We study the effective dynamics of black hole horizons in Einstein-Maxwell theory in a large number of spacetime dimensions D. We demonstrate that black hole dynamics may be recast as a closed 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, a charge density function and a divergence free velocity field. We determine the equations that govern membrane dyanamics at leading order in the large D expansion. Our derivation of the membrane equations assumes that the solution preserves SO(D − p − 2) isometry group where p is held fixed as D is taken to infinity. However our final membrane equations are fully geometrical; they make no reference to this isometry group, and in particular are independent of p and so may have a greater range of validity than their derivation.

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