  Motivated by gravitational wave observations of binary black hole mergers, we
present a procedure to compute the leading order nonlinear gravitational wave
interactions around a Kerr black hole. We describe the formalism used to derive
the equations for second order perturbations. We develop a procedure that
allows us to reconstruct the first order metric perturbation solely from
knowledge of the solution to the first order Teukolsky equation, without the
need of Hertz potentials. Finally, we illustrate this metric reconstruction
procedure in the asymptotic limit for the first order quasi-normal modes of
Kerr. In a companion paper, we present a numerical implementation of these
ideas.
