Local uniqueness of the Black String with small circle size

TL;DR

This paper proves local uniqueness and infinitesimal rigidity of the Black String solution in Ricci-flat metrics for small circle sizes, using elliptic PDE analysis and fixed point theorems.

Contribution

It establishes local uniqueness of the Black String with small circle size by applying fixed point theorems to the nonlinear elliptic PDE formulation.

Findings

Black String is infinitesimally rigid for small circle size

No other Ricci-flat metrics exist near the Black String in a certain neighborhood

Comparison with scalar field problem shows potential for global results

Abstract

In this article we study uniqueness of the Black String, i.e. the product of 4-dimensional Schwarzschild space with a circle of length L. In arXiv:2410.20967, this was reduced to a non-linear elliptic PDE, and we use this setup to show that for small L the Black String is infinitesimally rigid as a Ricci-flat metric. Using a fixed point theorem, we prove that this implies local uniqueness, i.e. there exist no other Ricci-flat metrics near the Black String, and we give bounds for the size of the neighborhood in which the Black String is unique. We compare this with the toy problem of a scalar field satisfying an elliptic equation that was already solved in arXiv:2410.20967 using different methods. In this case we can use the fixed point theorem method to prove not just a local but a global statement.