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

## TL;DR

This paper introduces a novel bulk metric in higher dimensions that, through parallel transport, reproduces the incompressible Navier-Stokes equations without relying on Einstein's equations, offering a new off-shell perspective on fluid-gravity duality.

## Contribution

It proposes a new bulk metric construction where parallel transport yields Navier-Stokes equations, providing an off-shell approach to fluid-gravity correspondence independent of Einstein's equations.

## Key findings

- Parallel transport reproduces Navier-Stokes equations.
- Incompressibility arises from vanishing expansion parameter.
- The metric encodes forcing terms as if a free fluid is transported.

## 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.