Black Holes as Incompressible Fluids on the Sphere

TL;DR

The authors study finite deformations of the $p+2$-dimensional Schwarzschild geometry that preserve a conformally round boundary metric and the mean curvature, and solve the vacuum Einstein equations in a near-horizon limit $λ\to 0$. They demonstrate that the leading dynamics of these deformations is governed by the incompressible Navier-Stokes equation on the $p$-sphere with viscosity $ν=1$, for a velocity field $v^i$ and pressure $P$ satisfying $\partial_t v^i + v^j \nabla_j v^i + \nabla^i P - (\nabla^2 v^i + R^i_j v^j) = 0$ and $\nabla_i v^i = 0$. A third-order obstruction under Dirichlet boundary conditions is resolved by allowing fluctuations in the conformal factor while keeping $K$ fixed, preserving the universal horizon-fluid behavior. The work highlights a conceptual link between global existence questions for $p$-dimensional incompressible NS and a form of cosmic censorship in $p+2$-dimensional GR, suggesting a novel bridge between fluid dynamics and gravitational singularity structure.

Abstract

We consider finite deformations of the p+2-dimensional Schwarzschild geometry which obey the vacuum Einstein equation, preserve the mean curvature and induced conformal metric on a sphere a distance $λ$ from the horizon and are regular on the future horizon. We show perturbatively that in the limit $λ$ approaches 0 the deformations are given by solutions of the nonlinear incompressible Navier-Stokes equation on the p-sphere. This relation provides a link between global existence for p-dimensional incompressible Navier-Stokes fluids and a novel form of cosmic censorship in p+2-dimensional general relativity.