Characterization of generalized quasi-Einstein manifolds and modified gravity
Uday Chand De, Hülya Bağdatli Yilmaz
TL;DR
The paper analyzes generalized quasi-Einstein manifolds $(GQE)_n$ and their realization as spacetimes under a parallel unit time-like vector field. It develops a geometric classification showing that such spacetimes are PF, with a Ricci tensor of the form $R_{ij}=\lambda g_{ij}+\gamma\varphi_i\varphi_j$ and constant scalar curvature, placing them in Gray's $\mathfrak{B}$ and $\mathfrak{B}'$ subspaces and yielding $div\,\mathcal{C}=0$; in 4D these spacetimes are GRW and RW, conformally flat, and exhibit quasi-constant curvature. The work then incorporates $\mathcal{F}(\mathfrak{R})$-gravity to derive explicit relations for the effective energy-momentum components in $(GQE)_4$ with a parallel unit timelike field, showing $p$ and $\sigma$ in terms of $\mathcal{F}'(R)$, $\mathcal{F}(R)$, $R$, and $\lambda$, and identifying conditions under which dark-energy behavior is possible (or precluded) and radiation-era possibilities for specific $\mathcal{F}(R)$. Finally, it translates these results into energy-condition constraints, providing curvature-based inequalities that govern NEC, WEC, SEC, and DEC in this modified gravity setting, thereby linking generalized quasi-Einstein geometry to cosmologically relevant constraints.
Abstract
In this work, a detailed examination of a specific case of a generalized quasi-Einstein manifold (GQE)n is provided. It begins by exploring generalized quasi-Einstein spacetimes under certain conditions. The analysis then focuses on cases that admit a parallel time-like vector field. Among the findings, it is demonstrated that such spacetimes can be categorized as generalized Robertson-Walker spacetimes, Robertson-Walker spacetimes, and quasi-constant curvature spacetimes. Additionally, the physical implications of these results are discussed. It is also investigated (GQE)4 spacetimes, which accept F(R)-gravity and feature a parallel unit time-like vector field. Finally, various energy conditions are analyzed based on the results related to F(R)-gravity.