Vortex geometry for the equatorial slice of the Kerr black hole

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.