Trapped surfaces as boundaries for the constraint equations

TL;DR

This paper investigates the use of trapped surfaces as boundary conditions for Einstein's vacuum constraint equations, establishing existence and uniqueness of solutions and exploring implications for black hole collisions.

Contribution

It introduces a novel boundary condition based on trapped surfaces for Einstein's equations and proves the well-posedness of the resulting boundary value problem.

Findings

Existence of solutions in the exterior region under certain assumptions

Uniqueness of solutions for the boundary value problem

Relevance to black hole collision studies

Abstract

Trapped surfaces are studied as inner boundary for the Einstein vacuum constraint equations. The trapped surface condition can be written as a non linear boundary condition for these equations. Under appropriate assumptions, we prove existence and uniqueness of solutions in the exterior region for this boundary value problem. We also discuss the relevance of this result for the study of black holes collisions.