A note on time-symmetric hypersurfaces in the Schwarzschild geometry

TL;DR

This paper proves that any smooth, inextensible, time-symmetric spacelike hypersurface in the maximal Schwarzschild spacetime must intersect the bifurcation sphere, highlighting a geometric constraint in black hole spacetimes.

Contribution

It establishes a new geometric property of hypersurfaces in Schwarzschild spacetime, specifically their intersection with the bifurcation sphere, under smoothness and inextensibility conditions.

Findings

Any $C^k$, $k extgreater{}2$, inextensible, time-symmetric hypersurface intersects the bifurcation sphere.

The result applies to hypersurfaces embedded in the maximal Schwarzschild geometry.

Provides insight into the geometric structure of black hole horizons.

Abstract

In this note we show that any inextensible time-symmetric space-like hypersurface of differentiability class $C^k$, $k\geq 2$ isometrically embedded in the maximal Schwarzschild geometry must intersect the bifurcation sphere.