Incompressible Navier–Stokes equation from Einstein–Maxwell and Gauss–Bonnet–Maxwell theories

Abstract

The dual fluid description for a general cutoff surface at radius r=rc outside the horizon in the charged AdS black brane bulk space–time is investigated, first in the Einstein–Maxwell theory. Under the non-relativistic long-wavelength expansion with parameter ϵ, the coupled Einstein–Maxwell equations are solved up to O(ϵ2). The incompressible Navier–Stokes equation with external force density is obtained as the constraint equation at the cutoff surface. For non-extremal black brane, the viscosity of the dual fluid is determined by the regularity of the metric fluctuation at the horizon, whose ratio to entropy density η/s is independent of both the cutoff rc and the black brane charge. Then, we extend our discussion to the Gauss–Bonnet–Maxwell case, where the incompressible Navier–Stokes equation with external force density is also obtained at a general cutoff surface. In this case, it turns out that the ratio η/s is independent of the cutoff rc but dependent on the charge density of the black brane.