Topology change in General Relativity and the black-hole black-string transition

TL;DR

This paper explores the phase transition between black holes and black strings in higher-dimensional General Relativity, using Morse theory and local models to understand topology changes and stability across dimensions.

Contribution

It introduces a minimalistic phase diagram constrained by Morse theory, proposing a topology-changing merger transition and analyzing stability variations with dimension.

Findings

Morse theory constrains black hole/black string phase diagrams

A local cone model illustrates the merger transition

Stability of the cone depends on spacetime dimension

Abstract

In the presence of compact dimensions massive solutions of General Relativity may take one of several forms including the black-hole and the black-string, the simplest relevant background being R^{3+1} * S^1. It is shown how Morse theory places constraints on the qualitative features of the phase diagram, and a minimalistic diagram is suggested which describes a first order transition whose only stable phases are the uniform string and the black-hole. The diagram calls for a topology changing ``merger'' transition in which the black-hole evolves continuously into an unstable black-string phase. As evidence a local model for the transition is presented in which the cone over S^2 * S^2 plays a central role. Horizon cusps do not appear as precursors to black hole merger. A generalization to higher dimensions finds that whereas the cone has a tachyon function for d=5, its stability depends interestingly on the dimension - it is unstable for d<10, and stable for d>10.