# Unbound motion on a Schwarzschild background: Practical approaches to frequency domain computations

## TL;DR

This paper presents practical methods for frequency domain computations of gravitational perturbations caused by unbound motion on a Schwarzschild black hole background, improving speed and accuracy.

## Contribution

It develops a general procedure to derive higher-order master functions that enhance convergence and computational efficiency in frequency domain analyses.

## Key findings

- Higher-order master functions converge faster at large distances.
- The new approach improves speed and accuracy of frequency domain codes.
- The method facilitates analysis of unbound particle motion around black holes.

## Abstract

Gravitational perturbations due to a point particle moving on a static black hole background are naturally described in Regge-Wheeler gauge. The first-order field equations reduce to a single master wave equation for each radiative mode. The master function satisfying this wave equation is a linear combination of the metric perturbation amplitudes with a source term arising from the stress-energy tensor of the point particle. The original master functions were found by Regge and Wheeler (odd parity) and Zerilli (even parity). Subsequent work by Moncrief and then Cunningham, Price and Moncrief introduced new master variables which allow time domain reconstruction of the metric perturbation amplitudes. Here I explore the relationship between these different functions and develop a general procedure for deriving new higher-order master functions from ones already known. The benefit of higher-order functions is that their source terms always converge faster at large distance than their lower-order counterparts. This makes for a dramatic improvement in both the speed and accuracy of frequency domain codes when analyzing unbound motion.