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 → ∞ 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 <sup>d</sup> 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 determine the 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.