The Hoop Conjecture in Spherically Symmetric Spacetimes

TL;DR

This paper establishes general conditions for the formation of trapped surfaces in spherically symmetric spacetimes, broadening understanding of gravitational collapse without restrictive assumptions, and introduces properties of the Misner-Sharp energy.

Contribution

It provides novel, general criteria for trapped surface existence applicable to arbitrary slices, and suggests extensions to nonspherical cases via a modified Jang equation.

Findings

Conditions for trapped surfaces under matter concentration

Positivity and monotonicity of Misner-Sharp energy

Applicability to arbitrary spacelike slices

Abstract

We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any auxiliary assumptions such as maximality, time-symmetry, or special extrinsic foliations, and most importantly they can easily be generalized to the nonspherical case once an existence theory for a modified version of the Jang equation is developed. Moreover, our methods also yield positivity and monotonicity properties of the Misner-Sharp energy.