Bilinear products and the orthogonality of quasinormal modes on hyperboloidal foliations

TL;DR

This paper investigates the properties of bilinear products for black-hole quasinormal modes on hyperboloidal slices, addressing divergence issues and proposing regularisation methods, with explicit calculations for scalar perturbations of Schwarzschild black holes.

Contribution

It introduces regularisation procedures for bilinear products of QNMs on hyperboloidal foliations and defines excitation factors within this framework.

Findings

Bilinear integrals are divergent due to boundary behavior.

Regularisation methods can produce finite, well-defined bilinear forms.

Explicit computation of excitation factors for Schwarzschild scalar QNMs.

Abstract

We explore the properties of bilinear products for black-hole quasinormal modes (QNMs) formulated on hyperboloidal foliations. We find that, although QNM solutions are smooth and finite on future-directed hyperboloids, the integrand of the bilinear form with respect to which the modes are orthogonal is still divergent. This is a result of the reflection (equivalently, CPT) transformation required in the definition of the products, which modifies the behaviour of the integrand at the boundaries. We present several regularisation procedures that yield a finite and well-defined bilinear form. In addition, we examine an alternative definition of the bilinear products that incorporates flux contributions, discussing its advantages and limitations. Finally, we define the QNM excitation factors and coefficients within the hyperboloidal framework in terms of the bilinear products, and compute them explicitly for a choice of mode numbers and constant initial data. For concreteness, we work with the QNMs associated to scalar perturbations of the Schwarzschild family of spacetimes.