Geometric Flows and Black Holes

TL;DR

This paper explores how geometric flows can be applied to black-hole problems, revealing that under certain conditions, these flows share solutions like the Schwarzschild black hole, linking mathematics and physics.

Contribution

It demonstrates that geometric evolution equations can satisfy the Birkhoff theorem and unify black-hole solutions across different flows in vacuum spacetime.

Findings

Geometric flows satisfy Birkhoff theorem under certain conditions.

Spherically symmetric solutions are shared by Einstein, Ricci, and hyperbolic flows.

Schwarzschild solution appears in multiple geometric flow contexts.

Abstract

Motivated by the newest progress in geometric flows both in mathematics and physics, we apply the geometric evolution equation to study some black-hole problems. Our results show that, under certain conditions, the geometric evolution equations satisfy the Birkhoff theorem, and surprisingly, in the case of spherically symmetric metric field, the Einstein equation, the Ricci flow, and the hyperbolic geometric flow in vacuum spacetime have the same black-hole solutions, especially in the case of $\Lambda=0$, they all have the Schwarzschild solution. In addition, these results can be generalized to a kind of more general geometric flow.