From Navier-Stokes To Einstein

TL;DR

The paper constructs a precise map between solutions of the incompressible Navier–Stokes equations in $p+1$ dimensions and dual vacuum Einstein solutions in $p+2$ dimensions, using a flat intrinsic boundary on $\\Sigma_c$ whose extrinsic curvature encodes the fluid stress tensor. It shows that a near-horizon limit aligns the gravitational expansion with the hydrodynamic ($\\epsilon$) expansion, so the Einstein equations reproduce $p+1$-dimensional NS dynamics with viscosity $\\eta=r_c$, and, for $p=2$, the bulk geometry is Petrov type II. By presenting nonlinear solutions in both the $\\epsilon$-expansion and the near-horizon expansion and proving their equivalence (via $\\lambda=\\epsilon^2/r_c$), the work provides a mathematically precise realization of holographic fluid–horizon duality and a concrete bridge between horizon dynamics and fluid dynamics. The results generalize the membrane paradigm and parallel AdS/CFT approaches to flat-space gravity, highlighting possible deeper holographic connections between black hole horizons and dissipative hydrodynamics.

Abstract

We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in $p+1$ dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in $p+2$ dimensions. The dual geometry has an intrinsically flat timelike boundary segment $Σ_c$ whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a "near-horizon" limit in which $Σ_c$ becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For $p=2$, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.

Figures (2)

• Figure 1: This figure depicts the Einstein geometry holographically dual to a fluid. The accelerated boundary hypersurface at radius $r=r_c$ is intrinsically flat but the extrinsic curvature is given by the fluid stress tensor. This extrinsic curvature leads to gravity waves which propagate radially inward. The leading-order condition that these waves do not cross the past horizon ${\cal H}^-$ of at $\tau=-\infty$ or produce singularities on the future horizon ${\cal H}^+$ at $r=0$ is the non-linear incompressible Navier-Stokes equation for the fluid.
• Figure 2: On the surface, prior to $\tau=\tau_*$, all initial data is trivial. At $\tau=\tau_*$, a gravitational shock wave arrives. The shock forces the fluid on , and consequently the $v_i$ is nontrivial on after $\tau_*$.