Quasi-normal modes and shadows of scale-dependent regular black holes

TL;DR

The paper investigates how a scale-dependent regular black hole with parameter $\epsilon$ behaves under scalar and Dirac perturbations and in shadow imaging. By combining a sixth-order WKB QNM analysis with ray-traced thin-disk shadows, it demonstrates that $\epsilon$ modifies the real part of QNM frequencies and the shadow structure, while preserving stability (negative $\mathrm{Im}\,\omega$). It also establishes and tests a QNM–shadows correspondence in the eikonal limit, enabling inference of photon-sphere properties from QNM data and vice versa. The results support the potential of multimessenger tests to constrain quantum-corrected BH metrics, with implications for future detectors such as LISA and ngEHT.

Abstract

In this paper we investigate how a regular scale-dependent black hole, characterized by a single extra parameter $ε$, behaves under perturbations by a test field (quasi-normal modes) and under light imaging (shadows) in a four-dimensional space-time background. On the quasi-normal modes side, we study how it responds to scalar and Dirac perturbations. To do this, we implement the well known WKB semi-analytic method of 6th order for obtaining the quasi-normal frequencies. We discuss the behavior of the real and imaginary parts of the quasi-normal modes for different values of the parameter $ε$ and the overtone $n$ and multipole $\ell$ numbers. On the black hole imaging side, we ray-trace the geometry and illuminate it with a thin-accretion disk. Choosing $ε=1.0$ we compute the size of the central brightness depression and generate full images of the black hole. We discuss the features (i.e. luminosity) of successive photon rings through the Lyapunov exponent of nearly-bound, unstable geodesics. Furthermore we use the correspondence (in the limit $\ell \gg n$) between quasi-normal mode frequencies and unstable bound light orbits to infer the numerical values of the latter using the former and find a remarkable accuracy of the correspondence in providing the right numbers. Our results support the usefulness of this correspondence in order to perform cross-tests of black holes using these two messengers.

Figures (7)

• Figure 1: Scalar and Dirac effective potentials for different values of the parameters $\{M, \epsilon, \xi \text{ (or } \ell)\}$. First Row: Scalar effective potential for a fixed mass $M=1$ varying the parameters $\ell$ and $\epsilon$. Second Row: Dirac effective potential for a fixed mass $M=1$ varying the parameters $\xi$ and $\epsilon$.
• Figure 2: QNMs for massless scalar perturbations. Left Panel: QNMs for $M=1$, varying $\epsilon$ from 0.0 to 1.0 for different values of the parameter $\ell$ for the fundamental mode $(n=0)$. Right Panel: QNMs for $M=1$, varying $\epsilon$ from 0.0 to 1.0 for different values of the parameter $\ell$ for the first excited mode $(n=1)$.
• Figure 3: Real/imaginary part of scalar QNM frequencies, $\omega_R$/$\omega_{I}$, against the scale-dependent parameter $\epsilon$. Top Left Panel:$\text{Re}(\omega)$ vs $\epsilon$ for $M=1$, varying $\ell$ from 6 to 9 for the fundamental mode $(n=0)$. Top Right Panel:$-\text{Im}(\omega)$ vs $\epsilon$ for $M=1$, varying $\ell$ from 6 to 9 for the fundamental mode $(n=0)$. Down Left Panel:$\text{Re}(\omega)$ vs $\epsilon$ for $M=1$, varying $\ell$ from 6 to 9 for the first exited mode $(n=1)$. Down Right Panel:$-\text{Im}(\omega)$ vs $\epsilon$ for $M=1$, varying $\ell$ from 6 to 9 for the first excited mode $(n=1)$.
• Figure 4: QNMs for massless Dirac perturbations. Left Panel: QNMs for $M=1$, varying $\epsilon$ from 0.0 to 1.0 for different values of the parameter $\xi$ for the fundamental mode $(n=0)$. Right Panel: QNMs for $M=1$, varying $\epsilon$ from 0.0 to 1.0 for different values of the parameter $\xi$ for the first exited mode $(n=1)$.
• Figure 5: Real/imaginary part of Dirac quasinormal frequencies, $\omega_R$/$\omega_{I}$, against the scale-dependent parameter $\epsilon$. Top Left Panel:$\text{Re}(\omega)$ vs $\epsilon$ for $M=1$, varying $\xi$ from 6 to 9 for the fundamental mode $(n=0)$. Top Right Panel:$-\text{Im}(\omega)$ vs $\epsilon$ for $M=1$, varying $\xi$ from 6 to 9 for the fundamental mode $(n=0)$. Down Left Panel:$\text{Re}(\omega)$ vs $\epsilon$ for $M=1$, varying $\xi$ from 6 to 9 for the first exited mode $(n=1)$. Down Right Panel:$-\text{Im}(\omega)$ vs $\epsilon$ for $M=1$, varying $\xi$ from 6 to 9 for the first exited mode $(n=1)$.