Pseudo generalized Ricci-recurrent spacetimes with certain applications to modified gravity

Let $M^{n}$ ($n\geq 4$) be a semi (or, pseudo)-Riemannian manifold and $g$ be its semi-Riemannian metric with signature $(p,m)$, such that $p+m=n$. $M^{n}$ equipped with $g$ is referred to as a Lorentzian manifold [1] if $g$ is a Lorentzian metric with signature $(n-1,1)$ or, $(1,n-1)$. Time-oriented Lorentzian manifolds are Lorentzian manifolds that accept a globally time-like vector field, also physically referred to as spacetime. A generalized Robertson-Walker (shortly, GRW) spacetime is a Lorentzian manifold of dimension $n$ $\left(n\geq 4\right)$ which can be shaped by a warped product $-I\times_{\varPsi^{2}}\stackrel{{\scriptstyle\ast}}{{M}}$, $\stackrel{{\scriptstyle\ast}}{{M}}$ stands for $\left(n-1\right)$-dimensional Riemannian manifold, $I\in\mathbb{R}$ (set of real numbers) is an open interval, and $\varPsi>0$ is the warping function. This notion was presented by Alías et al. [2] in $1995$. In particular, GRW spacetime becomes Robertson-Walker ( shortly, RW) spacetime if we assume that $\stackrel{{\scriptstyle\ast}}{{M}}$ is a three dimensional Riemannian manifold of constant sectional curvature. For more details about GRW spacetimes, we refer ([2]-[7]).

A $4$-dimensional Lorentzian manifold $\mathrm{M}^{4}$ is described as a perfect fluid spacetime (shortly, PFS) if the Ricci tensor $\mathcal{R}_{lk}$ fulfills

$\alpha$, $\beta$ being scalars (not simultaneously zero) and the velocity vector $u_{k}$ stands for a unit time-like vector {that is, $u_{k}u^{k}=-1$, $u^{k}=g^{lk}u_{l}$}. The matter field in general relativity (in short, GR) is demonstrated by $T_{lk}$, which is called the energy momentum tensor (in short, EMT). Since heat conduction terms and stress terms corresponding to viscosity does not occur, the fluid is referred to as perfect [8]. The EMT [1] for a PFS has the following form

$p$, $\sigma$ being the isotropic pressure and the energy density, respectively. If $\sigma=p$, then the PFS is named stiff matter fluid. If $\sigma+p=0$, $p=0$ and $\sigma=3p$, then the PFS is called the dark energy epoch of the Universe, dust matter fluid and radiation era [9], respectively. Without the cosmological constant, the Einstein’s field equations (shortly, EFEs) are stated as

in which $\kappa$ and $\mathcal{R}$ stand for the gravitational constant and Ricci scalar, respectively. Equations (1.1), (1.2) and (1.3) infer that

In $\mathrm{M}^{4}$, the Weyl conformal curvature tensor $\mathcal{C}^{l}_{ijk}$ is demonstrated by

in which $\mathcal{R}^{l}_{ijk}$ denotes the curvature tensor and $\mathcal{R}^{l}_{k}=\mathcal{R}_{jk}g^{lj}$.

It is well-known that [10]

It is important to note that the Weyl geometry initially presented as a unifying theory of gravity and electromagnetic [11] that could challenge Einstein’s theory of gravity which is the main motivation behind its investigation in this article.

For the purpose of determining Einstein’s field equations, the study of spacetime symmetries is important. Symmetries are characteristic of geometry and reveal physics. There are many symmetries in matter and spacetime geometry. The metric equations are useful because they make many solutions easier. In GR, their primary utilization is that they classify solutions of Einstein’s field equations.

In 1952, Patterson [12] invented the idea of Ricci-recurrent manifolds. In 1995, De et al. [13] introduced the concept of generalized Ricci-recurrent manifolds. In a recent paper [14] Mallick et al. studied generalized Ricci-recurrent manifolds with applications to relativity. The significance of the Generalized Ricci-recurrent structure and its interaction with the modified $f(\mathcal{R})$-theory [15] and the modified Gauss-Bonnet $f(\mathcal{R},G)$-theory [16] are well established. On the other hand, pseudo $Z$-symmetric spacetimes have been investigated by Mantica and Suh [17] and Ozen[18] have studied $m$-projectively flat spacetimes. Also, in [19] $\Psi$-conformally symmetric spacetimes have been investigated. Moreover, in [20], we have studied $\psi$-conharmonically symmetric spacetimes. As well, many authors have looked at the spacetime of general relativity in various methods; for additional information, see ( [21], [22]). Inspired by foregoing studies we introduce and characterize a novel spacetime named pseudo generalized Ricci-recurrent spacetimes.

