Foliation of the Kottler-Schwarzschild-De Sitter Spacetime by Flat Spacelike Hypersurfaces

TL;DR

This paper develops a method to foliate all cases of the Kottler-Schwarzschild-de Sitter spacetime with flat spacelike hypersurfaces using a non-singular time coordinate, extending coordinate regularity techniques from black hole geometries.

Contribution

It introduces a non-singular time coordinate to achieve flat foliation of the entire KSSdS spacetime, including extreme cases lacking regular coordinates.

Findings

Successful foliation of all KSSdS cases with flat spacelike hypersurfaces.

Extension of regular coordinate techniques to extreme KSSdS spacetime.

Demonstration of a non-singular coordinate approach for complex spacetime geometries.

Abstract

There exist Kruskal like coordinates for the Reissner-Nordstrom (RN) black hole spacetime which are regular at coordinate singularities. Non existence of such coordinates for the extreme RN black hole spacetime has already been shown. Also the Carter coordinates available for the extreme case are not manifestly regular at the coordinate singularity, therefore, a numerical procedure was developed to obtain free fall geodesics and flat foliation for the extreme RN black hole spacetime. The Kottler-Schwarzschild-de Sitter (KSSdS) spacetime geometry is similar to the RN geometry in the sense that, like the RN case, there exist non-singular coordinates when there are two distinct coordinate singularities. There are no manifestly regular coordinates for the extreme KSSdS case. In this paper foliation of all the cases of the KSSdS spacetime by flat spacelike hypersurfaces is obtained by introducing a non-singular time coordinate.