An action for and hydrodynamics from the improved Large D membrane

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, 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. 4 The constant λ in (1.4) is proportional to (minus of) the usual cosmological constant. We have chosen the normalization of λ to ensure that AdS D with radius 1 √ λ is a solution the equations (1.4) when λ is positive, while de Sitter space with radius 1 √ −λ solves (1.4) when λ is negative. Upon setting λ = 0 (1.4) reduces to the usual (flat space) vacuum Einstein equations.

Large D membrane for higher derivative gravity and black hole second law

“…Since, the entropy current is conserved in the stationary configuration we can compute the entropy which is the charge corresponding to this current on any arbitrary space-like slice of the membrane world-volume. Since, the membrane worldvolume has a timelike Killing vector field, we can choose a 'Kaluza-Klein' like coordinate system (used also in [26]), in which the induced metric on the membrane world-volume is given by…”

“…In the above we have used some results associated with the Kaluza-Klein type metric from section 6 of [26]. Using the fact that we are in the stationary configuration, the entropy for EGB gravity in the Kaluza-Klein type coordinates then evaluates to…”

“…Ideally we should be computing the boundary stress tensor sourced by the membranes in AdS space (see [26] for similar analysis in Einstein-Hilbert gravity). But this match at linear order in α is in similar spirit to the match of the corresponding ratio in [26].…”

A leading-order comparison between fluid-gravity and membrane-gravity dualities

“…After implementing the correct gauge transformation, we finally get a field redefinition of the fluid variables (i.e., fluid velocity and the temperature) in terms of membrane velocity and its shape 4 . We hope such a rewriting would lead to some new ways to view fluid and membrane dynamics and more ambitiously to a new duality between fluid and membrane dynamics in large number of dimensions, where gravity has no role to play (See [19], [16] for a similar discussion on such field redefinition and rewriting of fluid equations though in [19] the authors have taken the large D limit in a little different way than ours).…”

“…To handle non-trivial dynamics, we have to take recourse of perturbation. Two such important perturbation schemes, which can handle dynamical fluctuations around static solutions even at non-linear level, are 'derivative expansion' [1][2][3][4][5][6] and expansion in inverse powers of dimension [7][8][9][10][11][12][13][14][15][16]. The first one generates 'black-hole' type solutions (i.e., space-time with singularity shielded behind the horizon) that are in one to one correspondence with the solutions of relativistic Navier Stokes equation 1 whereas the second one generates similar 'black hole type' solutions, but dual to the dynamics of a codimension one membrane embedded in the asymptotic geometry 2 .…”