The shear diffusion coefficient for generalized theories of gravity

TL;DR

This paper investigates how shear diffusion coefficients in generalized gravity theories relate to boundary hydrodynamics via gauge-gravity duality, revealing modifications to Einstein theory relations and emphasizing horizon-localized expressions.

Contribution

It derives a local horizon expression for shear diffusion coefficients in generalized gravity theories and shows modifications to the Einstein relation between shear viscosity and entropy ratio.

Findings

Shear diffusion coefficients are equal at the horizon and boundary.

The Einstein relation between shear viscosity and entropy ratio is modified.

Diffusion coefficients can be expressed using polarization-specific gravitational couplings.

Abstract

Near the horizon of a black brane in Anti-de Sitter (AdS) space and near the AdS boundary, the long-wavelength fluctuations of the metric exhibit hydrodynamic behaviour. The gauge-gravity duality then relates the boundary hydrodynamics for generalized gravity to that of gauge theories with large finite values of 't Hooft coupling. We discuss, for this framework, the hydrodynamics of the shear mode in generalized theories of gravity in d+1 dimensions. It is shown that the shear diffusion coefficients of the near-horizon and boundary hydrodynamics are equal and can be expressed in a form that is purely local to the horizon. We find that the Einstein-theory relation between the shear diffusion coefficient and the shear viscosity to entropy ratio is modified for generalized gravity theories: Both can be explicitly written as the ratio of a pair of polarization-specific gravitational couplings but implicate differently polarized gravitons. Our analysis is restricted to the shear-mode fluctuations for simplicity and clarity; however, our methods can be applied to the hydrodynamics of all gravitational and matter fluctuation modes.