Axial gravitational perturbations and echo-like signals of a hairy black hole from gravitational decoupling

Abstract

We study axial gravitational perturbations of a hairy black hole constructed in the framework of gravitational decoupling and investigate the geometric origin of echo-like late-time signals in this spacetime. We derive the odd-parity master equation and the corresponding effective potential, and we compute the quasinormal-mode spectrum by using frequency-domain and time-domain methods. We show that, in a suitable region of parameter space, the axial potential develops a double-peak structure that supports a trapping cavity and gives rise to echo-like late-time waveforms. Rather than imposing near-horizon reflectivity by hand, the delayed pulses therefore arise dynamically from the geometry of the effective potential. We also clarify that the parameter region exhibiting echoes need not coincide with the region in which the weak energy condition is satisfied everywhere outside the event horizon, and this distinction must be taken into account when interpreting the physical status of the solution. Our results provide a useful framework for probing black-hole hair through gravitational-wave ringdown and for exploring possible observational departures from the standard no-hair paradigm.

Figures (7)

• Figure 1: Metric function $f(r)$ for several choices of $\alpha$ and $\beta$ with $M=1$. In the left panel, $\beta=0.20$ is fixed and $\alpha$ changes. In the right panel, $\alpha=0.30$ is fixed and $\beta$ changes. The blue and purple curves correspond to cases with one positive root. The red curve marks the extremal case with two positive roots and one degenerate horizon. The green curve shows the case with three distinct positive roots.
• Figure 2: Parameter-space structure in the $(\beta,\alpha)$ plane for the hairy black hole with $M=1$. The shaded area shows the region with three distinct positive roots. The red curves represent the extremal boundaries, on which two positive roots merge into a degenerate horizon. Outside the shaded region, the metric function has a single positive root.
• Figure 3: Effective potential $V(r_*)$ (left panel) and semilogarithmic time-domain profiles (right panel) for axial gravitational perturbations with $M=1$, $l=2$, and $\beta=0.2$. The three curves correspond to $\alpha=0.1$, $0.355$, and $0.404$.
• Figure 4: Effective potential as a function of the tortoise coordinate $r_*$ (left panel), and the corresponding time-domain waveform as a function of time $t$ (right panel). The parameters are fixed as $M=1$, $\ell=2$, and $\beta=0.25$. From top to bottom, the three cases correspond to $\alpha=0.45$ (top), $0.455$ (middle), and $0.46$ (bottom), respectively.
• Figure 5: Effective potential as a function of the tortoise coordinate $r_*$ (left panel), and the corresponding time-domain waveform as a function of time $t$ (right panel). The parameters are fixed as $M=1$, $\ell=2$, and $\alpha=0.45$. From top to bottom, the three curves correspond to $\beta=0.241$ (top), $0.245$ (middle), and $0.249$ (bottom), respectively.