Massive vector fields on the Schwarzschild spacetime: quasinormal modes and bound states

TL;DR

This paper investigates the behavior of massive vector fields around Schwarzschild black holes, computing their quasinormal modes and bound states using numerical and analytical methods, with implications for hidden photon phenomenology.

Contribution

It introduces a comprehensive numerical and analytical framework for analyzing massive vector fields on Schwarzschild spacetime, including spectra computation and near-horizon analysis.

Findings

Computed quasinormal mode spectra for massive vector fields

Identified quasi-bound states unique to massive fields

Provided analytical approximations in the small-mass limit

Abstract

We study the propagation of a massive vector or Proca field on the Schwarzschild spacetime. The field equations are reduced to a one-dimensional wave equation for the odd-parity part of the field and two coupled equations for the even-parity part of the field. We use numerical techniques based on solving (scalar or matrix-valued) three-term recurrence relations to compute the spectra of both quasi-normal modes and quasi-bound states, which have no massless analogue, complemented in the latter case by a forward-integration method. We study the radial equations analytically in both the near-horizon and far-field regions and use a matching procedure to compute the associated spectra in the small-mass limit. Finally, we comment on extending our results to the Kerr geometry and its phenomenological relevance for hidden photons arising e.g. in string theory compactifications.