A Scale-Critical Trapped Surface Formation Criterion: A New Proof via Signature for Decay Rates

TL;DR

This paper introduces a new proof method for trapped surface formation in gravitational collapse, utilizing decay rate signatures and rescaling techniques to extend previous scale-critical theorems.

Contribution

It presents a novel ansatz and a systematic approach based solely on decay rate signatures, providing a new proof and extension of existing theorems in gravitational collapse.

Findings

Proof of trapped surface formation in a far-from-center region

Extension of An-Luk's scale-critical theorem

Methodology based on decay rate signatures

Abstract

For gravitational collapse, we observe a correspondence between region close to past null infinity and region close to central singularity. In line with this philosophy, we construct a new ansatz, with which we first present a 40-page self-contained proof of trapped surface formation in a far-from-center region. A systematical approach with only signature for decay rates is used. Then by a rescaling argument, we convert this result into a reproof and an extension of a scale-critical theorem proved by An-Luk.