Unstable ‘black branes’ from scaled membranes at large D

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×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.