Quasinormal spectrum and the black hole membrane paradigm

TL;DR

This paper demonstrates that the quasinormal spectrum of black holes encodes properties of the membrane paradigm, showing a universal, purely imaginary frequency in the long wavelength limit, supporting the universality of shear viscosity in gauge-gravity duality.

Contribution

It analytically links quasinormal modes to membrane paradigm properties and confirms the universality of shear viscosity to entropy density ratio in gravity dual theories.

Findings

The lowest quasinormal frequency matches membrane paradigm predictions.

The dispersion relation is identical to membrane diffusion results.

The spectrum exhibits a universal, purely imaginary frequency in the long wavelength limit.

Abstract

The membrane paradigm approach to black hole physics introduces the notion of a stretched horizon as a fictitious time-like surface endowed with physical characteristics such as entropy, viscosity and electrical conductivity. We show that certain properties of the stretched horizons are encoded in the quasinormal spectrum of black holes. We compute analytically the lowest quasinormal frequency of a vector-type perturbation for a generic black hole with a translationally invariant horizon (black brane) in terms of the background metric components. The resulting dispersion relation is identical to the one obtained in the membrane paradigm treatment of the diffusion on stretched horizons. Combined with the Buchel-Liu universality theorem for the membrane's diffusion coefficient, our result means that in the long wavelength limit the black brane spectrum of gravitational perturbations exhibits a universal, purely imaginary quasinormal frequency. In the context of gauge-gravity duality, this provides yet another (third) proof of the universality of shear viscosity to entropy density ratio in theories with gravity duals.