Abstract
The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics. This striking connection has been explored in several dynamics-based approaches and has surfaced in various forms over the past four decades. In a recent construction, it has been shown that the manifold properties of geometric duals are in fact intimately connected to the dynamics of incompressible fluids, thus bypassing the conventional on-shell standpoints. Following such a prescription, we construct the geometrical description that effectively captures the dynamics of an incompressible NS fluid with respect to a uniformly rotating frame. We propose the gravitational dual(s) described by bulk metric(s) in $(p+2)$-dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in $(p+1)$-dimensions. We argue that free fluid flows on manifold(s) described by the proposed metric(s) can be effectively considered as an equivalent theory of non-relativistic viscous fluid dynamics with respect to (w.r.t) a uniform rotating frame. We also present suggestive insights as to how space-time rotation parameters encode information pertaining to the inertial effects in the corresponding fluid dual.
Highlights
The nonrelativistic incompressible Navier-Stokes (NS) equation [1,2]ð1:1Þ and the Einstein field equations of gravity Gab 1⁄4 8πG c4T ab ð1:2Þ are two of the most important and well-studied differential equations in physics and mathematics
The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics
We propose the gravitational dual(s) described by bulk metric(s) in (p þ 2) dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in (p þ 1) dimensions
Summary
T ab ð1:2Þ are two of the most important and well-studied differential equations in physics and mathematics. The spacetime is endowed with a general bulk stress-energy tensor and an event horizon This cutoff surface approach has been applied in various cases; see Refs. In the previous cutoff surface approach the underlying physics is that there exists a nontrivial map between the fluid side and the gravity side constrained by their dynamical equations of motion. The constraint of the incompressibility condition on the fluid side is shown to naturally arise from the vanishing of the expansion parameter corresponding to the bulk velocity field It is for this reason that the projection of the parallel transport equation of the bulk velocity field on the cutoff slice is so important in this framework. The uppercase latin letters denote the transverse coordinates intrinsic to the hypersurface (i.e., the angular sector of the metric) and they run from A; B 1⁄4 2; ...; p þ 1 as the labels 0 and 1 have been chosen for time and radial coordinates, respectively
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