Gravitational decoupling and regular hairy black holes: Geodesic stability, quasinormal modes, and thermodynamic properties

TL;DR

This work analyzes a regular hairy black hole produced via gravitational decoupling, focusing on geodesic stability through Lyapunov exponents, quasinormal modes, and thermodynamics. It shows that the hair parameter $\beta$ moves the ISCO and photon sphere inward and can mimic Kerr-like near-extremal spin in a finite range, while perturbations remain stable. QNM spectra, computed with WKB and in the eikonal limit, reveal slower decay at larger $\beta$ and a direct link to unstable null geodesics, tying gravitational dynamics to observable ringdown signals. Thermodynamics in both Bekenstein–Hawking and Rényi frameworks uncovers a minimum horizon radius, nontrivial temperature behavior, and stability features distinct from Schwarzschild, with Rényi statistics offering further insight into the system's thermodynamic viability. Collectively, the results provide potentially observable signatures in geodesic dynamics, shadows, and Hawking emission that could test gravitational decoupling hairy black holes in strong-field regimes.

Abstract

The stability of geodesic orbits around a regular hairy black hole, in the gravitational decoupling setup, is investigated by employing Lyapunov exponents, which quantify the divergence rate of nearby trajectories in dynamical systems. Both timelike and null geodesics are addressed, probing the effect of the hair parameter on orbital stability. Deviations from the Schwarzschild solution have a significant influence on orbit stability, potentially providing observational signatures. Quasinormal modes of regular hairy black holes are calculated, and their thermodynamic properties are discussed. Both the Rényi and the Bekenstein-Hawking entropies are reported, deepening our understanding of gravitational dynamics in the strong-field regime, contributing to ongoing approaches to modified gravity.

Figures (15)

• Figure 1: Metric coefficient \ref{['hrbh']} of the regular hairy black hole \ref{['main_metric']}. The plot displays the existence of two horizons, corresponding to the zeroes of $f$, for $0<\beta<0.39$. The radial coordinate is rescaled and expressed in units of the Schwarzschild mass.
• Figure 2: Effective potential \ref{['potential_eff']} for timelike geodesics around the regular hairy black hole corresponding to: (a) Schwarzschild, (b) $\beta = 0.05$, (c) $\beta = 0.20$, and (d) $\beta = 0.39$. The radial coordinate is rescaled and expressed in units of the Schwarzschild mass.
• Figure 3: Lyapunov exponents $\lambda_p$ for a circular timelike orbit around the regular hairy black hole \ref{['main_metric']}. The exponents with respect to proper time diverge near $r_c = 3$ for massive orbits, indicating absolute geodesic instability and immediate growth of perturbations, which is a sign of extreme chaotic behavior. This behavior is important for understanding the dynamics of matter accretion onto the black hole. The radius of circular orbit $r_c$ is rescaled and expressed in units of the Schwarzschild mass.
• Figure 4: Lyapunov exponents $\lambda_0$ for a circular timelike orbit around the regular hairy black hole \ref{['main_metric']}. We observe that in the case of $\beta_{\rm crit} \approx 0.39$, a black hole with a single event horizon and no singularity, there is a breakdown of real values of the Lyapunov exponents near the minimal radius of the event horizon. This indicates that geodesics are unstable throughout the entire interior region, although the nature of the instability changes near the horizon. The radius of circular orbit $r_c$ is rescaled and expressed in units of the Schwarzschild mass.
• Figure 5: Effective potential \ref{['null_potential']} for timelike geodesics around the regular hairy black hole corresponding to: (a) Schwarzschild, (b) $\beta = 0.05$, (c) $\beta = 0.20$, and (d) $\beta = 0.39$. The radial coordinate is rescaled and expressed in units of the Schwarzschild mass.