Radiation outer boundary conditions and near-to-far field signal transformations for the Bardeen-Press equation

TL;DR

This paper develops exact boundary conditions and teleportation kernels for the Bardeen-Press equation, enabling accurate long-time simulations of gravitational waves without unphysical artifacts.

Contribution

It introduces and implements exact radiation outer boundary conditions and near-to-far field transformations for the Bardeen-Press equation, improving simulation accuracy.

Findings

Boundary and teleportation kernels are well approximated by exponential sums.

Kernel-based boundary conditions eliminate unphysical late-time growth.

The method achieves correct late-time decay rates in simulations.

Abstract

Several theoretical and astrophysical problems - including gravitational-wave modeling for extreme mass-ratio inspirals - require accurate time-domain solutions of the spin-weight $s=-2$ Teukolsky equation in Boyer-Lindquist coordinates. Because such simulations are performed on finite computational domains, they typically introduce an artificial outer boundary where nontrivial boundary conditions must be imposed. If these conditions are inaccurate, then spurious reflections and slowly-growing unphysical modes may corrupt long-time evolutions. We develop and implement exact radiation outer boundary conditions for the Bardeen-Press equation (a harmonic moment of the $a=0$ Teukolsky equation), making the artificial boundary transparent at any finite radius. We also construct near-to-far field teleportation kernels that map field data recorded at finite radius $r_1$ to the data reaching $r_2 > r_1$. The possible choice $r_2 = \infty$ corresponds to asymptotic waveform evaluation, that is propagation of the data to future null infinity. We show that both boundary and teleportation kernels are well approximated by exponential sums, with associated error bounds. Implemented in a time-domain solver, our kernel-based boundary conditions eliminate unphysical late-time growth and give the correct late-time decay rates, affording efficient long-duration simulations for waveform modeling and related blackhole perturbation calculations.