Energy, entropy and the Ricci flow

TL;DR

This paper explores the application of Ricci flow techniques from differential geometry to general relativity, analyzing the evolution of geometric quantities related to black holes and energy in asymptotically flat spacetimes.

Contribution

It introduces a novel approach to study the evolution of horizons and mass in general relativity using Ricci flow, linking geometric analysis with physical concepts.

Findings

Derived an inequality relating area and Hawking mass evolution.

Developed a physical intuition through spherical symmetry case.

Proposed a maximum principle for Ricci flow in asymptotically flat spaces.

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.