Taking the Null-Hypersurface Limit in the Parikh-Wilczek Membrane Approach

We consider subtleties of the horizon (null-hypersurface) limit in the Parikh-Wilczek Membrane Approach to Black Holes. Specifically, we refine the correspondence between the projected Einstein equations of gravity with matter and the Raychaudhuri-Damour-Navier-Stokes (RDNS) equations of relativistic hydrodynamics. For a general configuration of gravity with matter we obtain additional terms in the hydrodynamic equations, which include very specific combinations of the contracted logarithmic derivatives of a parameter (the regularization function) determining the proximity of a stretched membrane to the black hole horizon. Nevertheless, direct computations of the new terms for exact (Schwarzschild and Kerr) black hole solutions prompt the standard form of the RDNS equations, due to the non-expanding horizon property of these solutions. Therefore, the reduction of the extended RDNS equations to their classical form may be viewed as an additional consistency condition in the exact black hole solutions hydrodynamics, and may serve as a non-trivial test for various viable approximations of spacetime metrics. We compare in detail the Parikh-Wilczek Membrane Approach with the Gourgoulhon-Jaramillo method of a null-hypersurface description, as well as give the link of the obtained results to our previous work on the Kerr black holes.