Vortex geometry for the equatorial slice of the Kerr black hole

TL;DR

This paper explores the formal analogy between the geometry of the equatorial slice of a Kerr black hole and the acoustic geometry of a rotating fluid vortex, providing insights into black hole spacetime and potential experimental analogs.

Contribution

It identifies the most general acoustic geometry compatible with fluid dynamics and shows how the Kerr black hole's equatorial slice can be represented in this vortex form.

Findings

The acoustic geometry for a collapsing/expanding vortex is derived.

A coordinate transformation maps the Kerr equatorial slice to a vortex form.

The entire Kerr spacetime cannot be represented as a perfect-fluid acoustic geometry.

Abstract

The spacetime geometry on the equatorial slice through a Kerr black hole is formally equivalent to the geometry felt by phonons entrained in a rotating fluid vortex. We analyse this situation in some detail: First, we find the most general ``acoustic geometry'' compatible with the fluid dynamic equations in a collapsing/expanding perfect-fluid line vortex. Second, we demonstrate that there is a suitable choice of coordinates on the equatorial slice through a Kerr black hole that puts it into this vortex form; though it is not possible to put the entire Kerr spacetime into perfect-fluid ``acoustic'' form. Finally, we briefly discuss the implications of this formal equivalence; both with respect to gaining insight into the Kerr spacetime and with respect to possible vortex-inspired experiments.