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 SD-p -2 x R-p,R-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.