Near-horizon Geodesic Instabilities and Anomalous Decay of Quasinormal Modes in Weyl Black Holes

TL;DR

This work analyzes the stability of Weyl geometry around an exact black hole solution, connecting geodesic chaos, quantified by Lyapunov exponents, to the quasinormal mode spectrum of a test scalar field in an asymptotically de Sitter–like background. By deriving the photon-sphere QNMs with a WKB approach and examining both massless and massive perturbations, the authors establish a geodesic–QNMs correspondence in this modified gravity setting and reveal how the unstable geodesics control both oscillation and decay rates. They uncover an anomalous decay regime, with a critical scalar mass separating regimes where longer-lived modes are associated with higher or lower angular momentum, and show that the Lyapunov exponent governs the decay width of the QNMs and the effective potential’s angular-width relation. Overall, the results illuminate the interplay between geodesic stability and ringdown in Weyl gravity, suggesting observational signatures and guiding future explorations of modified gravity black holes and their perturbations.

Abstract

We study the stability of the Weyl geometry considering an exact black hole solution. By calculating the geodesics of massless and massive scalar fields orbiting outside the Weyl black hole background and using the Lyapunov exponent, we show that geodesic instabilities, characterized by the Lyapunov exponent, appear in the asymptotically de Siter-like spacetime. Calculating the photon sphere's quasinormal modes (QNMs) of a scalar field perturbing the Weyl black hole, we find a relation connecting the QNMs with the Lyapunov exponent in the asymptotically de Siter-like spacetime. Furthermore, we study the anomalous decay rate of the QNMs connecting their behavior with the Lyapunov exponent.

Figures (11)

• Figure 1: Behavior of the scalar field $\Phi(r)$ as a function of the radial coordinate $r$ (in units of $r_g$, where $r_g$ is the gravitational radius). The scalar field follows the relation $\Phi(r) = C_1/(r + C_2)^2$ with $C_1 = 65.2$ and $C_2 = 15r_g$, satisfying the boundary condition $\Phi(r) \to 0$ at infinity. The plot shows the monotonic decrease of the scalar field strength as we move away from the black hole horizon.
• Figure 2: Behavior of the radial component of the Weyl vector field $\omega_1(r)$ as a function of the radial coordinate $r$ (in units of $r_g$). The Weyl vector field follows the relation $\omega_1(r) = -2/[\alpha(r + C_2)]$ with $C_2 = 15r_g$, derived from the scalar field equation $\Phi' = \alpha\Phi\omega_1$. The plot shows the inverse relationship between the Weyl vector field strength and radial distance, with $\omega_1(r)$ approaching zero asymptotically far from the black hole. The field is plotted in natural units where the Weyl gauge coupling $\alpha$ sets the scale.
• Figure 3: Plot of the effective potential of photons. Here we have used the values $L=0.1$, $\delta=0.01$, $r_g=0.1$, and $C_3=-0.01$. The plot shows that the value of the photon-sphere radius is $r_{ps}\approx 0.1508$, where the effective potential is maximum and it is independent of the cosmological constant. In addition, we find $r_{\Lambda}\approx 13.7701$.
• Figure 4: The Lyapunov exponent $\lambda_0^2$ as a function of $C_3$, and $\delta$. Here, $r_g=1$.
• Figure 5: The Lyapunov exponent for massive particles $\lambda_m^2$ as a function of $C_3$, and $\delta$. Here, $m=1$, $r_g=1$, and $L=5$.