Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2007-11-03
Class.Quant.Grav.25:035012,2008
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
24 pages, two figures, paper improved in response to referees' comments, to appear in Classical and Quantum Gravity
Scientific paper
10.1088/0264-9381/25/3/035012
The Ricci flow is a heat equation for metrics, which has recently been used to study the topology of closed three manifolds. In this paper we apply Ricci flow techniques to general relativity. We view a three dimensional asymptotically flat Riemannian metric as a time symmetric initial data set for Einstein's equations. We study the evolution of the area A and Hawking mass M of a two dimensional closed surface under the Ricci flow. The physical relevance of our study derives from the fact that, in general relativity the area of apparent horizons is related to black hole entropy and the Hawking mass of an asymptotic round 2-sphere is the ADM energy.We begin by considering the special case of spherical symmetry to develop a physical feel for the geometric quantities involved. We then consider a general asymptotically flat Riemannian metric and derive an inequality which relates the evolution of the area of a closed surface S to its Hawking mass. We suggest that there may be a maximum principle which governs the long term existence of the asymptotically flat Ricci flow.
Chowdhury Sutirtha Roy
Samuel Joseph
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