Marginally trapped surfaces in a perturbed Schwarzschild spacetime

TL;DR

This paper investigates the properties of marginally trapped surfaces in a perturbed Schwarzschild spacetime, establishing existence and uniqueness results for nearly spherically symmetric incoming null hypersurfaces using a new geometric method.

Contribution

It introduces a novel method for analyzing spacelike surfaces in double null coordinates and proves the existence and uniqueness of marginally trapped surfaces in perturbed Schwarzschild spacetimes.

Findings

Existence of a unique marginally trapped surface for nearly spherically symmetric incoming null hypersurfaces.

Development of a general geometric method for spacelike surfaces in double null coordinates.

Application of the method to study horizon-related structures in Lorentzian manifolds.

Abstract

The concept of a marginally trapped surface is important in the theory of general relativity. In the Schwarzschild black hole spacetime, its event horizon is foliated by marginally trapped surfaces. In a more general black hole spacetime, the concept of a marginally trapped surface is closely related to various sorts of horizon, for example, the apparent horizon, the trapping boundary, the isolated horizon and the dynamical horizon. In this paper, we study the set of marginally trapped surfaces in a perturbed Schwarzschild spacetime. We show that for every incoming null hypersurface which is nearly spherically symmetric, there exists a unique embedded marginally trapped surface. In order to prove this result, we develop a general method to study the geometry of spacelike surfaces in a double null coordinate system, which can be applied to study other problems for spacelike surfaces in a Lorentzian manifold.