Separable geodesic action slicing in stationary spacetimes

TL;DR

This paper introduces a class of spacetime slicings based on geodesic actions in stationary spacetimes, leading to observer-based coordinates with desirable properties like unit lapse, and explains their relation to known coordinate systems such as Painlevé-Gullstrand.

Contribution

It provides a new perspective on slicing stationary spacetimes using separable geodesic equations, generalizing known coordinates and revealing geometric properties of observer trajectories.

Findings

Slicings with orthogonal geodesic trajectories represent freely falling observers.

The approach reproduces Painlevé-Gullstrand coordinates for Schwarzschild.

In static spherically symmetric spacetimes, the slicing can be made intrinsically flat.

Abstract

A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent freely falling observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function. This explains some of the properties of the original Painlev\'e-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the G\"odel spacetime. For the static spherically symmetric case the slicing can be chosen to be intrinsically flat with spherically symmetric geodesic observers, leaving all the gravitational field information in the shift vector field.