Energy, entropy and the Ricci flow

Abstract

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.

24 pages, two figures, paper improved in response to referees' comments, to appear in Classical and Quantum Gravity

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• On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
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• A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity
• Constructing Lifshitz spaces using the Ricci flow
• Higher order geometric flows on three dimensional locally homogeneous spaces
• RG-2 flow, mass and entropy
• Modified Ricci flow and asymptotically non-flat spaces
• Stochastic Einstein equations
• Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow
• A New Geometric Flow on 3-Manifolds: the $K$-Flow
• The Higgs mechanism and geometrical flows for two-manifolds