Painlevé-gullstrand Coordinates for the Kerr Solution

We construct a coordinate system for the Kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the Painlevé-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. Introduction The Painlevé-Gullstrand coordinate system for the Schwarzschild solution [Pai21, Gul22] is a particularly simple horizon-penetrating coordinate system, admitting interesting physical interpretations [HL08, Vis05]. It was inexplicably overlooked for a long time, but its importance has increased in recent years due to its relation with analogue gravity models [BLV05]. In this paper we construct a generalization of the Painlevé-Gullstrand coordinate system for the Kerr solution [Ker63], based on the zero angular momentum observers dropped from infinity. This is quite different from the generalization considered in [HL08], where the Doran form of the Kerr metric [Dor00] was used. The paper is organized as follows. In the first section we present the new coordinate system (but relegate the details of its derivation to an appendix so as not to interrupt the flow of the paper). In the second section we interpret this coordinate system as describing space flowing on a Riemannian 3-manifod, with the stationary limit given by 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. In the third section we address the question of what is meant by the flow of space. In particular, we show that the acceleration of free-falling objects is generally not in the direction of this flow. We also show that motions close to the flow of space can be obtained from a classical conservative system with a magnetic term. Finally, in the fourth section we compare the new coordinate system with the Doran coordinate system. We use the Einstein summation convention with latin indices i, j, . . . running from 1 to 3. Bold face letters u,v, . . . represent vectors on the space manifold. This work was partially supported by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER and by grant POCI/MAT/58549/2004. 1