Unit-lapse versions of the Kerr spacetime

Abstract

Abstract The Kerr spacetime is perhaps the most astrophysically important of the currently known exact solutions to the Einstein field equations. Whenever spacetimes can be put in unit-lapse form it becomes possible to identify some very straightforward timelike geodesics, (the ‘rain’ geodesics), making the physical interpretation of these spacetimes particularly clean and elegant. The most well-known of these unit-lapse formulations is the Painlevé–Gullstrand form of the Schwarzschild spacetime, though there is also a Painlevé–Gullstrand form of the Lense–Thirring (slow rotation) spacetime. More radically there are also two known unit-lapse forms of the Kerr spacetime—the Doran and Natário metrics—though these are not precisely in Painlevé–Gullstrand form. Herein we shall seek to explicate the most general unit-lapse form of the Kerr spacetime. While at one level this is ‘merely’ a choice of coordinates, it is a strategically and tactically useful choice of coordinates, thereby making the technically challenging but astrophysically crucial Kerr spacetime somewhat easier to deal with. While in the current article we focus on the ‘rain’ geodesics, it should be noted that the explicit unit-lapse metrics we present are also useful for looking at other more complicated geodesics in the Kerr spacetime.