Impact of symmetry inheritance on conformally flat spacetime

TL;DR

This paper explores how curvature inheritance symmetry influences conformally flat spacetimes, showing that such spacetimes reduce to Einstein spacetimes under certain symmetries and conditions, with implications for dark matter and de Sitter space.

Contribution

It proves that conformally flat spacetimes with curvature inheritance symmetry are Einstein spacetimes and analyzes their properties in the context of Einstein's field equations and cosmological models.

Findings

Conformally flat spacetime reduces to Einstein spacetime with curvature inheritance symmetry.

Energy-momentum tensor is covariantly constant in such spacetimes.

Certain conditions lead to dark matter-like equations of state and de Sitter spacetime examples.

Abstract

The goal of this research paper is to investigate curvature inheritance symmetry in conformally flat spacetime. Curvature inheritance symmetry in conformally flat spacetime is shown to be a conformal motion. We have proven that a conformally flat spacetime reduces to Einstein spacetime if admits curvature inheritance symmetry. A few results on conformally flat spacetimes that obey Einstein's field equation with or without a cosmological constant, if admits the curvature inheritance symmetry. The energy-momentum tensor is to be covariantly constant in a 4-dimensional relativistic perfect fluid spacetime which is also conformally flat spacetime, admits curvature inheritance, and obeys Einstein's field equations in the presence of a cosmological constant. Moreover, it is also obtained that such spacetimes with perfect fluid satisfy the the vacuum-like equation of state consecutively it is dark matter. Finally, in the third part of the article, the case compatible with all Theorems from Theorem \ref{Th2.1} to Theorem \ref{Th2.5n} is shown. On the other hand, it has also been emphasized that it is an example of de Sitter spacetime. It has been demonstrated that this spacetime also has a conformal killing vector.