Quasi-Einstein structures on f-associated standard static space-time

Abstract

In a generalized quasi-Einstein f-associated standard static space-time, the Laplacian Δ of the warping function f adheres to the Laplacian equation. The Ricci tensor of the base manifold satisfies the generalized Ricci-Hessian equation. The base manifold is shown to be quasi-Einstein if the Hessian $H^f$ of the warping function f is proportional to the metric tensor, and it is proven to be a generalized quasi-Einstein manifold if $H^f$ takes on the form of a perfect fluid tensor. The scalar curvature of both the space-time and its base manifold are provided. The Ricci curvature of the base manifold, the Laplacian of the warping function and the scalar curvature of the base manifold in a quasi-Einstein f-associated SSST are described in two scenarios: when the generator is space-like and when it is time-like. In the first scenario, it is demonstrated that the space-time is Ricci simple and the warping function remains constant if the base manifold is constant. In a generalized quasi-Einstein standard static space-time, the base manifold is a generalized quasi-Einstein manifold if the Hessian $H^f$ is proportional the metric tensor. A specific example of an f-associated standard static space-time is also provided. The equation of state takes the form $p=w\sigma$ where w ranges from −1 to 0, representing a transition from dark energy-dominated space-times to non-relativistic, matter-dominated ones.