Kerr Black Holes within the Membrane Paradigm

TL;DR

This paper explores the membrane paradigm for Kerr black holes, deriving transport coefficients of the near-horizon fluid and analyzing their behavior for different black hole spins, with implications for holography and black hole physics.

Contribution

It extends the membrane paradigm to rotating Kerr black holes, calculating complex transport coefficients and analyzing their properties across different black hole configurations.

Findings

Viscosities remain constant across Kerr solutions.

Transport coefficients exhibit poles at specific radial coordinates.

Qualitative behavior varies between slowly rotating and extremal Kerr black holes.

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.