The Jang equation, apparent horizons, and the Penrose inequality

TL;DR

This paper analyzes the Jang equation in spherical symmetry, showing its limitations in deriving the Penrose inequality and in horizon formation criteria, thus questioning its applicability in general relativity proofs.

Contribution

It demonstrates that the Jang equation approach cannot establish the Penrose inequality in spherical symmetry and highlights its limited validity for horizon formation criteria.

Findings

Jang equation reduces to a first order in spherical symmetry

Proposed Penrose inequality derivation via Jang equation fails in spherical symmetry

Analytic criteria based on Jang equation have limited validity for trapped surface conjecture

Abstract

The Jang equation in the spherically symmetric case reduces to a first order equation. This permits an easy analysis of the role apparent horizons play in the (non)existence of solutions. We demonstrate that the proposed derivation of the Penrose inequality based on the Jang equation cannot work in the spherically symmetric case. Thus it is fruitless to apply this method, as it stands, to the general case. We show also that those analytic criteria for the formation of horizons that are based on the use of the Jang equation are of limited validity for the proof of the trapped surface conjecture.