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In mathematical [[general relativity]], the '''Penrose inequality''', first conjectured by Sir [[Roger Penrose]], estimates the mass of a [[spacetime]] in terms of the total area of its [[black holes]] and is a generalization of the [[positive mass theorem]]. The '''Riemannian Penrose inequality''' is an important special case. Specifically, if (''M'', ''g'') is an [[asymptotically flat]] Riemannian [[3-manifold]] with nonnegative [[scalar curvature]] and [[ADM mass]] ''m'', and ''A'' is the area of the outermost [[minimal surface]] (possibly with multiple [[Connected component (topology)|connected components]]), then the Riemannian Penrose inequality asserts
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: <math>m \geq \sqrt{\frac{A}{16\pi}}.</math>
 
This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, [[space-like]], [[totally geodesic]] [[submanifold]]
of a (3&nbsp;+&nbsp;1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (''M'',&nbsp;''g'') having nonnegative scalar curvature is equivalent to the spacetime obeying the [[dominant energy condition]].
 
This inequality was first proved by [[Gerhard Huisken]] and [[Tom Ilmanen]] in 1997 in the case where ''A'' is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined [[inverse mean curvature flow]], which they developed. In 1999, [[Hubert Bray]] gave the first complete proof of the above inequality using a conformal [[geometric flow|flow]] of metrics. Both of the papers were published in 2001.
 
== Physical Motivation ==
The original physical argument that led Penrose to conjecture such an inequality invoked the [[Hawking area theorem]] and the [[Cosmic censorship hypothesis]].
 
== Case of Equality ==
Both the Bray and Huisken-Ilmanen proofs of the Riemannian Penrose inequality state that under the hypotheses, if
 
: <math>m = \sqrt{\frac{A}{16\pi}},</math>
then the manifold in question is isometric to a slice of the [[Schwarzschild spacetime]] outside of the outermost minimal surface.
 
== Penrose Conjecture ==
More generally, Penrose conjectured that an inequality as above should hold for spacelike submanifolds of spacetimes that are not necessarily time-symmetric. In this case, nonnegative scalar curvature is replaced with the [[dominant energy condition]], and one possibility is to replace the minimal surface condition  with an [[apparent horizon]] condition. Proving such an inequality remains an open problem in general relativity, called the '''Penrose Conjecture.'''
 
== References ==
*Bray, H. "Proof of the Riemannian Penrose inequality using the positive mass theorem", ''Journal of Differential Geometry'', '''59''', (2001) 177–267.
* Bray, H., and Chru&#347;ciel, P. (2003) "The Penrose Inequality", ESI preprint 1390, http://arxiv.org/abs/gr-qc/0312047
*{{Citation | last1=Huisken | first1=G. | last2=Ilmanen | first2=T. | title=The Riemannian Penrose inequality | url=http://dx.doi.org/10.1155/S1073792897000664 | doi=10.1155/S1073792897000664 | id={{MR|1486695}} | year=1997 | journal=International Mathematics Research Notices | issn=1073-7928 | issue=20 | pages=1045–1058}}
* Huisken, G., and Ilmanen, T. "The inverse mean curvature flow and the Riemannian Penrose inequality", ''Journal of Differential Geometry'', '''59''', (2001), 353–437.
 
[[Category:Riemannian geometry]]
[[Category:Geometric inequalities]]
[[Category:General relativity]]
[[Category:Theorems in geometry]]

Latest revision as of 00:14, 12 February 2014

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