time with $\kappa\mu_{\infty}={\mathrm{R}}/4$ and negative pressure $\kappa p_{\infty}=-{\mathrm{R}}/4$. Then, asimptotically in the future $p=-\mu$. The case with spatial curvature $R^{*}=0$ and its cosmological implications are studied in [36].

$\bullet$ In $\mathcal{I}$ the solution (25) gives the dependence in the cosmological time of pressure and density, via the warping function:

[TABLE]

Elimination of $\beta a(t)^{2}$ gives a phantom-type equation of state $p=-\frac{5}{3}\mu+\rm{const}$, studied by Caldwell [37].

$\bullet$ In $\mathcal{I}\oplus\mathcal{B}$ the RW space-time is unrestricted.

$\bullet$ In ${\mathcal{I}}^{\perp}$, the Ricci tensor contains the Weyl term, then it is not perfect-fluid, and the GRW space-time is not RW. However, the condition $\nabla_{k}{\mathrm{R}}=0$ gives the time evolution of the eigenvalue of the Ricci tensor $\xi(t)=\alpha a(t)^{-4}+\frac{1}{4}{\mathrm{R}}$.

Acknowledgments

The third author was supported by grant Proj. NRF-2018-R1D1A1B-05040381 from the National Research Foundation of Korea.

We thank the referee for his valuable suggestions, that helped us in improving and clarifying the final form of the paper.