A membrane paradigm at large D

Understand

We study $SO(d+1)$ invariant solutions of the classical vacuum Einstein equations in $p+d+3$ dimensions.

• In the limit $d \to \infty$ 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.