Second-order perturbations of Kerr black holes: Formalism and reconstruction of the first-order metric

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 quasinormal modes of Kerr. In a companion paper [J. L. Ripley et al., Phys. Rev. D 103, 104018 (2021)] we present a numerical implementation of these ideas.