Near the horizon of a black brane solution in anti-de Sitter space, the long-wavelength fluctuations of the metric exhibit hydrodynamic behavior. 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 confirms that $\eta /s$ is equal to $1/4\pi$ 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.

• Received 1 September 2008

DOI:https://doi.org/10.1103/PhysRevD.79.021901