If the covariant derivative of $\mathcal{R}_{lk}$ is in the following form:

where $u_{i}$, $v_{i}$ and $w_{i}$ are non-zero covariant vectors, $\mathcal{D}$ is the $\left(0,2\right)$-type structure tensor of the manifold obeying

then we named such an $n$-dimensional manifold as pseudo generalized Ricci-recurrent manifold, denoted by $P\left(GR_{n}\right)$. If $w_{i}=0$, then $P\left(GR_{n}\right)$ reduces to a generalized Ricci-recurrent manifold $\left(GR_{n}\right)$ [13]. For $v_{i}=w_{i}=0$, $P\left(GR_{4}\right)$ becomes a Ricci-recurrent manifold $\left(R_{n}\right)$ [12]. If $\mathcal{R}_{lk}$ obeys (1.7) and $u_{i}$ is a unit time-like vector, then $\mathrm{M}^{4}$ is called a $P\left(GR_{4}\right)$ spacetime.

Multiplying (1.7) with $g^{lk}$ and using (1.8), we reach

The physical motivation for researching various spacetime models in general relativity and cosmology is to gain additional insight into particular phases of the universe’s evolution, which can be divided into the following three stages:
(i) The initial stage, (ii) The intermediate stage and (iii) The final stage.

The initial stage concerns viscous fluid whereas the intermediate stage concerns non-viscous fluid and both are admitting heat flux. The final stage equipped with thermal equilibrium tells about the perfect fluid stage. In our current study, we select the final stage.

The scientific world as a whole accepts the idea that our Cosmos is currently going through an accelerated phase. EFEs are not adequate to determine the late-time inflation of the cosmos without supposing the existence of certain unseen components that could account for the dark energy and dark matter origins. It is the primary source of inspiration for the extension to acquire higher order field equations of gravity.

According to GR theory, it is commonly accepted that energy conditions (ECs) are essential resources for studying black holes and wormholes in various modified gravities ([23]-[27]). The Raychaudhuri equations [28], which methodically produce the ECs, express the intriguing nature of gravity through the positivity condition $\mathcal{R}_{lk}u^{l}u^{k}\geq 0$, $u^{l}$ is a null vector. The last geometric criterion is the same as the null energy condition (NEC) $T_{lk}u^{l}u^{k}\geq 0$. Certainly, the weak energy condition (WEC) reflects that $T_{lk}u^{l}u^{k}\geq 0,$ for every time-like vector $u^{l}$ and preserves a positive local energy density. In addition, a spacetime fulfills the dominant energy condition (DEC) if $T_{lk}u^{l}v^{k}\geq 0$ holds for every two co-oriented time-like vectors $u$ and $v$ and strong energy condition (SEC) [29] if $R_{lk}u^{l}u^{k}\geq 0$ holds for all time-like vectors $u$. An overview of the four primary ECs is given here. A number of additional, lesser-known point wise ECs exist that express different restrictions on the stress energy tensor. A trace energy condition (TEC) was also present in the past. This implies that for a perfect fluid, $\sigma-3p\geq 0$. For a number of years in the 1950s and 1960s, this was thought to be a physically feasible state. Now, opinions have changed. Particularly, the TEC is violated with the finding of stiff equations of state for matter found in neutron stars [30]. We bring it up here as a specific illustration of an EC.

Interestingly, the idea of $f(\mathcal{R})$-gravity appears as a spontaneous extension of Einstein’s theory of gravity. Here, the function $f(\mathcal{R})$, in which $\mathcal{R}$ denotes the Ricci scalar, modifies the Hilbert-Einstein action term. The aforementioned theory was invented by Buchdahl [31], and Starobinsky [32] has demonstrated its validity through research on cosmic inflation. Through the introduction of certain couplings between the geometrical quantities and the matter sector, the $f(\mathcal{R})$ theory of gravity has been further generalized. The non-minimal coupling between the matter lagrangian density and the curvature invariant has been proved in [33], which is known as the $f(\mathcal{R},L_{m})$ theory of gravity. The corresponding Lagrangian can be modified by incorporating an analytic function of $T_{jk}T^{jk}$ in this generalization procedure for the $f(\mathcal{R},L_{m})$ theory. $f(\mathcal{R},T^{2})$ gravity or energy-momentum squared gravity is the result of selecting the corresponding Lagrangian. In 2014, Katirci and Kavuk [34] originally put forward this theory, which allows the existence of a term in the action functional that is proportional to $T_{jk}T^{jk}$. The $f(\mathcal{R},G)$-gravity theory [25] was one of these modified theories. It was developed by changing the previous Ricci scalar $\mathcal{R}$ by a function of $\mathcal{R}$ and $G$, the Gauss-Bonnet invariant. The $f(\mathcal{R},T)$-gravity theory, discovered by Harko et al. [26], was another modified theory. This is an extension of $f\left(\mathcal{R}\right)$-gravity ([23], [24]) in which the trace $T$ of the EMT is directly linked to any arbitrary function of $\mathcal{R}$. Several functional forms of $f(\mathcal{R})$ were previously provided in the following works: ([35]-[39]) which shows that this modified theory has a number of cosmological applications.

