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

TL;DR

This work analyzes black-hole and black-string phases in a five-dimensional spacetime with a compact extra dimension, using Morse theory to constrain the phase diagram and arguing for a topology-changing merger between the BH and uniform string. It introduces the cone over $S^2\times S^2$ as a local model for the topology change, then studies Ricci-flat deformations to reveal a dimension-dependent stability, with a critical dimension $d=10$ separating unstable ($d<10$) from stable ($d>10$) regimes. The resulting picture is a simple, hysteretic first-order-like phase diagram in which only the BH and uniform string are stable, an unstable non-uniform string emanates from the GL point and terminates at a merger, and a localized kink in the action marks the topology-changing transition. The analysis connects horizon topology change to a geometric transition and discusses implications for no-hair theorems and potential quantum-gravity interpretations, as well as directions for future Lorentzian and higher-dimensional investigations.

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.