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 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.

Boosted black string bombs

“…The spectrum of quasi-bound states can be computed using a simple numerical method, which takes advantage of the fact that the solutions decay exponentially far away from the black hole horizon, and was employed in this context in [24]. Near the horizon of the black hole, x ≪ 1 ,the mode functions can be expanded in a Taylor series of the form:…”

“…The superradiant growth of particle number in a given bound state is, due to Pauli blocking, an exclusive property of bosonic fields and recently it has been explicitly shown to occur for massive vector fields [23]. The associated growth rate is larger than for massive scalar fields, as conjectured in [24], so that the black hole bomb effect can also be used to place stringent bounds on the mass of the photon and as a probe of ultra-light hidden photons in string compactifications and related scenarios.…”

Numerical evolutions of spherical Proca stars

“…This is indeed confirmed by a Fourier analysis (inset). Such an analysis can be compared with the frequencies of the first few monopole quasibound states of a Proca field on a Schwarzschild BH [41] -Table II 1 -and yields the following conclusions:…”

The decoupling problem of the Proca equation; and treatment of Dirac, Maxwell and Proca fields on the resulting pp-wave spacetimes

“…The longitudinal degree of freedom acquired by spin 1 field when the mass term is introduced, prevents the full decoupling of different components of the field in black hole space-times. For that reason authors have recently employed numerical techniques to solve separable but coupled wave equations for massive vector fields in Schwarzschild background [12,13], where one degree of freedom can be decoupled only.…”