Geometrical inequalities bounding angular momentum and charges in General Relativity

TL;DR

This paper reviews geometrical inequalities in General Relativity that constrain angular momentum and charge based on mass and size, highlighting differences between black holes and ordinary objects.

Contribution

It summarizes recent results on bounds for angular momentum and charges, emphasizing the mathematical conditions for these inequalities in black holes and ordinary objects.

Findings

Inequalities restrict black hole rotation speed and charge.

Mathematical conditions for bounds are identified.

Challenges remain in estimating parameters for ordinary objects.

Abstract

Geometrical inequalities show how certain parameters of a physical system set restrictions on other parameters. For instance, a black hole of given mass can not rotate too fast, or an ordinary object of given size can not have too much electric charge. In this article we are interested in bounds on the angular momentum and electromagnetic charges, in terms of total mass and size. We are mainly concerned with inequalities for black holes and ordinary objects. The former are the most studied systems in this context in General Relativity, and where most results have been found. Ordinary objects, on the other hand, present numerous challenges and many basic questions concerning geometrical estimates for them are still unanswered. We present the many results in these areas. We make emphasis in identifying the mathematical conditions that lead to such estimates, both for black holes and ordinary objects.