Reducing triangular systems of ODEs with rational coefficients, with applications to coupled Regge-Wheeler equations

TL;DR

This paper presents a systematic method to find rational solutions to inhomogeneous rational ODE systems and applies it to analyze the reducibility of coupled Regge-Wheeler equations relevant in black hole perturbation theory.

Contribution

It introduces a finite-dimensional linear algebra approach for solving rational ODE systems and determines conditions for their reduction to diagonal form, specifically applied to Regge-Wheeler equations.

Findings

Method effectively finds all rational solutions to the ODE systems.

Decides when systems can be reduced to diagonal form using rational differential operators.

Reproduces and extends previous identities in black hole perturbation analysis.

Abstract

We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black hole spacetime. Our systematic approach reproduces and complements identities that have been previously found by trial and error methods.