Ricci flow and black holes

Abstract

Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to general relativity, where gradient flow is Ricci flow, and focus on the example of four-dimensional Euclidean gravity with boundary S1 × S2, representing the canonical ensemble for gravity in a box. At high temperature the action has three saddle points: hot flat space and a large and small black hole. Adding a time direction, these also give static five-dimensional Kaluza–Klein solutions, whose potential energy equals the four-dimensional action. The small black hole has a Gross–Perry–Yaffe-type negative mode, and is therefore unstable under Ricci flow. We numerically simulate the two flows seeded by this mode, finding that they lead to the large black hole and to hot flat space respectively, in the latter case via a topology-changing singularity. In the context of string theory these flows are world-sheet renormalization group trajectories. We also use them to construct a novel free energy diagram for the canonical ensemble.