Double null coordinates for Kerr spacetime

Abstract

We present a definition for a pair of null coordinates that are naturally adapted to the horizons and future null infinity of Kerr spacetime, and that are generated by the center-of-mass sections at future null infinity. They are a smooth, round family of null hypersurfaces which foliate the Kerr spacetime in an outgoing and an ingoing sense, respectively, and they have a regular extension across the horizons. Because of Kerr’s peculiar geometry, the construction involves a nonlinear differential equation for a scalar function related to Carter’s constant, whose solution cannot be expressed in terms of simple analytic functions. We present the numerical solution of this scalar for a particular choice of the geometrical parameters. In this setting, there naturally appears a two-dimensional spacelike family of round surfaces that are parametrized by , which are the intersections of both null coordinates, where can be thought of as the tortoise coordinate extension for the Kerr spacetime. The surfaces are axially symmetric, but they have an dependence in Boyer-Lindquist coordinates. They can also be characterized in a complete geometrical way by their Gaussian and extrinsic curvature scalars, which we were able to compute by the use of Geroch-Held-Penrose formalism. We compare our definition with other previous attempts in the literature, and we show that all of them have divergent behavior at the axis of symmetry. Thus, our construction presents the first double null coordinate system which makes possible computations over all of the Kerr spacetime.