FIG 2 - uploaded by Benjamin Tippett

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# In this diagram we look at ρ − | J | on the equator of an radius σ = 1 . 4 black hole, as a function of the parameters a and b from K ij . From this we set a ≪ 0 and b small.

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The Penrose inequality has so far been proven in cases of spherical symmetry and in cases of zero extrinsic curvature. The next simplest case worth exploring would be non-spherical, non-rotating black holes with non-zero extrinsic curvature. Following Karkowski et al.'s construction of prolate black holes, we define initial data on an asymptoticall...

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## Citations

... In this review, numerical results have been mentioned only very tangentially. Further details can be found in [102], [103], [90], [52], [104], [105], [97], [98] and [153]. ...

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 review, 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.