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.Comment: 12 page