Present status of the Penrose inequality

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.

54 pages, 3 figures, to be submitted as a Topical Review to Classical and Quantum Gravity

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• The Gauss-Bonnet-Chern mass for graphic manifolds
• A formalism for the calculus of variations with spinors
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• On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass
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