Riemannian Penrose inequality

Riemannian Penrose inequality

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 the most 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 components), then the Riemannian Penrose inequality asserts

: m geq sqrt{frac{A}{16pi.

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifoldof a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of ("M", "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 inverse mean curvature flow, which they developed. In 1999, Hubert Bray gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001.

References

*Bray, H. "Proof of the Riemannian Penrose inequality using the positive mass theorem", Journal of Differential Geometry, 59, (2001) 177-367.
* Huisken, G., and Ilmanen, T. "The inverse mean curvature flow and the Riemannian Penrose inequality", Journal of Differential Geometry, 59, (2001), 353-437.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • List of inequalities — This page lists Wikipedia articles about named mathematical inequalities. Inequalities in pure mathematics =Analysis= * Askey–Gasper inequality * Bernoulli s inequality * Bernstein s inequality (mathematical analysis) * Bessel s inequality *… …   Wikipedia

  • Contributors to general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

  • List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia

  • Inverse mean curvature flow — In the field of differential geometry in mathematics, inverse mean curvature flow (IMCF) is an example of a geometric flow of hypersurfaces a Riemannian manifold (for example, smooth surfaces in 3 dimensional Euclidean space). Intuitively, a… …   Wikipedia

  • Gerhard Huisken — 2006 Gerhard Huisken (* 20. Mai 1958 in Hamburg) ist ein deutscher Mathematiker. Inhaltsverzeichnis 1 Leben …   Deutsch Wikipedia

  • Mass in general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Black hole — For other uses, see Black hole (disambiguation). Simulated view of a black hole (center) in front of the Large Magellanic Cloud. Note the gravitat …   Wikipedia

  • Entropy — This article is about entropy in thermodynamics. For entropy in information theory, see Entropy (information theory). For a comparison of entropy in information theory with entropy in thermodynamics, see Entropy in thermodynamics and information… …   Wikipedia

  • List of geometry topics — This is list of geometry topics, by Wikipedia page.*Geometric shape covers standard terms for plane shapes *List of mathematical shapes covers all dimensions *List of differential geometry topics *List of geometers *See also list of curves, list… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”