Kerr/Fluid Duality and Singularity of Solutions to the Fluid Equation

TL;DR

This paper explores the duality between fluid dynamics and black hole horizons, deriving equations for viscous fluids on spheroidal surfaces and analyzing conditions for smooth solutions and caustic formation.

Contribution

It derives a fluid equation dual to perturbations of Kerr and Schwarzschild black holes and analyzes the conditions for solution smoothness related to null geodesic congruences.

Findings

The fluid equation is dual to black hole horizon perturbations.

Null geodesic expansion scalar can become negative, leading to caustics.

Positive semi-definite expansion scalar is necessary for smooth solutions.

Abstract

An equation for a viscous incompressible fluid on a spheroidal surface which is dual to the perturbation around the near-near horizon extreme Kerr (n-NHEK) black hole is derived. It is also shown that an expansion scalar $\theta$ of a congruence of null geodesics on the null horizon of the perturbed n-NHEK spacetime, which is dual to a viscous incompressible fluid, is not positive semi-definite, even if initial conditions on the velocity are smooth. Unless initial conditions are elaborated, caustics of null congruence will occur on the horizon in the future. A similar result is obtained for a perturbed Schwarzschild black hole spacetime which is dual to a viscous incompressible fluid on $S^2$. An initial condition that $\theta$ be positive semi-definite at any point on $S^2$ is a necessary condition for the existence of smooth solutions to incompressible Navier-Stokes (NS) equation on $S^2$.