Self-force with (3+1) codes: a primer for numerical relativists

TL;DR

This paper demonstrates the first successful implementation of a (3+1) numerical relativity code for self-force calculations, enabling more integrated analyses of extreme-mass-ratio inspirals using existing tools.

Contribution

It revisits and validates a (3+1) method for self-force calculations, showing its viability with two independent codes and achieving high accuracy.

Findings

Accurately computed self-force components to within 1%.

Confirmed consistency between self-force and energy flux.

First successful (3+1) self-force calculation.

Abstract

Prescriptions for numerical self-force calculations have traditionally been designed for frequency-domain or (1+1) time-domain codes which employ a mode decomposition to facilitate in carrying out a delicate regularization scheme. This has prevented self-force analyses from benefiting from the powerful suite of tools developed and used by numerical relativists for simulations of the evolution of comparable-mass black hole binaries. In this work, we revisit a previously-introduced (3+1) method for self-force calculations, and demonstrate its viability by applying it to the test case of a scalar charge moving in a circular orbit around a Schwarzschild black hole. Two (3+1) codes originally developed for numerical relativity applications were independently employed, and in each we were able to compute the two independent components of the self-force and the energy flux correctly to within $< 1%$. We also demonstrate consistency between $t$-component of the self-force and the scalar energy flux. Our results constitute the first successful calculation of a self-force in a (3+1) framework, and thus open opportunities for the numerical relativity community in self-force analyses and the perturbative modeling of extreme-mass-ratio inspirals.