Geometric flows and black hole entropy

TL;DR

This paper explores the relationship between Perelman's Ricci flow entropy and black hole geometric entropy, finding no direct connection but identifying a similar flow potentially relevant to black hole thermodynamics.

Contribution

It introduces a new flow related to Ricci flow that appears connected to black hole geometric entropy, offering potential new insights in black hole physics.

Findings

Perelman's entropy is not linked to geometric entropy.

A similar flow may be connected to black hole entropy.

Potential applications to Penrose inequality.

Abstract

Perelman has given a gradient formulation for the Ricci flow, introducing an ``entropy function'' which increases monotonically along the flow.We pursue a thermodynamic analogy and apply Ricci flow ideas to general relativity. We investigate whether Perelman's entropy is related to (Bekenstein-Hawking)geometric entropy as familiar from black hole thermodynamics. From a study of the fixed points of the flow we conclude that Perelman entropy is not connected to geometric entropy. However, we notice that there is a very similar flow which DOES appear to be connected to geometric entropy. The new flow may find applications in black hole physics suggesting for instance, new approaches to the Penrose inequality.