The ratio of shear viscosity to entropy density in generalized theories of gravity

TL;DR

The paper addresses the viscosity-to-entropy-density ratio $\eta/s$ in generalized theories of gravity within the AdS/CFT framework, arguing it equals the ratio of two horizon gravitational couplings: $\eta/s = \frac{1}{4\pi} \frac{(\kappa_{rt})^2}{(\kappa_{xy})^2}$. The approach is a local computation using the Wald entropy formalism and horizon decoupling of the highest-helicity graviton polarization $h^x{}_y$, with the relevant couplings extracted from $\frac{1}{(\kappa_{\mu\nu})^2} = \mp \frac{1}{4} \left( \frac{\delta \mathscr{L}}{\delta R_{ab}^{\;\;cd}} \right)^{(0)} \hat{\epsilon}_{ab}\hat{\epsilon}^{cd}$. The framework shows that theories related to Einstein gravity by field redefinitions (e.g., $F(R)$ gravity) yield $\eta/s = \frac{1}{4\pi}$, while higher-derivative terms involving the Riemann tensor can modify the ratio; matter couplings not explicitly involving the Riemann tensor do not affect $\eta/s$. The results reproduce known corrections for Riemann-squared and Gauss-Bonnet gravity and extend to Lovelock theories, providing a simple, horizon-local criterion for $\eta/s$ across a broad class of gravitational theories.

Abstract

Near the horizon of a black brane solution in Anti-de Sitter space, the long-wavelength fluctuations of the metric exhibit hydrodynamic behaviour. For Einstein's theory, the ratio of the shear viscosity of near-horizon metric fluctuations eta to the entropy per unit of transverse volume s is eta/s=1/4 pi. We propose that, in generalized theories of gravity, this ratio is given by the ratio of two effective gravitational couplings and can be different than 1/4 pi. Our proposal implies that eta/s is equal for any pair of gravity theories that can be transformed into each other by a field redefinition. In particular, the ratio is 1/4 pi for any theory that can be transformed into Einstein's theory; such as F(R) gravity. Our proposal also implies that matter interactions -- except those including explicit or implicit factors of the Riemann tensor -- will not modify eta/s. The proposed formula reproduces, in a very simple manner, some recently found results for Gauss-Bonnet gravity. We also make a prediction for eta/s in Lovelock theories of any order or dimensionality.