Echoes from Hairy Black Holes

Abstract

We study the waveforms of time signals produced by scalar perturbations in static hairy black holes, in which the perturbations can be governed by a double-peak effective potential.The inner potential peak would give rise to echoes, which provide a powerful tool to test the Kerr hypothesis. The waveforms are constructed in the time and frequency domains, and we find that the late-time waveforms are determined by the long-lived and sub-long-lived quasinormal modes, which are trapped in the potential valley and near the smaller peak, respectively. When the distance between the peaks is significantly larger than the width of the peaks, a train of decaying echo pulses is produced by the superposition of the long-lived and sub-long-lived modes. In certain cases, the echoes can vanish and then reappear. When the peaks are close enough, one detects far fewer echo signals and a following sinusoid tail, which is controlled by the long-lived or sub-long-lived mode and hence decays very slowly.

Gravitational Lensing by Black Holes with Multiple Photon Spheres

“…The existence of two photon spheres can significantly affect the optical appearance of HBHs illuminated by the surrounding accretion disk, e.g., producing bright rings of different radus in the black hole images [88] and noticeably increasing the flux of the observed images [89]. Moreover, the effective potential for a scalar perturbation in the HBHs with two photon spheres was shown to exhibit a double-peak structure, leading to long-lived quasinormal modes [90] and echo signals [91]. It is worth noting that the existence of two photon spheres outside the event horizon has also been reported for dyonic black holes with a quasi-topological electromagnetic term [92] and black holes in massive gravity [93,94].…”