Painlevé–Gullstrand coordinates versus Kerr spacetime geometry

Abstract

AbstractWe discuss the tension between the possible existence of Painlevé–Gullstrand coordinate systemsversusthe 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 distinguishstrongandweakPainlevé–Gullstrand coordinate systems, andconformalvariants thereof, cataloguing what we know can and cannot be done—sometimes we can makeexplicit globalstatements, sometimes we must resort toimplicit localstatements. 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 factorizedunimodularform; 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.