Conformal nonlinear fluid dynamics from gravity in arbitrary dimensions

Abstract

We generalize recent work to construct a map from the conformal Navier Stokes equations with holographically determined transport coefficients, in d spacetime dimensions, to the set of asymptotically locally AdS d+1 long wavelength solutions of Einstein's equations with a negative cosmological constant, for all d > 2. We find simple explicit expressions for the stress tensor (slightly generalizing the recent result by Haack and Yarom (arXiv:0806.4602) ), the full dual bulk metric and an entropy current of this strongly coupled conformal fluid, to second order in the derivative expansion, for arbitrary d > 2. We also rewrite the well known exact solutions for rotating black holes in AdS d+1 space in a manifestly fluid dynamical form, generalizing earlier work in d = 4. To second order in the derivative expansion, this metric agrees with our general construction of the metric dual to fluid flows.

Nonlinear magnetohydrodynamics from gravity

“…In the context of the AdS/CFT correspondence, this construction also provides an efficient means of computing transport coefficients in the dual CFT. The details of this connection have been worked out for pure gravity in arbitrary dimensions [1,[3][4][5], Einstein-Maxwell theory in 4 + 1 dimensions [6][7][8], and gravity coupled to a scalar field in 4 + 1 dimensions [9]. There is also an extensive earlier literature on the study of transport coefficients within the AdS/CFT correspondence; see [10] for a review and further references.…”

Hydrodynamic vortices and their gravity duals

“…Recently however one example of a provable, at least as an asymptotic expansion, gravitational holographic duality has been discovered in Refs. [1,2]. This is a map between black brane solutions of Einstein's equations with a negative cosmological constant and conformal fluid flows in one less dimension.…”

“…It was proven in Refs. [1,2] that the constraint components of Einstein's equations, intuitively the integrability conditions in a null direction, are related to the conservation of the stress tensor via the map. Therefore solutions of Einstein's equations automatically yield solutions to the hydrodynamic equations.…”

“…This failure is due to the fact that it misses a universal element of turbulence, intermittency: inside of every turbulent flow there is a region with laminar flow. Many phenomenological models have been developed which include some intermittency by hand, and so are able to roughly numerically fit the higher n-point functions, with a precision which depends on how many parameters are included 2 . As a first approach we will omit intermittency altogether.…”

“…[1,2]. To apply this map to turbulent flows, one must first find the stationary vortex solutions in a conformal hydrodynamic theory which are the analogues of the vortices discussed above in the context of incompressible fluids.…”