Hawking emission of massive vector fields by Kerr black holes

Abstract

We compute, for the first time, the Hawking emission spectrum of massive vector (Proca) fields by spinning Kerr black holes, determining the associated greybody factors and the resulting mass and spin loss functions. We show, in particular, that the scalar (longitudinal) polarization of the Proca field has a spectrum approaching that of a free scalar field in the massless limit (in which it becomes a pure gauge mode), although we find substantial differences for finite mass. The contribution of the two vector (transverse) polarization modes coincides, as expected, with the one obtained by Page for the Maxwell field in the massless limit. The black hole's evaporation rate is dominated by the scalar mode for slowly spinning black holes and by the two vector modes as the black hole approaches extremality. As for other fields, we find that Proca Hawking emission is Boltzmann-suppressed for Hawking temperatures $T_H\lesssim |μ-Ω_H|$, where $μ$ is the field mass and $Ω_H$ is the angular velocity of the black hole's horizon. This implies that highly spinning black holes can efficiently emit massive vector fields at temperatures parametrically below the field's mass. Finally, we also find that superradiant emission is more pronounced for massive vector fields, with a maximum amplification factor of $\simeq 7\%$ (compared to $\simeq 4\%$ for massless photons).