A Charged Membrane Paradigm at Large D

We study the effective dynamics of black hole horizons in Einstein-Maxwell theory in a large number of spacetime dimensions $D$. We demonstrate that horizon dynamics may be recast as a well posed initial value problem for the motion of a codimension one non gravitational membrane moving in flat space. The dynamical degrees of freedom of this membrane are its shape, charge density and a divergence free velocity field. We determine the equations that govern membrane dynamics at leading order in the large $D$ expansion. Our derivation of the membrane equations assumes that the solution preserves an SO$(D-p-2)$ isometry with $p$ held fixed as $D$ is taken to infinity. However we are able to cast our final membrane equations into a completely geometric form that makes no reference to this symmetry algebra.