The Penrose Inequality

Abstract

In 1973, R. Penrose presented an argument that the total mass of a space-time which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16\pi }$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. This inequality was first established by G. Huisken and T. Ilmanen in 1997 for a single black hole and then by one of the authors (H.B.) in 1999 for any number of black holes. The two approaches use two different geometric flow techniques and are described here. We further present some background material concerning the problem at hand, discuss some applications of Penrose-type inequalities, as well as the open questions remaining.

29 pages, latex2e, minor misprints corrected, final version to appear in "The Einstein Equations and the Large Scale Behavior of Gravitational Fields (50 years of the Cauchy problem in general relativity)", H. Friedrich and P.T. Chruściel, Editors, Birkhaeuser, 2004

References in corpus (6)

• Topological Censorship and Higher Genus Black Holes
• Towards the classification of static vacuum spacetimes with negative cosmological constant
• Collapsing Shells and the Isoperimetric Inequality for Black Holes
• Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem
• On the Penrose inequality
• Trapped surfaces as boundaries for the constraint equations

Cited by in corpus (2)

• Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness
• Notes on several phenomenological laws of quantum gravity