Viscoelastic hydrodynamics of charged black holes

TL;DR

The paper shows that small-amplitude, long-wavelength perturbations of a planar AdS black hole charged under two-form potentials are governed by relativistic viscoelastic hydrodynamics, yielding explicit expressions for transport coefficients including a negative shear modulus in parts of parameter space and identifying an incoherent-density diffusion-driven instability. Through a fluid/gravity analysis, it derives five key transport coefficients $(p,a,\sigma,\zeta,\eta)$ (with $\,\zeta=0$) and provides analytic and numerical results for the shear modulus $G$ and shear viscosity $\eta$, including low-temperature, low-density, and self-dual-point limits. The study also reveals a dual scalar description—diffusive heat transport under no-flux boundary conditions—obtained by a Hodge duality map, with the heat-diffusion mode characterized as the collective excitation of the viscoelastic fluid. The results clarify the stability conditions of the higher-form black brane, identify the primary instability channel, and establish a precise holographic link between viscoelastic hydrodynamics and diffusive hydrodynamics, offering a framework for extending to more general backgrounds and non-linear regimes.

Abstract

We study the dynamics of an isotropic, planar AdS black hole charged under a pair of two-form gauge potentials. We prove that long wavelength, small amplitude perturbations of this state are governed by the relativistic theory of viscoelastic hydrodynamics. We use this effective theory to identify instabilities in certain regions of parameter space, and show that the dominant instability is always driven by fluctuations of an incoherent density of the higher-form charge. The spacetime we study also arises as a solution to gravity coupled to a pair of massless scalar fields, where part of its dynamics is governed by a simpler effective theory of heat diffusion. We derive this simpler description directly from viscoelastic hydrodynamics, demonstrating that the heat diffusion mode is the collective excitation of the viscoelastic fluid under no-flux boundary conditions.

Figures (4)

• Figure 1: Plots of $\theta/r_h$ (left) and $\Theta_1(r_h)$ (right) as a function of $m/r_h$. Note that the right hand edge of each plot $m=\sqrt{6}r_h$ corresponds to the limit $m/T\rightarrow\infty$. The solid lines are numerical solutions of \ref{['eq:Theta1Eq']} while the dashed lines are the analytic approximations at low density (equations \ref{['eq:lowdensitytheta']} and \ref{['eq:lowdensityTheta1']}) and low temperature (equation \ref{['eq:highdensitythetaTheta1']}). The circle is the result \ref{['eq:thetaselfdual']} for $m=\sqrt{2}r_h$.
• Figure 2: The squared speed $v_\perp^2$ and attenuation $\Gamma_\perp$ of the hydrodynamic modes obtained using the transport coefficients calculated in Section \ref{['sec:holography']}. Results are shown for $\mathcal{M}/r_h=10$ (short-dashed), $\mathcal{M}/r_h=5$ (long-dashed) and $\mathcal{M}/r_h=2$ (solid). The circles are the results of Grozdanov:2018ewh from explicit numerical computation of black hole quasinormal modes.
• Figure 3: The squared speed $v_\parallel^2$ and attenuation $\Gamma_\parallel$ of the hydrodynamic modes obtained using the transport coefficients calculated in Section \ref{['sec:holography']}. Results are shown for $\mathcal{M}/r_h=10$ (short-dashed), $\mathcal{M}/r_h=5$ (long-dashed) and $\mathcal{M}/r_h=3$ (solid). The attenuation is not shown for the latter case: it has a pole where the speed vanishes.
• Figure 4: The hydrodynamic diffusivity $D_\parallel$ obtained using the transport coefficients calculated in Section \ref{['sec:holography']}. Results are shown for $\mathcal{M}/r_h=10$ (short-dashed), $\mathcal{M}/r_h=5$ (long-dashed).