Unstable `black branes' from scaled membranes at large $D$

TL;DR

This paper shows that the effective black brane equations describing Gregory-Laflamme instability are a special limit of the membrane equations for black holes at large D, unifying two approaches to black hole dynamics.

Contribution

It demonstrates that black brane equations are a limit of membrane equations, linking two frameworks for large D black hole dynamics.

Findings

Black brane equations match a limit of membrane equations.

The approach captures Gregory-Laflamme instability.

Unifies membrane and black brane descriptions.

Abstract

It has recently been demonstrated that the dynamics of black holes at large $D$ can be recast as a set of non gravitational membrane equations. These membrane equations admit a simple static solution with shape $S^{D-p-2} \times R^{p,1}$. In this note we study the equations for small fluctuations about this solution in a limit in which amplitude and length scale of the fluctuations are simultaneously scaled to zero as $D$ is taken to infinity. We demonstrate that the resultant nonlinear equations, which capture the Gregory- Laflamme instability and its end point, exactly agree with the effective dynamical `black brane' equations of Emparan Suzuki and Tanabe. Our results thus identify the `black brane' equations as a special limit of the membrane equations and so unify these approaches to large $D$ black hole dynamics.