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

Understand

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.