Incompressible Navier-Stokes Equation from Einstein-Maxwell and Gauss-Bonnet-Maxwell Theories

TL;DR

This paper derives the incompressible Navier-Stokes equations with external forces from Einstein-Maxwell and Gauss-Bonnet-Maxwell theories, analyzing the dual fluid behavior at various cutoff surfaces in charged AdS black brane backgrounds.

Contribution

It extends the fluid/gravity correspondence to include charged black branes and Gauss-Bonnet corrections, deriving the dual fluid equations at arbitrary cutoff surfaces.

Findings

Viscosity to entropy density ratio /s is independent of cutoff in Einstein-Maxwell case.

In Gauss-Bonnet-Maxwell case, /s depends on the black brane charge density.

The dual fluid obeys the incompressible Navier-Stokes equation with external force at the cutoff surface.

Abstract

The dual fluid description for a general cutoff surface at radius r=r_c 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 \epsilon, the coupled Einstein-Maxwell equations are solved up to O(\epsilon^2). 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 \eta/s is independent of both the cutoff r_c 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 \eta/s is independent of the cutoff r_c but dependent on the charge density of the black brane.