Shear Viscosity from Effective Couplings of Gravitons

TL;DR

This work demonstrates that the shear viscosity η of holographic field theories is fully determined by the effective coupling of transverse gravitons evaluated on the black hole horizon. By computing η via the Kubo formula and analyzing the graviton effective action, the authors derive a horizon-coupling formula valid for Einstein and Gauss-Bonnet gravities with minimally coupled matter, then apply it to AdS Gauss-Bonnet gravity with Maxwell F^4 corrections. They show η/s remains 1/4π in Einstein gravity but generically deviates in Gauss-Bonnet theories, with corrections controlled by the horizon data and parameterized by λ and ε (the F^4 coupling). Causality constraints bound λ and characterize how higher-derivative and matter corrections influence the viscosity-to-entropy ratio, revealing that the universal bound is specific to Einstein gravity with minimal coupling and can be violated (or restored toward 1/4π) in more general theories. This horizon-based approach provides a robust framework for assessing transport properties in holographic models with various higher-curvature and matter couplings.

Abstract

We calculate the shear viscosity of field theories with gravity duals using Kubo-formula by calculating the Green function of dual transverse gravitons and confirm that the value of the shear viscosity is fully determined by the effective coupling of transverse gravitons on the horizon. We calculate the effective coupling of transverse gravitons for Einstein and Gauss-Bonnet gravities coupled with matter fields, respectively. Then we apply the resulting formula to the case of AdS Gauss-Bonnet gravity with $F^4$ term corrections of Maxwell field and discuss the effect of $F^4$ terms on the ratio of the shear viscosity to entropy density.

Figures (1)

• Figure 1: $P(f)=8\varepsilon f^3+f$ with $\varepsilon>0$ and $\varepsilon<0$. The red curve denotes $P(f)$ as a function of $f$ in the case of $\varepsilon>0$ and the blue curve for the case of $\varepsilon<0$. The fact that $P(f)\rightarrow 0$ when $r\rightarrow \infty$ and the boundary condition $f \rightarrow 0$ as $r \rightarrow \infty$ gives that when $P(f)\rightarrow 0$, $f$ must approach to zero, too. Thus the physical part of the curve of $P(f)$ must start from the origin in the figure. For $Q>0$, the curve of $P(f)$ must be in the above of $f$-axis. Thus we have $0<f<\infty$ with $\varepsilon>0$ and $0<f\leq f_{\rm max}$ for $\varepsilon<0$. The behavior of $P(f_+)$ as a function of $f_+$ is the same as $P(f)$.