On Hyperboloidal Foliations in the Study of Black Hole Quasinormal Modes

TL;DR

This paper shows that hyperboloidal foliation techniques for studying black hole quasinormal modes are mathematically equivalent to the continued fraction method, with efficiency improvements mainly due to numerical grid choices rather than boundary behavior.

Contribution

It establishes the equivalence between hyperboloidal foliation and continued fraction approaches in analyzing black hole quasinormal modes, clarifying the source of computational efficiency.

Findings

Hyperboloidal and continued fraction methods are mathematically equivalent.

Efficiency gains are mainly due to Chebyshev grid usage.

Boundary behavior does not significantly affect method performance.

Abstract

In this work, we demonstrate that the hyperboloidal foliation technique, applied to the study of black hole quasinormal modes, where the spatial boundary is shifted from spacelike infinity to the future event horizon and null infinity, is effectively equivalent to the continued fraction approach, in which the asymptotic wave function typically diverges at both ends of spatial infinity. Specifically, a given hyperboloidal slicing, corresponding to a particular choice of coordinates, always uniquely determines a scheme for extracting the asymptotic form of the wave function at the spatial boundary. Owing to the mathematical equivalence, it follows that the efficiency and precision observed using the hyperboloidal approach should be attributed, not to avoiding the pathological behavior at the spatial boundaries, but primarily to other factors, such as the use of Chebyshev grids.