Late time dynamics of scalar perturbations outside black holes. I. A shelltoy model - art. no. 044016

We present a new analytic approach for the study of late time evolution of linear test-fields, propagating on the exterior of black holes. This method provides a calculation scheme applicable to Kerr black holes (for which ca se no analytic calculation of the late time tails has been presented so far ). In this paper we develop the new technique and apply it to the case of m assless scalar waves evolving on the background geometry of a static spheri cally symmetric thin shell with a Schwarzschild exterior. The late time beh avior of the scalar field at null infinity is calculated, and is explicitly related to the form of (quite arbitrary) initial data. This reproduces the well-known late time power-law decaying tails. In an accompanying paper we apply our approach to the complete Schwarzschild black hole geometry, wher e we obtain the familiar inverse-power late time tails at null infinity, as well as at time-like infinity and along the event horizon. A calculation o f the late time power-law tails in the Kerr geometry, based on the same app roach, will be presented in a forthcoming paper. [S0556-2821(99)06002-6].