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 extremization 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 that are everywhere approximately ‘parallel’ to the boundary, and demonstrate that the boundary ‘shadow’ of this membrane dynamics is boundary hydrodynamics 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.
Highlights
It has recently been demonstrated that the dynamics of black holes in a large number of dimensions is ‘dual’ to the motion of a probe membrane1 propagating without back reaction on any background that solves Einstein’s equations.2 The degrees of freedom of this probe membrane are its shape and a velocity field (D − 2 degrees of freedom) that lives on its world volume
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
Plugging the explicit form of the membrane stress tensor, (1.7) into the general formula (5.43), we find that the boundary stress tensor dual to our membrane -accurate to second order in derivatives — is given by
Summary
It has recently been demonstrated that the dynamics of black holes in a large number of dimensions is ‘dual’ to the motion of a probe membrane propagating without back reaction on any background that solves Einstein’s equations. The degrees of freedom of this probe membrane are its shape (one degree of freedom) and a velocity field (D − 2 degrees of freedom) that lives on its world volume. In the rest of this paper we first present our improved version of the leading order large D membrane stress tensor of [12] We use this stress tensor to study of the properties of the membrane in equilibrium. The membrane equations presented in this paper are just the first term in a systematically improvable approximation to black hole dynamics Given this fact it is somewhat surprising that the membrane equations presented in this paper turn out — in simple situations — to reproduce black hole physics better than we had the right to expect, getting some results exactly right even at finite values of D — as we explain below
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