A membrane paradigm at large D

We study $SO(d+1)$ invariant solutions of the classical vacuum Einstein equations in $p+d+3$ dimensions. In the limit $d\to \mathrm{\infty }$ with $p$ held fixed we construct a class of solutions labelled by the shape of a membrane (the event horizon), together with a velocity' field that lives on this membrane. We demonstrate that our metrics can be corrected to nonsingular solutions at first sub-leading order in $\frac{1}{d}$ if and only if the membrane shape and velocity’ field obey equations of motion which we determine. These equations define a well posed initial value problem for the membrane shape and this `velocity’ and so completely determinethe dynamics of the black hole. They may be viewed as governing the non-linear dynamics of the light quasi normal modes of Emparan, Suzuki and Tanabe.