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Black Brane Viscosity and the Gregory-Laflamme Instability

Joan Camps, Roberto Emparan, Nidal Haddad

TL;DR

This work shows that long-wavelength fluctuations of neutral vacuum black p-branes in asymptotically flat space can be described by an effective relativistic hydrodynamics on the brane worldvolume, as captured by the blackfold framework. Through a derivative expansion of the Einstein equations, the authors compute the intrinsic transport coefficients, finding η/s = 1/(4π) and ζ = 2η(1/p − c_s^2) with c_s^2 = −1/(n+1), and they show these saturate generic bounds. The unstable sound mode in the effective fluid maps to the Gregory-Laflamme instability, and including viscous damping yields a simple, accurate dispersion relation that matches numerical GL data, especially in the large-n limit; a conjectured exact large-n form is presented. The results illustrate the practicality of intrinsic blackfold hydrodynamics in reproducing key gravitational phenomena and outline future directions to incorporate extrinsic worldvolume effects and curved backgrounds.

Abstract

We study long wavelength perturbations of neutral black p-branes in asymptotically flat space and show that, as anticipated in the blackfold approach, solutions of the relativistic hydrodynamic equations for an effective p+1-dimensional fluid yield solutions to the vacuum Einstein equations in a derivative expansion. Going beyond the perfect fluid approximation, we compute the effective shear and bulk viscosities of the black brane. The values we obtain saturate generic bounds. Sound waves in the effective fluid are unstable, and have been previously related to the Gregory-Laflamme instability of black p-branes. By including the damping effect of the viscosity in the unstable sound waves, we obtain a remarkably good and simple approximation to the dispersion relation of the Gregory-Laflamme modes, whose accuracy increases with the number of transverse dimensions. We propose an exact limiting form as the number of dimensions tends to infinity.

Black Brane Viscosity and the Gregory-Laflamme Instability

TL;DR

This work shows that long-wavelength fluctuations of neutral vacuum black p-branes in asymptotically flat space can be described by an effective relativistic hydrodynamics on the brane worldvolume, as captured by the blackfold framework. Through a derivative expansion of the Einstein equations, the authors compute the intrinsic transport coefficients, finding η/s = 1/(4π) and ζ = 2η(1/p − c_s^2) with c_s^2 = −1/(n+1), and they show these saturate generic bounds. The unstable sound mode in the effective fluid maps to the Gregory-Laflamme instability, and including viscous damping yields a simple, accurate dispersion relation that matches numerical GL data, especially in the large-n limit; a conjectured exact large-n form is presented. The results illustrate the practicality of intrinsic blackfold hydrodynamics in reproducing key gravitational phenomena and outline future directions to incorporate extrinsic worldvolume effects and curved backgrounds.

Abstract

We study long wavelength perturbations of neutral black p-branes in asymptotically flat space and show that, as anticipated in the blackfold approach, solutions of the relativistic hydrodynamic equations for an effective p+1-dimensional fluid yield solutions to the vacuum Einstein equations in a derivative expansion. Going beyond the perfect fluid approximation, we compute the effective shear and bulk viscosities of the black brane. The values we obtain saturate generic bounds. Sound waves in the effective fluid are unstable, and have been previously related to the Gregory-Laflamme instability of black p-branes. By including the damping effect of the viscosity in the unstable sound waves, we obtain a remarkably good and simple approximation to the dispersion relation of the Gregory-Laflamme modes, whose accuracy increases with the number of transverse dimensions. We propose an exact limiting form as the number of dimensions tends to infinity.

Paper Structure

This paper contains 10 sections, 57 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Hydrodynamic perturbations of black branes
  3. Preliminaries
  4. Black branes and their effective stress tensor
  5. Perturbations
  6. Asymptotic infinity
  7. Solving the perturbation equations
  8. Viscous stress tensor
  9. Damped unstable sound waves and the Gregory-Laflamme instability
  10. Discussion

Figures (2)

  • Figure 1: Left: dispersion relation $\Omega(k)$, eq. \ref{['GLvisc']}, for unstable sound waves in the effective black brane fluid (normalized relative to the thickness $r_0$). Right: $\Omega(k)$ for the unstable Gregory-Laflamme mode for black branes (numerical data courtesy of P. Figueras). For black $p$-branes in $D$ spacetime dimensions, the curves depend only on $n=D-p-3$.
  • Figure 2: Dispersion relation $\Omega(k)$ of unstable modes for $n=100$: the solid line is our analytic approximation eq. \ref{['GLvisc']}; the dots are the numerical solution of the Gregory-Laflamme perturbations of black branes (numerical data courtesy of P. Figueras).