The Gregory-Laflamme Instability of the Schwarzschild Black String Exterior

TL;DR

This paper provides a rigorous mathematical proof of the Gregory-Laflamme instability for the five-dimensional Schwarzschild black string, demonstrating the existence of exponentially growing linearized gravitational modes.

Contribution

It introduces a novel variational approach to prove the instability, reducing the problem to a Schrödinger eigenvalue equation and establishing the presence of unstable modes.

Findings

Existence of exponentially growing mode solutions

Reduction of Einstein equations to a Schrödinger eigenvalue problem

Proof of instability for the five-dimensional Schwarzschild black string

Abstract

In this paper, a direct rigorous mathematical proof of the Gregory-Laflamme instability for the five-dimensional Schwarzschild black string is presented. Under a choice of ansatz for the perturbation and a gauge choice, the linearised vacuum Einstein equation reduces to an ODE problem for a single function. In this work, a suitable rescaling and change of variables is applied which casts the ODE into a Schr\"odinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution which has admissible behavior at the future event horizon and spacelike infinity. After the addition of a pure gauge solution, this gives rise to a regular exponentially growing mode solution of the linearised vacuum Einstein equation in harmonic/transverse-traceless gauge.