On the formation of trapped surfaces

TL;DR

This paper simplifies and extends Christodoulou's groundbreaking proof on the formation of trapped surfaces in vacuum spacetimes, reducing technical requirements and broadening initial conditions.

Contribution

It provides a simpler, more localized proof for the formation of trapped surfaces, enlarging initial conditions and reducing derivative requirements.

Findings

Enlarged admissible initial conditions for trapped surface formation

Reduced the derivative requirement from two to one

Proof can be localized with respect to angular sectors

Abstract

In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.