Painleve-Gullstrand Coordinates for the Kerr Solution

TL;DR

This paper introduces a new coordinate system for the Kerr black hole solution, extending Painleve-Gullstrand coordinates from Schwarzschild to Kerr, providing insights into the flow of space and horizon structure.

Contribution

It develops a generalized coordinate system for Kerr spacetime based on zero angular momentum observers, enhancing understanding of space flow and horizon properties.

Findings

Kerr metric expressed as flowing space on a curved 3-manifold

Stationary limit and horizons identified via flow speed

Comparison with Doran coordinates shows similarities and differences

Abstract

We construct a coordinate system for the Kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the Painleve-Gullstrand coordinate system for the Schwarzschild solution. The Kerr metric can then be interpreted as describing space flowing on a (curved) Riemannian 3-manifold. The stationary limit arises as the set of points on this manifold where the speed of the flow equals the speed of light, and the horizons as the set of points where the radial speed equals the speed of light. A deeper analysis of what is meant by the flow of space reveals that the acceleration of free-falling objects is generally not in the direction of this flow. Finally, we compare the new coordinate system with the closely related Doran coordinate system.