Painleve-Gullstrand coordinates versus Kerr spacetime geometry

We discuss the tension between the possible existence of Painleve-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 Painleve-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.

Automated summary

This article discusses the tension between the possible existence of Painleve-Gullstrand coordinate systems and the explicit geometrical features of the Kerr spacetime. It distinguishes between strong and weak Painleve-Gullstrand coordinate systems, and explores what can and cannot be done in these systems. For the Kerr spacetime, the best achievable option seems to be setting the lapse function to unity and representing spatial slices with a factorized unimodular form. The article concludes by examining to what extent this construction can lead to implementing a suitable analogue spacetime model for laboratory simulations of the Kerr spacetime.