Kerr Black Holes within the Membrane Paradigm

Abstract

We consider the membrane viewpoint a l\`a Parikh-Wilczek on the Kerr solution for a rotating black hole. Computing the stress-energy tensor of a close-to-the-horizon stretched membrane and comparing it to the stress-tensor of a viscous fluid, we recover transport coefficients in terms of the Kerr geometry. Viscosities of the dual fluid remain constant, while the rest of the transport coefficients become complex functions of radial and angle coordinates. We study the qualitative behavior of the pressure, expansion, and energy/momentum densities for two specific black holes: the slowly rotating black hole, with the angular momentum of one percent of the black hole mass squared, and the extremal Kerr black hole. For the Kerr solution in the Boyer-Lindquist coordinates, these transport coefficients generally have poles at different values of the radial coordinate in the range between the horizon and the Schwarzschild radius of the black hole, in dependence on the fixed angle direction. We briefly discuss our findings in the context of a relation between the Membrane Paradigm and the AdS/CFT correspondence, the KSS bound violation, the coordinate choice, and a non-stationary extension of the Kerr solution.