Geometric inequalities for axially symmetric black holes

TL;DR

This paper reviews recent advances in geometric inequalities related to axially symmetric black holes in General Relativity, highlighting their significance in understanding gravitational collapse and the cosmic censorship conjecture.

Contribution

It summarizes recent results and proof ideas concerning geometric inequalities for axially symmetric black holes, emphasizing their physical and geometrical importance.

Findings

Inequalities hold for Kerr-Newman black holes.

Some inequalities extend to dynamical black holes.

Open problems in the field are identified.

Abstract

A geometric inequality in General Relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the Kerr-Newman black hole satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse, they are closed related with the cosmic censorship conjecture. Axially symmetric black holes are the natural candidates to study these inequalities because the quasi-local angular momentum is well defined for them. We review recent results in this subject and we also describe the main ideas behind the proofs. Finally, a list of relevant open problem is presented.