Perturbations of the Kerr spacetime in horizon-penetrating coordinates

We derive the Teukolsky equation for perturbations of a Kerr spacetime when the spacetime metric is written in either ingoing or outgoing Kerr-Schild form. We also write explicit formulae for setting up the initial data for t he Teukolsky equation in the time domain in terms of a 3-metric and an extr insic curvature, The motivation of this work is to have in place a formalis m to study the evolution in the 'close limit' of two recently proposed solu tions to the initial-value problem in general relativity that are based on Kerr-Schild slicings. A perturbative formalism in horizon-penetrating coord inates is also very desirable in connection with numerical relativity simul ations using black hole 'excision'.