On a conformal Schwarzschild-de Sitter spacetime

TL;DR

This paper explores a conformal Schwarzschild-de Sitter spacetime derived from the C-metric, analyzing its properties, extremal case, weak-field limit, and geodesic structure, revealing connections to Rindler and de Sitter geometries.

Contribution

It provides a detailed investigation of the conformal Schwarzschild-de Sitter spacetime, including source stress tensor, extremal case, and its relation to Rindler and de Sitter geometries, with new insights into its properties.

Findings

Source stress tensor computed and analyzed.

Extremal case with nonstatic metric studied.

Connection to Rindler and de Sitter geometries established.

Abstract

On the basis of the C-metric, we investigate the conformal Schwarzschild - deSitter spacetime and compute the source stress tensor and study its properties, including the energy conditions. Then we study its extremal version ($b^{2} = 27m^{2}$, where $b$ is the deS radius and $m$ is the source mass), when the metric is nonstatic. The weak-field version is analyzed in several frames, and the metric becomes flat with the special choice $b = 1/a$, $a$ being the constant acceleration of the Schwarzschild-like mass or black hole. This form is Rindler's geometry in disguise and is also conformal to a de Sitter metric where the acceleration plays the role of the Hubble constant. In its time dependent version, one finds that the proper acceleration of a static observer is constant everywhere, in contrast with the standard Rindler case. The timelike geodesics along the z-direction are calculated and proves to be hyperbolae.