  A key source for LISA will be the inspiral of compact objects into
supermassive black holes. Recently Mino has shown that in the adiabatic limit,
gravitational waveforms for these sources can be computed using for the
radiation reaction force the gradient of one half the difference between the
retarded and advanced metric perturbations. Using post-Newtonian expansions, we
argue that the resulting waveforms should be sufficiently accurate for signal
detection with LISA. Data-analysis templates will require higher accuracy,
going beyond adiabaticity; this remains a significant challenge. We describe an
explicit computational procedure for obtaining waveforms based on Mino's
result, for the case of a point particle coupled to a scalar field. We derive
an expression for the time-averaged time derivative of the Carter constant, and
verify that the expression correctly predicts that circular orbits remain
circular while evolving via radiation reaction. The derivation uses detailed
properties of mode expansions, Green's functions and bound geodesic orbits in
the Kerr spacetime, which we review in detail. This paper is about three
quarters review and one quarter new material. The intent is to give a complete
and self-contained treatment of scalar radiation reaction in the Kerr
spacetime, in a single unified notation, starting with the Kerr metric and
ending with formulae for the time evolution of all three constants of the
motion that are sufficiently explicit to be used immediately in a numerical
code.
