A Membrane Paradigm: Next to leading order story and instability and membrane hydrodynamics in finite D

It has recently been demonstrated that black hole dynamics can be reformulated as the equations of motion of a codimension one membrane roughly the black hole event horizon These membrane equations have previously been determined at leading order in the large D expansion In this talk we implement this duality to next order in 1 D and thereby determine the leading corrections to the membrane equations of motion and the metric Our results obey several consistency checks in particular upon linearizing our equations about a spherical membrane we reproduce the known quasinormal mode frequencies of large D Schwarschild black holes including first subleading corrections in 1 D Our membrane equations which admit a static solution in the sphere times black brane manifold captures the Gregory Laflamme instability with small fluctuations in scaled amplitude and length scale This result thus identify the black brane equations as a special limit of the membrane equations We will generalize this membrane paradigm equations to finite D and understand membrane fluid correspondence