Late-Time Dynamics of Scalar Fields on Rotating Black Hole Backgrounds

The purpose of the studies presented in this paper is to point out the need for further work towards a definitive answer to the following question: How do perturbations of a rotating black hole get radiated away? Using the Teukolsky formalism [1], we can search for an answer to that question by studying the evolution of a complex spin-weighted wavefunction that is defined in terms of curvature perturbations of a fixed black hole background geometry. The late-time dynamics of gravitational perturbations is governed by features very similar to those occurring for scalar test-fields [2,6]. In the special case of vanishing spin-weight, the Teukolsky equation reduces to the linear wave equation for a scalar test field Φ on a Kerr background, which has a somewhat simpler structure than the Teukolsky equation. For the sake of simplicity, the present discussion is confined to the evolution of scalar test fields. For a spherically symmetric, asymptotically flat background geometry, the angular variables can be separated off, and with an appropriate radial coordinate x the wave equation for a scalar test field Φ reduces to a (1 + 1) dimensional wave equation with an effective potential V (x),