Uniqueness of Flat Spherically Symmetric Spacelike Hypersurfaces Admitted by Spherically Symmetric Static Spactimes

TL;DR

This paper proves that spherically symmetric static spacetimes have a unique foliation by flat, spherically symmetric spacelike hypersurfaces, which is important for black hole physics studies.

Contribution

It establishes the uniqueness of flat, spherically symmetric spacelike hypersurfaces in spherically symmetric static spacetimes, filling a gap in the understanding of their foliation structure.

Findings

Existence of flat, spherically symmetric hypersurfaces in such spacetimes.

Uniqueness of these hypersurfaces up to translation along the timelike Killing vector.

Guarantee of unique flat foliation for black hole physics applications.

Abstract

It is known that spherically symmetric static spacetimes admit a foliation by {\deg}at hypersurfaces. Such foliations have explicitly been constructed for some spacetimes, using different approaches, but none of them have proved or even discussed the uniqueness of these foliations. The issue of uniqueness becomes more important due to suitability of {\deg}at foliations for studying black hole physics. Here {\deg}at spherically symmetric spacelike hy- persurfaces are obtained by a direct method. It is found that spherically symmetric static spacetimes admit {\deg}at spherically symmetric hypersurfaces, and that these hypersurfaces are unique up to translation under the time- like Killing vector. This result guarantees the uniqueness of {\deg}at spherically symmetric foliations for such spacetimes.