An Action for and Hydrodynamics from the improved Large D membrane

TL;DR

This paper improves the membrane stress tensor in large D black hole dynamics, ensuring consistent finite D behavior, and links membrane solutions to boundary hydrodynamics and black hole thermodynamics.

Contribution

The authors develop an improved membrane stress tensor that remains valid at finite D and establish a connection between membrane dynamics, boundary hydrodynamics, and black hole thermodynamics.

Findings

The improved stress tensor reduces to previous results at large D.

Static spherical membranes reproduce Schwarzschild black hole thermodynamics at finite D.

Boundary hydrodynamics is derived from membrane dynamics with explicit second-order stress tensor.

Abstract

It has recently been demonstrated that black hole dynamics at large D is dual to the motion of a probe membrane propagating in the background of a spacetime that solves Einstein's equations. The equation of motion of this membrane is determined by the membrane stress tensor. In this paper we `improve' the membrane stress tensor derived in earlier work to ensure that it defines consistent probe membrane dynamics even at finite $D$ while reducing to previous results at large D. Our improved stress tensor is the sum of a Brown York term and a fluid energy momentum tensor. The fluid has an unusual equation of state; its pressure is nontrivial but its energy density vanishes. We demonstrate that all stationary solutions of our membrane equations are produced by the extremisation of an action functional of the membrane shape. Our action is an offshell generalization of the membrane's thermodynamical partition function. We demonstrate that the thermodynamics of static spherical membranes in flat space and global AdS space exactly reproduces the thermodynamics of the dual Schwarzschild black holes even at finite D. We study the long wavelength dynamics of membranes in AdS space, and demonstrate that the boundary `shadow' of this membrane dynamics is boundary hydrodynamics with with a definite constitutive relation. We determine the explicit form of shadow dual boundary stress tensor upto second order in derivatives of the boundary temperature and velocity, and verify that this stress tensor agrees exactly with the fluid gravity stress tensor to first order in derivatives, but deviates from the later at second order and finite D.