Painlevé–Gullstrand coordinates versus Kerr spacetime geometry

We discuss the tension between the possible existence of Painlevé–Gullstrand coordinate systems versus the explicit geometrical features of the Kerr spacetime; a subject of interest to Professor Thanu Padmanabhan in the weeks immediately preceding his unexpected death. We shall carefully distinguish strong and weak Painlevé–Gullstrand coordinate systems, and conformal variants thereof, cataloguing what we know can and cannot be done—sometimes we can make explicit global statements, sometimes we must resort to implicit local statements. For the Kerr spacetime the best that seems to be achievable is to set the lapse function to unity and represent the spatial slices with a 3-metric in factorized unimodular form; this arises from considering the Doran version of Kerr spacetime in Cartesian coordinates. We finish by exploring the (limited) extent to which this construction might possibly lead to implementing an “analogue spacetime” model suitable for laboratory simulations of the Kerr spacetime.