Relativistic CFT hydrodynamics from the membrane paradigm

We use the membrane paradigm to analyze the horizon dynamics of a uniformly boosted black brane in a (d + 2)-dimensional asymptotically Anti-de-Sitter space-time and a Rindler acceleration horizon in (d + 2)-dimensional Minkowski space-time. We show that in these cases the horizon dynamics is governed by the relativistic CFT hydrodynamics equations. The fluid velocity and temperature correspond to the normal to the horizon and to the surface gravity, respectively. The second law of thermodynamics for the fluid is mapped into the area increase theorem of General Relativity. The analysis is applicable, in general, to perturbations around a stationary horizon, when the scale of variations of the macroscopic fields is much larger than the inverse of the temperature. We show that the non-relativistic limit of our analysis yields the incompressible Navier-Stokes equations.