CFT dual of the AdS Dirichlet problem: fluid/gravity on cut-off surfaces

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.

Section: The Relationmentioning

confidence: 63%

An action for and hydrodynamics from the improved Large D membrane

Dandekar

Kundu

Mazumdar

et al. 2018

“…Another instance of the fluid/gravity correspondence in asymptotically flat spacetimes can be found in the blackfold approach [17,18]. Further developments were reported in [19][20][21][22][23][24][25][26][27][28].…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

The relativistic fluid dual to vacuum Einstein gravity

Compère

McFadden

Skenderis

et al. 2012

We present a construction of a (d + 2)-dimensional Ricci-flat metric corresponding to a (d + 1)-dimensional relativistic fluid, representing holographically the hydrodynamic regime of a (putative) dual theory. We show how to obtain the metric to arbitrarily high order using a relativistic gradient expansion, and explicitly carry out the computation to second order. The fluid has zero energy density in equilibrium, which implies incompressibility at first order in gradients, and its stress tensor (both at and away from equilibrium) satisfies a quadratic constraint, which determines its energy density away from equilibrium. The entire dynamics to second order is encoded in one first order and six second order transport coefficients, which we compute. We classify entropy currents with non-negative divergence at second order in relativistic gradients. We then verify that the entropy current obtained by pulling back to the fluid surface the area form at the null horizon indeed has a non-negative divergence. We show that there are distinct near-horizon scaling limits that are equivalent either to the relativistic gradient expansion we discuss here, or to the non-relativistic expansion associated with the Navier-Stokes equations discussed in previous works. The latter expansion may be recovered from the present relativistic expansion upon taking a specific non-relativistic limit.

“…Fluid dynamics on finite cutoff hypersurfaces for black holes in holography and beyond was considered before in refs. [76][77][78][79][80][81] and here we focus on the charge dynamics to understand its role, via semi-holography [43,45,82], in calculations of Schwinger-Keldysh effective actions.…”

Section: The Near-horizon Region and The Schwinger-keldysh Effective mentioning

confidence: 99%

Holographic Schwinger-Keldysh effective field theories

Heller

Pinzani-Fokeeva

2019

We construct a holographic dual of the Schwinger-Keldysh effective action for the dissipative low-energy dynamics of relativistic charged matter at strong coupling in a fixed thermal background. To do so, we use a mixed signature bulk spacetime whereby an eternal asymptotically anti-de Sitter black hole is glued to its Euclidean counterpart along an initial time slice in a way to match the desired double-time contour of the dual field theory. Our results are consistent with existing literature and can be regarded as a fully-ab initio derivation of a Schwinger-Keldysh effective action. In addition, we provide a simple infrared effective action for the near horizon region that drives all the dissipation and can be viewed as an alternative to the membrane paradigm approximation.