Unit-lapse versions of the Kerr spacetime

TL;DR

This paper explores the most general unit-lapse coordinate form of the Kerr spacetime, aiming to simplify its analysis and interpretation by identifying straightforward timelike geodesics, thus aiding astrophysical applications.

Contribution

It derives the most general unit-lapse form of the Kerr spacetime, providing a strategic coordinate choice that simplifies the analysis of this complex solution.

Findings

Explicit form of the most general unit-lapse Kerr spacetime

Identification of straightforward timelike geodesics in this form

Enhanced interpretability of Kerr spacetime in astrophysical contexts

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 Painleve-Gullstrand form of the Schwarzschild spacetime, though there is also a Painleve-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 Natario metrics -- though these are not precisely in Painleve-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.