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.

• The paper demonstrates that axial gravitational perturbations in hairy black holes yield echo-like signals through a double-peak effective potential structure.
• It employs numerical methods such as Chebyshev-collocation, high-order WKB, and Prony analysis to detail shifts in quasinormal mode frequencies and damping rates.
• The study reveals that echo phenomena arise naturally from the spacetime geometry without artificial boundaries, offering new prospects for gravitational-wave observations.

Axial Gravitational Perturbations and Echoes in a Hairy Black Hole from Gravitational Decoupling

Introduction

The paper presents a systematic analysis of axial (odd-parity) gravitational perturbations in a class of spherically symmetric "hairy black holes" generated through the extended geometric deformation (EGD) approach—a variant of gravitational decoupling. The focus is on the dynamical and geometric origin of gravitational-wave echo-like signals, examining how the effective potential structure derived from the underlying geometric hair supports such features. The analysis combines QNM spectroscopy (frequency and time domain) and rigorous energy condition constraints on the configuration.

Hairy Black Hole Construction via Gravitational Decoupling

Within the EGD framework, the total energy-momentum tensor is split between the usual Einstein sector and an auxiliary anisotropic source $\Theta_{\mu\nu}$. The canonical seed geometry is Schwarzschild, which is deformed by introducing a radial-temporal function $h(r)$ controlling the non-trivial hair. The weak energy condition (WEC) outside the event horizon imposes nonlinear differential constraints on $h(r)$, parameterized by a hair amplitude $\alpha$ and a length scale $\beta$. When specialized to static, spherically symmetric metrics with imposed Schwarzschild-like $g_{tt}g_{rr}=-1$, the full solution is

$$ds^2 = \left(1-\frac{2M}{r}\right)h(r)dt^2 - \left[\left(1-\frac{2M}{r}\right)h(r)\right]^{-1}dr^2 - r^2d\Omega^2,$$

with $h(r)$ specified to satisfy WEC positivity. The causal structure (number and order of horizons) is governed by the roots of a transcendental equation involving $\alpha$ and $\beta$.

Crucially, the parameter space featuring non-trivial horizon structures, particularly multi-root and extremal cases, yields a window wherein the effective axial potential can depart substantially from the canonical Regge–Wheeler profile.

Master Equation and Axial Potential Structure

Axial (odd-parity) gravitational perturbations are formulated in the Regge–Wheeler gauge and decompose into a master Schrödinger-type equation,

$h(r)$0

where $h(r)$1 is the tortoise coordinate and the effective potential is

$h(r)$2

In the Schwarzschild limit ($h(r)$3), this reduces to the standard Regge–Wheeler potential. However, for finite $h(r)$4, $h(r)$5 develops a richer structure, including—under certain conditions—a double-peak profile, which is essential for the emergence of a trapping cavity supporting echo modes.

Quasinormal Mode Analysis

The QNM spectrum is computed using three numerical schemes: the Chebyshev-collocation pseudospectral method, high-order WKB with Padé resummation, and Prony analysis on time-domain waveforms. Across methods, results converge for low overtone fundamental modes.

Key quantitative findings include:

• The real part (oscillation frequency) of the fundamental QNM increases monotonically with the hair parameter $h(r)$6.
• The imaginary part (damping rate) exhibits non-monotonic dependence: enhanced damping for small $h(r)$7, then suppressed as $h(r)$8 increases and the horizon structure reorganizes.
• For certain critical values of $h(r)$9 and $h(r)$0, the effective axial potential transitions to a double-peak regime, indicating the onset of a trapping cavity.

This regime is characterized by the emergence of time-delayed secondary pulses (echoes) in the late-time ringdown signal.

Echo-Like Signal Emergence

The key claim is that echo-like signals emerge dynamically in regions of parameter space where the effective potential develops a double-peak profile with an intervening cavity. The analysis demonstrates that the cavity structure and the resulting echo phenomenology depend on barrier height, width, and separation, controlled by $h(r)$1 and $h(r)$2.

Unlike echo models requiring ad hoc reflective surfaces near the horizon, echoes here are a direct consequence of the spacetime's geometry—specifically, its anisotropic hair sector. The echo timescale and pulse structure are tunable via the parameters, offering a compelling scenario for gravitational-wave phenomenology beyond the standard no-hair ringdown.

The analysis further clarifies that the echo-supporting parameter region does not in general coincide with the set of solutions satisfying the WEC globally outside the event horizon; this distinction must be maintained in phenomenological interpretations.

Implications and Outlook

This work provides a robust framework for probing the phenomenology of black hole hair in gravitational-wave ringdown, directly linking new physics in the near-horizon geometry to observable signatures such as modified QNM spectra and genuine echo trains. The dynamical emergence of echoes without the imposition of non-standard boundary conditions is a significant theoretical outcome, and this scenario can be contrasted with echoes arising in alternative frameworks, e.g., non-trivial boundary conditions or exotic horizonless objects ("Primary hairs may create echoes" (Konoplya et al., 18 Aug 2025)).

From a theoretical perspective, the results strengthen the case for using ringdown and late-time waveform analysis as discriminants of beyond-Kerr black hole solutions, offering a test-bed for gravitational decoupling and other viable hair-generation mechanisms. Practically, the possibility of distinguishing these echo signals in observational gravitational-wave datasets informs future strategies for testing general relativity and energy condition-violating configurations.

Further development should focus on a comprehensive parameter scan of WEC-satisfying solutions, explicit analysis of polar perturbations, the influence of other field perturbations (e.g., scalar, electromagnetic), and the generalization to rotating backgrounds. Prospects for direct connection to LIGO–Virgo–KAGRA data analyses are especially pertinent, given the parameter dependence of the echo signal's physical viability.

Conclusion

The study rigorously establishes that in a class of spherically symmetric hairy black holes generated via gravitational decoupling, the axial perturbation sector admits dynamical gravitational-wave echoes in a specific regime of parameter space. The existence and properties of these echoes are controlled entirely by the geometry of the effective potential, without recourse to artificial boundary conditions. The framework advanced herein provides a precise pathway for both theoretical exploration and observational constraint of near-horizon hair via gravitational-wave spectroscopy and late-time signal analysis.