Narrow Results

This paper explores the characteristics of pseudo Ricci symmetric space-times within the vacuum solutions of $f(R)$-gravity. It is proved that a pseudo Ricci symmetric space-time satisfying the vacuum solutions of $f(R)$-gravity can be described as a perfect fluid space-time and its associated vector field ${\lambda }_i$ is torse-forming, Riemann compatible, Weyl compatible, and annihilates the Weyl tensor. Consequently, it is demonstrated that such a space-time can be classified as a generalized Friedman–Robertson–Walker space-time. In the case of four dimensions, it specifically corresponds to a Friedman–Robertson–Walker space-time.

A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

Recently, it is proven that generalized Robertson–Walker space-times in all orthogonal subspaces of Gray’s decomposition except 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.

In this paper, we classify proper non-static cylindrically symmetric (CS) perfect fluid space-times via conformal vector fields (CVFs) in the $f(R)$ gravity. In order to classify the space-times, we use the algebraic and direct integration approaches. In the process of classification, there exist 23 cases for which the considered space-times become proper non-static. By studying each case in detail, we find that the conformal vector fields are of dimensions two, three and fifteen in the $f(R)$ gravity.

In this paper, initially we solve the Einstein field equations (EFEs) for a static spherically (SS) symmetric perfect fluid space-times in the $f(T,B)$ gravity with the aid of some algebraic techniques. The extracted solutions are then utilized in order to get conformal vector fields (CVFs). It is important to mention that the adopted techniques enable us to obtain various classes of space-times with viable $f(T,B)$ gravity models which already exist in the literature. Excluding all such classes, we find that there exist three cases for which the space-times admit proper CVFs, whereas in rest of the cases, CVFs become KVFs. We have also highlighted some physical implications of our obtained results.

The aim of this paper is to classify non-conformally flat static plane symmetric (SPS) perfect fluid solutions via proper conformal vector fields (CVFs) in $f(T)$ gravity. For this purpose, first we explore some SPS perfect fluid solutions of the Einstein field equations (EFEs) in $f(T)$ gravity. Second, we utilize these solutions to find proper CVFs. In this study, we found 16 cases. A detailed study of each case reveals that in three of these cases, the space-times admit proper CVFs whereas in the rest of the cases, either the space-times become conformally flat or they admit proper homothetic vector fields (HVFs) or Killing vector fields (KVFs). The dimension of CVFs for non-conformally flat space-times in $f(T)$ gravity is four, five or six.

In this paper, we present a novel curvature tensor called the ${𝒦}$-curvature tensor, which is formed by combining the Weyl tensor and the ${{𝒲}}_2$-curvature tensor. It is proved that a semi-Riemannian manifold with traceless ${𝒦}$-curvature tensor is Einstein, while a ${𝒦}$-curvature flat semi-Riemannian manifold exhibits constant sectional curvature. We show that a space-time with traceless ${𝒦}$-curvature tensor or ${𝒦}$-curvature flat represents dark energy era. A semi-Riemannian manifold with divergence-free ${𝒦}$-curvature tensor is Weyl harmonic. In a space-time where the ${𝒦}$-curvature tensor is of divergence-free, both the isotropic pressure and energy density are constants. Under certain conditions, it is shown that a pseudo-${𝒦}$-symmetric semi-Riemannian manifold reduces to a pseudo-Riemannian manifold. It is demonstrated that pseudo-${𝒦}$-symmetric space-times can be regarded as generalized Robertson–Walker (GRW) space-times. Finally, a concrete example of pseudo-${𝒦}$-symmetric semi-Riemannian manifolds is presented.