Abstract
We extend our study of all-order linearly resummed hydrodynamics in a flat space~\cite{1406.7222,1409.3095} to fluids in weakly curved spaces. The underlying microscopic theory is a finite temperature $\mathcal{N}=4$ super-Yang-Mills theory at strong coupling. The AdS/CFT correspondence relates black brane solutions of the Einstein gravity in asymptotically \emph{locally} $\textrm{AdS}_5$ geometry to relativistic conformal fluids in a weakly curved 4D background. To linear order in the amplitude of hydrodynamic variables and metric perturbations, the fluid's energy-momentum tensor is computed with derivatives of both the fluid velocity and background metric resummed to all orders. We extensively discuss the meaning of all order hydrodynamics by expressing it in terms of the memory function formalism, which is also suitable for practical simulations. In addition to two viscosity functions discussed at length in refs.~\cite{1406.7222,1409.3095}, we find four curvature induced structures coupled to the fluid via new transport coefficient functions. In ref.~\cite{0905.4069}, the latter were referred to as gravitational susceptibilities of the fluid. We analytically compute these coefficients in the hydrodynamic limit, and then numerically up to large values of momenta.
Highlights
Where ε, uμ and gμν are the fluid energy density, four-velocity and metric tensor
We extend our study of all-order linearly resummed hydrodynamics in a flat space [1, 2] to fluids in weakly curved spaces
Our construction is limited to linear order in amplitudes of the fluid velocity and boundary metric perturbation, but exact in terms of the derivative expansion
Summary
We provide a clarification about what we mean by all-order resummed hydrodynamics. Relation for the dissipation tensor we resumm all orders in the gradient expansion, including infinite number of time derivatives This means the effective dynamical equations in principle require an infinite set of initial conditions or, equivalently, the hydrodynamics at hand is a theory of infinite number of degrees of freedom. In order to solve the dynamical equation (2.3), it is not sufficient to provide the initial condition for magnetization only, but we need the “history” current JH at all times, equivalent to providing infinitely many additional initial conditions. Our main goal is to compute the stress-energy tensor with all the derivative terms linear in the fluid dynamical variables and boundary metric perturbation resummed. These variables can be expanded to linear order uμ(xα) =. Once the metric corrections are found from the Einstein equations, it is straightforward to derive the stress-energy tensor from (C.1), (C.2)
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