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

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 quasinormal modes and quasibound 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.