Regularization of second-order scalar perturbation produced by a point-particle with a nonlinear coupling

TL;DR

This paper develops a regularization method for second-order scalar perturbations caused by a point particle with nonlinear coupling in curved spacetime, addressing divergence issues in standard solutions relevant for gravitational wave modeling.

Contribution

It introduces a novel regularization technique for second-order scalar perturbations with nonlinear sources, applicable as a toy-model for gravitational perturbation calculations.

Findings

Regularized solutions differ from standard retarded solutions.

Method successfully removes divergences in the toy-model.

Provides a foundation for more accurate gravitational perturbation analysis.

Abstract

Accurate calculation of the motion of a compact object in a background spacetime induced by a supermassive black hole is required for the future detection of such binary systems by the gravitational-wave detector LISA. Reaching the desired accuracy requires calculation of the second-order gravitational perturbations produced by the compact object. At the point particle limit the second-order gravitational perturbation equations turn out to have highly singular source terms, for which the standard retarded solutions diverge. Here we study a simplified scalar toy-model in which a point particle induces a nonlinear scalar field in a given curved spacetime. The corresponding second-order scalar perturbation equation in this model is found to have a similar singular source term, and therefore its standard retarded solutions diverge. We develop a regularization method for constructing well-defined causal solutions for this equation. Notably these solutions differ from the standard retarded solutions, which are ill-defined in this case.