Long-lived ringing of Bronnikov--Kim brane-world black holes and wormholes

Abstract

Massive-field perturbations on black-hole and wormhole backgrounds can form long-lived quasinormal ringing; this behaviour is not confined to brane-world spacetimes, but it is also not universal across compact-object geometries. Recent work on a brane-world black hole reported an unusual limiting behaviour in which the real oscillation frequency tends to zero as the scalar mass grows. We test how general this phenomenon is in the Bronnikov--Kim effective brane-world geometries, treating both the black-hole and wormhole sides of the same families. The comparison uses two complementary numerical treatments: Leaver's continued-fraction method for the black-hole side, after reducing the wave equation for the second Bronnikov--Kim family to rational form, and Chebyshev pseudospectral collocation for regular wormhole throats. The black hole of the second Bronnikov--Kim family shows the standard quasi-resonant pattern: the damping rates of the scalar branches considered here become very small, but the real frequencies remain finite. Representative wormholes from the first and second Bronnikov--Kim families also develop increasingly long-lived massive scalar modes, and extending the second-family wormhole scan to larger field mass reduces the damping by more than a factor of four relative to the massless case. Thus the robust brane-world effect found here is the appearance of long-lived massive-field ringing in both black holes and wormholes, whereas the vanishing of the real frequency is metric- or branch-dependent.