Geometrical inequalities bounding angular momentum and charges in General Relativity

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.

Invited review for Living Reviews in Relativity

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• Comments on Penrose inequality with angular momentum for outermost apparent horizons
• Refined inequalities for loosely trapped surface/attractive gravity probe surface
• New spinorial approach to mass inequalities for black holes in general relativity
• A Penrose-Type Inequality with Angular Momentum and Charge for Axisymmetric Initial Data
• Four types of attractive gravity probe surfaces
• Two mass conjectures on axially symmetric black hole--disk systems
• Bounding Horizon Area by Angular Momentum, Charge, and Cosmological Constant in 5-Dimensional Minimal Supergravity
• On extensions of the Penrose inequality with angular momentum
• The Positive Mass Theorem with Angular Momentum and Charge for Manifolds with Boundary
• Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass
• CMC surfaces and area-charge inequality for a spheroidal ECD spacetime
• New spinorial mass-quasilocal angular momentum inequality for initial data with marginally future trapped surface