Narrow Results

This paper characterizes the Lorentzian manifolds endowed with a semi-symmetric non-metric $\rho$-connection (briefly, ssnmρc). First, the existence of semi-symmetric non-metric connection (ssnmc) on Lorentzian manifold is established, and it is shown that an $n$-dimensional Lorentzian manifold equipped with an ssnmρc is a generalized Robertson–Walker spacetime. We also establish the condition for a Lorentzian manifold together with an ssnmρc to be a Robertson–Walker spacetime. In this series, the properties of Ricci semi-symmetric Lorentzian manifold endowed with an ssnmρc, almost Ricci solitons and gradient Ricci solitons are explored. Finally, a non-trivial example of Lorentzian manifold admits an ssnmρc, which is constructed to verify some of our results.

The aim of this paper is to obtain the condition under which a pseudosymmetric spacetime to be a perfect fluid spacetime. It is proven that a pseudosymmetric generalized Robertson–Walker spacetime is a perfect fluid spacetime. Moreover, we establish that a conformally flat pseudosymmetric spacetime is a generalized Robertson–Walker spacetime. Next, it is shown that a pseudosymmetric dust fluid with constant scalar curvature satisfying Einstein’s field equations without cosmological constant is vacuum. Finally, we construct a nontrivial example of pseudosymmetric spacetime.

In this paper, we investigate almost Schouten solitons and almost gradient Schouten solitons in spacetimes of general relativity. At first, it is proven that if a generalized Robertson–Walker spacetime permits an almost Schouten soliton, then it becomes a perfect fluid spacetime as well as the spacetime represents a dark matter era. Besides this, we investigate almost gradient Schouten solitons in generalized Robertson–Walker spacetimes. Moreover, a spacetime obeying almost Schouten solitons whose potential vector field is a non-homothetic conformal vector field is of Petrov type $O$ or $N$.

In this paper, we investigate Ricci–Yamabe solitons (RYSs) and gradient Ricci–Yamabe solitons (gradient RYSs) in generalized Robertson–Walker (GRW) spacetimes. At first, we prove that if a GRW spacetime admits a RYS, then it becomes a perfect fluid spacetime (PFS) and the divergence of the Weyl tensor vanishes. Also, a GRW spacetime admitting a RYS is of Petrov type $I$, $D$ or $O$ and in case of four dimension, the spacetime turns into a Robertson–Walker spacetime. Next, we show that if a GRW spacetime of constant scalar curvature admits a gradient RYS, then it becomes a PFS and the divergence of the Weyl tensor vanishes.

In general, a perfect fluid spacetime is not a generalized Robertson–Walker spacetime and the converse is also not true. In this paper, it is shown that if a perfect fluid spacetime satisfies the critical point equation, then either the spacetime becomes a generalized Robertson–Walker spacetime and represents dark era or the vorticity of the fluid vanishes as well as the spacetime is expansion free. Besides, we prove that if a generalized Robertson–Walker spacetime with constant scalar curvature satisfies the critical point equation, then the spacetime becomes a perfect fluid spacetime. Next, the existence of critical point equation is established by a non-trivial example. Finally, we discuss the critical point equation in $f(r)$-gravity. For the model $f(r)=r-\alpha (1-e^{-\frac{r}{\alpha }})$ ($\alpha$ = constant and $r$ is the scalar curvature of the spacetime), various energy conditions in terms of the scalar curvature are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions.

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an $\eta$–$\rho$-Einstein soliton, then the integral curves generated by the velocity vector field $u$ are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient $\eta$–$\rho$-Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an $\eta$–$\rho$-Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient $\eta$–$\rho$-Einstein soliton in $f({ℛ})$-gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function $\psi$ remains invariant under the velocity vector $u$. Also, for two models $f({ℛ})=e^{\alpha {ℛ}}+\alpha log{ℛ}$ and $f({ℛ})={{ℛ}}^n$ ($\alpha$ = constant and ${ℛ}$ is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition.