Anti de Sitter black holes and branes in dynamical Chern-Simons gravity: perturbations, stability and the hydrodynamic modes

TL;DR

This work analyzes perturbations of Schwarzschild–AdS black holes and branes in dynamical Chern-Simons gravity, where a parity-violating CS term couples gravity to a dynamical scalar. The authors derive a coupled system for axial gravitational perturbations and the scalar field, and solve it efficiently via a Frobenius-series expansion to obtain quasinormal modes, establishing stability across the parameter space. In the holographic limit, they show that the CS coupling leaves the universal shear viscosity to entropy density ratio $\eta/s=1/(4\pi)$ and the relaxation time $\tau_\pi$ unchanged, while it induces parity-violating corrections at order $q^4$ in the shear mode, characterized by a Hall viscosity-like effect and a calculable coefficient $\chi$ with $\omega_\text{shear}$ acquiring a term $- i\chi\alpha^2 q^4/(L^2 r_h^3)$. They provide an explicit analytic expression for the CS-induced contribution $\omega_\alpha = i\chi$, together with a relation for the $q^4$ coefficient $\tau_3$ in terms of known GR quantities. The results illuminate how higher-derivative parity-violating corrections modify hydrodynamics in the dual $(2+1)$-dimensional fluid without disrupting the GR-based transport structure, and they suggest extensions to charged AdS black holes and higher dimensions.

Abstract

Dynamical Chern-Simons (DCS) theory is an extension of General Relativity in which the gravitational field is coupled to a scalar field through a parity violating term. We study perturbations of anti-de Sitter black holes and branes in such a theory, and show that the relevant equations reduce to a set of coupled ODEs which can be solved efficiently through a series expansion. We prove numerically that black holes and branes in DCS gravity are stable against gravitational and scalar perturbations in the entire parameter space. Furthermore, by applying the AdS/CFT duality, we relate black hole perturbations to hydrodynamic quantities in the dual field theory, which is a (2+1)-dimensional isotropic fluid with broken spatial parity. The Chern-Simons term does not affect the entropy to viscosity ratio and the relaxation time, but instead quantities that enter the shear mode at order q^4 in the small momentum limit, for example the Hall viscosity and other quantities related to second and third order hydrodynamics. We provide explicit corrections to the gravitational hydrodynamic mode to first relevant order in the couplings.

Figures (4)

• Figure 1: Hydrodynamic mode for Schwarzschild black branes as a function of the DCS coupling $\alpha$, for different values of the momentum $q$.
• Figure 2: First overtones for scalar and gravitational modes as functions of $\alpha$ for $q=3$. The scalar mode is roughly independent from $\alpha$. Different values of $q$ give the same qualitative result.
• Figure 3: Convergence plot for $\alpha=0$ (left panel) and $\alpha/r_h^2=1$ (right panel). In the left panel we show the relative difference between the QN frequencies computed with a truncation order $N\to\infty$ and those computed at a given $N$, $(|\omega_N|-|\omega_\infty|)/|\omega_\infty|$. In the right panel we also show the same convergence plot for the ratio $\rho=\sigma_0/q_0$.
• Figure 4: The quasinormal frequencies as function of DCS coupling $\alpha=0$ and (right panel) the norm of the ratio $\sigma_0/q_0$.