Present status of the Penrose inequality

TL;DR

This paper reviews the current state of the Penrose inequality in mathematical relativity, discussing key results, proofs, and recent advances related to black hole mass bounds and gravitational collapse.

Contribution

It summarizes major developments and proof strategies concerning the Penrose inequality, highlighting recent breakthroughs and ongoing research challenges.

Findings

Complete resolution of the Riemannian Penrose inequality

Proposal by Bray and Khuri for the general case

Summary of proof ideas and current knowledge

Abstract

The Penrose inequality gives a lower bound for the total mass of a spacetime in terms of the area of suitable surfaces that represent black holes. Its validity is supported by the cosmic censorship conjecture and therefore its proof (or disproof) is an important problem in relation with gravitational collapse. The Penrose inequality is a very challenging problem in mathematical relativity and it has received continuous attention since its formulation by Penrose in the early seventies. Important breakthroughs have been made in the last decade or so, with the complete resolution of the so-called Riemannian Penrose inequality and a very interesting proposal to address the general case by Bray and Khuri. In this paper, the most important results on this field will be discussed and the main ideas behind their proofs will be summarized, with the aim of presenting what is the status of our present knowledge in this topic.