The literature mentioned above makes it very clear that more attention needs to be paid to $f(\mathcal{R})$ gravity, and there are still a lot of unanswered questions. Inspired by the foregoing investigations, this article is focused to investigate $P\left(GR_{4}\right)$ GRW spacetime satisfying $f(\mathcal{R})$ gravity. The $f(\mathcal{R})$ gravity model, $f\left(\mathcal{R}\right)=\mathcal{R}-\mu\mathcal{R}_{c}\tanh(\frac{\mathcal{R}}{\mathcal{R}_{c}})$ [40] satisfies local gravity constraint [41]. In this case, $f_{\mathcal{R}\mathcal{R}}>0$ if $\mathcal{R}>0$ and $f_{\mathcal{R}}>0$ if $\mu<1$. The condition $f_{\mathcal{R}\mathcal{R}}>0$ is required for the consistency of local gravity tests [42], for the presence of the matter-dominated epoch [43] and for the stability of cosmological perturbations [44]. In this paper, we choose the model $f\left(\mathcal{R}\right)=\mathcal{R}-\mu\mathcal{R}_{c}\tanh(\frac{\mathcal{R}}{\mathcal{R}_{c}})$ in which $\mu$ and $\mathcal{R}_{c}$ are positive constants, which is chosen to explain different ECs. The level of theoretical variety introduced by these higher curvature components is not found in more basic theories like General Relativity. This variability may open up new avenues for understanding and predicting late-time cosmic acceleration. Higher curvature components in gravity theories may result in changed dynamics at late-times, which may enable more realistic modelling of the cosmic acceleration at late-times and providing a better fit to observational data. The inclusion of higher curvature terms may be inspired by their importance in the effective action of quantum gravity theories, like string theory [45]. Examining their consequences for late-time cosmology can provide insight into the interaction between quantum physics and gravity. Investigating late-time cosmology in these structures offers a special chance to test basic physics outside of general relativity. Higher curvature terms may affect the behaviour of cosmic structures or affect cosmological perturbations, among other particular cosmological effects. Studying these implications can provide important insights into the fundamentals of gravity theory. These ideas might provide an alternate explanation for cosmic acceleration to dark energy, which could help to solve some of the long-standing cosmological problems. Higher curvature terms can have an impact on gravitational wave generation and propagation. Examining late-time cosmology in these theories may have consequences for gravitational wave signal detection and interpretation.

This article is structured as: In Section 2, an example of a $P\left(GR_{4}\right)$ spacetime is illustrated. $P\left(GR_{4}\right)$ GRW spacetimes and $P\left(GR_{4}\right)$ PFS are investigated in next two Sections. Finally, a $P\left(GR_{4}\right)$ GRW spacetime in $f(\mathcal{R})$ gravity theory is considered.

The covariant derivative of non-vanishing Ricci tensor is written as

The one-forms we select are as follows:

and

for all $y\in\mathbb{R}^{4}$.

We consider $\mathcal{D}_{ij}$ as follows:

for all $y\in\mathbb{R}^{4}$. It is enough to examine at the subsequent equations in order to confirm the relation (1.7):

and

The other cases hold trivially.

By applying the same deduction, it is possible to demonstrate that equation (2.12) is likewise true.

By virtue of (2.3) and (2.7) we find

Equations (2.10) and (2.13) together yield

Clearly, the trace$\left(\mathcal{D}_{ij}\right)=0.$ Therefore, $\left(\mathbb{R}^{4},g\right)$ is a $P\left(GR_{4}\right)$ spacetime.

