Vortex analogue for the equatorial geometry of the Kerr black hole

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 example of 'analogue gravity' 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 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, and indicate ways in which more general analogue spacetimes might be constructed.