Construction of the second-order gravitational perturbations produced by a compact object

Accurate calculation of the gradual inspiral motion in an extreme mass-ratio binary system, in which a compact object inspiral towards a supermassive black hole requires calculation of the interaction between the compact object and the gravitational perturbations that it induces. These metric perturbations satisfy linear partial differential equations on a curved background space-time induced by the supermassive black hole. At the point-particle limit the second-order perturbations equations have source terms that diverge as r^-4, where r is the distance from the particle. This singular behavior renders the standard retarded solutions of these equations ill defined. Here we resolve this problem and construct well-defined and physically meaningful solutions to these equations. We recently presented an outline of this resolution [E. Rosenthal, Phys. Rev. D 72, 121503 (2005).]. Here we provide the full details of this analysis. These second-order solutions are important for practical calculations: the planned gravitational-wave detector LISA requires preparation of waveform templates for the potential gravitational waves. Construction of templates with desired accuracy for extreme mass-ratio binaries requires accurate calculation of the inspiral motion including the interaction with the second-order gravitational perturbations.