Quasinormal spectrum and the black hole membrane paradigm

TL;DR

This work shows that key stretched-horizon properties from the membrane paradigm are encoded in the quasinormal spectrum of black branes. By analytically solving for the lowest quasinormal modes of both vector and shear gravitational perturbations, it derives diffusive, purely imaginary frequencies with diffusion constants tied to the background metric, matching membrane-paradigm predictions. Using universality results and the gauge/gravity duality, it establishes a universal $\omega = -i q^2/(4\pi T)$ and reinforces the celebrated $\eta/s = 1/4\pi$ universality for gravity-dual theories. The findings underscore a deep link between horizon dynamics and holographic transport coefficients, offering a third line of evidence for universal hydrodynamic behavior in strongly coupled systems.

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.