Multiplying (1.7) with $u^{i}$ and using (1.8) infers

Since $\nabla_{k}\left(\mathcal{R}_{ij}u^{i}\right)=\left(\nabla_{k}\mathcal{R}_{ij}\right)u^{i}+\mathcal{R}_{ij}\left(\nabla_{k}u^{i}\right)$, (3.3) becomes

Equations (3.1), (3.2) and (3.4) together imply

Interchanging $j$ and $k$ in (3.5), we obtain

Subtracting (3.6) from (3.5), we have

Multiplying (3.7) with $u^{j}$, we acquire

that is,

From (3.5) and (3.9), it follows that

Hence, we reach:

In a $P\left(GR_{4}\right)$ GRW spacetime

in which

Multiplying equation (3.15) by $g^{il}$, we acquire

Thus the equation (3.14) is verified. Hence, we write:

The magnetic and electric parts of the Weyl tensor are given by

in which $u_{k}u^{k}=-1$ and $\tilde{C}_{kijl}=\frac{1}{2}{\epsilon}_{kilm}C^{lm}_{jl}$ is the dual [47]. These two tensors are traceless, symmetric and obey $E_{hk}u^{h}=0$ and $H_{hk}u^{h}=0$.

For $n=4$, from the definition of Weyl tensor, we provide

By hypothesis $P\left(GR_{4}\right)$ is a GRW spacetime.

Hence, using equation (3.15) in (3.19), we infer

From equation (3.1), we obtain

and hence we get

From the above, we can easily acquire

which entails

Multiplying the foregoing equation by $u^{k}$ yields

in which $f=u^{k}\varphi_{k}$.

Using equation (3.24) in (3.20), we get

Now, from equation (3.24), we provide

Also, from equation (3.15), we acquire

Hence, the last two equations jointly give

Using equation (3.28) in (3.25), we get $E_{ij}=0$.

Thus we can write

Let $P\left(GR_{4}\right)$ spacetime be a GRW spacetime. From the last two Theorems it follows that the velocity vector is Riemann compatible and the electric part of the spacetime under consideration vanishes. Hence, using Definition 1, we conclude that the velocity vector is Weyl compatible.

Also, we are aware that $H=0$, or the magnetic component of the Weyl tensor disappears when a spacetime’s vector field is Weyl compatible [46]. Therefore, the spacetime under consideration becomes conformally flat since $E=0$ and $H=0$ and hence is of petrov type O. Therefore we state:

The vorticity tensor $\mu_{lk}$ is described by

Therefore, using equations (4.7) and (4.9), we provide $\mu_{lk}=0$, that is, vorticity free.

The covariant gradient $\nabla_{k}u_{j}$ in a spacetime has the following standard decomposition [8]:

In view of the above equations, the last equation provides $\sigma_{kj}=0$, that is, shear free.

A spacetime is said to be stationary if $u_{i}$ is Killing and static ([48], [49], p. 283) for irrotational vector $u_{i}$. For a smooth vector $v$,

$\pounds$ denotes the Lie derivative. As $\nabla_{k}u_{i}=0$, then $\pounds_{u}g_{ki}=0$, means that $u_{i}$ is Killing. Moreover, $\nabla_{k}u_{i}=0$ gives $u_{i}$ is irrotational. Therefore, the spacetime is static.

Hence, it is possible to declare the following:

Here, we will look into a few characteristics of a $P\left(GR_{4}\right)$ GRW spacetime in $f(\mathcal{R})$ gravity.

Choose the Einstein-Hilbert action

in which $L_{m}$ is the matter Lagrangian density described as

In this case, $\kappa^{2}=8{\pi}G$, $G$ stands for the Newton’s constant and hence the modified formula can be written as [50]

The field equations can be expressed in the following form by applying the variation with $g^{ij}$

where $\square$ indicates the D’Alembert’s operator.

Taking trace of equation (5.1) provides

Subtracting the expression $\frac{\mathcal{R}f_{\mathcal{R}}(\mathcal{R})}{2}g_{ij}$ from (5.1), we infer

such that

in which

Using the equation (5.3), we acquire

which obeys the equations (5.4) and (5.5).

Multiplying equation (3.10) with $g^{jk}$, we reach

Utilizing (5.7) in (3.10), we get

Taking covariant derivative of the foregoing equation yields

Multiplying the above equation by $u^{k}$ provides

Again multiplying the previous equation by $u^{i}$ gives

where $u^{i}\nabla_{i}u_{j}=\dot{u_{j}}$ and $u^{i}\nabla_{i}\mathcal{R}=\dot{\mathcal{R}}$.

Using $u_{j}\dot{u_{j}}=0$, we readily conclude that the acceleration vector

Now transvecting equation (1.9) by $u^{i}$, we acquire

Covariant derivative of equation (1.9) produces

Interchanging $i$ and $j$ yields