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Relativistic Holographic Hydrodynamics from Black Hole Horizons

We consider the AdS/CFT correspondence in the hydrodynamic regime up to the second order in a derivative expansion. We demonstrate that the fluid conservation equations are equivalent to Einstein's constraint equations projected on different hyper-surfaces. We derive that result for hyper-surfaces of the form r=R(x) up to the first order in a derivative expansion of the metric. At the second order expansion, we introduce the notion of different black hole horizons, and focus on two particular horizon hyper-surfaces: the event horizon and the apparent horizon. We calculate the temperature and entropy current for the apparent horizon and show that the latter agrees with the area increase theorem for the black hole, and differs from the entropy current calculated for the event horizon.