The Incompressible Rindler fluid versus the Schwarzschild-AdS fluid

TL;DR

This paper compares the fluid dynamics derived from black hole backgrounds, specifically Rindler space and Schwarzschild-AdS, highlighting differences in compressibility and viscosity, and relating these to the membrane paradigm.

Contribution

It demonstrates that Rindler space describes an incompressible fluid while Schwarzschild-AdS exhibits zero bulk viscosity, clarifying their relation within the fluid/gravity correspondence.

Findings

Rindler space corresponds to an incompressible fluid.

Schwarzschild-AdS fluid has vanishing bulk viscosity.

Negative bulk viscosity observed in membrane paradigm.

Abstract

We study the proposal by Bredberg et al. (1006.1902), where the fluid is defined by the Brown-York tensor on a timelike surface at r=r_c in black hole backgrounds. We consider both Rindler space and the Schwarzschild-AdS (SAdS) black hole. The former describes an incompressible fluid, whereas the latter describes the vanishing bulk viscosity at arbitrary r_c, but these two results do not contradict with each other. We also find an interesting "coincidence" with the black hole membrane paradigm which gives a negative bulk viscosity. In order to show these results, we rewrite the hydrodynamic stress tensor via metric perturbations using the conservation equation. The resulting expressions are suitable to compare with the Brown-York tensor.