Effective metric in fluid-gravity duality through parallel transport: a proposal

Abstract

The incompressible Navier-Stokes (NS) equation is known to govern the hydrodynamic limit of essentially any fluid and its rich non-linear structure has critical implications in both mathematics and physics. The employability of the methods of Riemannian geometry to the study of hydrodynamical flows has been previously explored from a purely mathematical perspective. In this work, we propose a bulk metric in $(p+2)$-dimensions with the construction being such that the induced metric is flat on a timelike $r=r_c$ (constant) slice. We then show that the equations of {\it parallel transport} for an appropriately defined bulk velocity vector field along its own direction on this manifold when projected onto the flat timelike hypersurface requires the satisfaction of the incompressible NS equation in $(p+1)$-dimensions. Additionally, the incompressibility condition of the fluid arises from a vanishing expansion parameter $\theta$, which is known to govern the convergence (or divergence) of a congruence of arbitrary timelike curves on a given manifold. In this approach Einstein's equations do not play any role and this can be regarded as an {\it off-shell} description of fluid-gravity correspondence. We argue that {\it our metric effectively encapsulates the information of forcing terms in the governing equations as if a free fluid is parallel transported on this curved background}. We finally discuss the implications of this interesting observation and its potentiality in helping us to understand hydrodynamical flows in a probable new setting.

Few comments added, to appear in Phys. Rev. D

References in corpus (27)

• Nonlinear Fluid Dynamics from Gravity
• Thermodynamical Aspects of Gravity: New insights
• From AdS/CFT correspondence to hydrodynamics
• Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence
• From Navier-Stokes To Einstein
• The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity
• Forced Fluid Dynamics from Gravity
• Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces
• The holographic fluid dual to vacuum Einstein gravity
• Non-Relativistic Fluid Dual to Asymptotically AdS Gravity at Finite Cutoff Surface
• Action principle for the Fluid-Gravity correspondence and emergent gravity
• The Relativistic Rindler Hydrodynamics
• The Fluid/Gravity Correspondence: a new perspective on the Membrane Paradigm
• From Petrov-Einstein to Navier-Stokes in Spatially Curved Spacetime
• Incompressible Fluids of the de Sitter Horizon and Beyond
• Magnetohydrodynamics from gravity
• Incompressible Navier-Stokes Equations from Einstein Gravity with Chern-Simons Term
• Fluid/gravity duality with Petrov-like boundary condition in a spacetime with a cosmological constant
• Holographic charged fluid dual to third order Lovelock gravity
• Holographic Charged Fluid with Anomalous Current at Finite Cutoff Surface in Einstein-Maxwell Gravity
• Fluid/Gravity Correspondence For General Non-rotating Black Holes
• Higher Curvature Gravity and the Holographic fluid dual to flat spacetime
• The Petrov-like boundary condition at finite cutoff surface in Gravity/Fluid duality
• The Navier-Stokes equation and solution generating symmetries from holography
• Bulk Viscosity of dual Fluid at Finite Cutoff Surface via Gravity/Fluid correspondence in Einstein-Maxwell Gravity
• Dual Fluid for the Kerr Black Hole
• Fluid description of gravity on a timelike cut-off surface: beyond Navier-Stokes equation

Cited by in corpus (4)

• Fluid-gravity correspondence in the scalar-tensor theory of gravity: (in)equivalence of Einstein and Jordan frames
• Gravity dual of Navier-Stokes equation in a rotating frame through parallel transport
• Diffeomorphism symmetries near a timelike surface in black hole spacetime
• DC conductivities and Stokes flows in Dirac semimetals