  We derive a geometrical version of the Regge-Wheeler and Zerilli equations,
which allows us to study gravitational perturbations on an arbitrary
spherically symmetric slicing of a Schwarzschild black hole. We explain how to
obtain the gauge-invariant part of the metric perturbations from the amplitudes
obeying our generalized Regge-Wheeler and Zerilli equations and vice-versa. We
also give a general expression for the radiated energy at infinity, and
establish the relation between our geometrical equations and the Teukolsky
formalism. The results presented in this paper are expected to be useful for
the close-limit approximation to black hole collisions, for the Cauchy
perturbative matching problem, and for the study of isolated horizons.
