Spectral Analysis of Quasinormal Modes of Planck Stars

Abstract

We investigate the quasinormal modes (QNMs) of Planck stars within the framework of scale-dependent gravity (SDG). In our setup, the running parameter $α$ is fixed to a negative value by matching the effective Newtonian potential to the one-loop EFT result. As a consequence, the associated running Newton coupling does not realise the ultraviolet fixed point of asymptotically safe gravity, and the geometry should be interpreted as an SDG-inspired effective metric rather than a realisation of asymptotically safe gravity itself. We focus on the resulting renormalisation-group-improved Schwarzschild metric, which naturally yields a finite-size Planck-density core. Building on this background, we compute the QNM spectrum for scalar, electromagnetic, and gravitational perturbations using the Spectral Method (SM). This approach, known for its superior accuracy over high-order WKB schemes, enables the detection of fundamental modes, large families of overtones, and purely imaginary overdamped modes that are entirely missed in previous analysis. Our results reveal a robust Martini glass morphology of the oscillatory spectrum across perturbation sectors, nearly equally spaced overdamped modes with characteristic anomalous gaps, and the emergence, in the gravitational sector, of isolated overdamped modes separated from the main sequence by exceptionally large frequency intervals. These features, resolved here for the first time in the Planck-star context, underscore the importance of high-precision spectral techniques in probing subtle signatures of quantum-gravity-inspired black hole models.

Figures (3)

• Figure 1: Lapse function $F(r)$ for $\alpha > 0$: Horizons correspond to the zeros of $F(r)$. The curves are shown for a fixed $\alpha > 0$ and increasing values of the mass parameter $M$: $0 < M_{\text{red}} < M_{\text{green}} \equiv M_c < M_{\text{blue}}$. At the critical mass $M = M_c$ (green curve), the two horizons merge, corresponding to the extremal black hole configuration. For $M > M_c$ (blue curve), the lapse exhibits two distinct horizons $r_-$ and $r_+$, where $F(r_-) = F(r_+) = 0$.
• Figure 2: Lapse function $F(r)$ for $\alpha < 0$. In the physical region $r > 0$, the spacetime always contains a curvature singularity located at $r = r_0 > 0$ and a single event horizon at $r = r_h > r_0$, where $F(r_h) = 0$. The horizon thus always encloses the singularity, ensuring the absence of naked singularities.
• Figure 3: Horizon mass function $M(r_h)$ (red dashed line), singularity mass function $M(r_0)$ (blue dot–dashed line), and Schwarzschild mass function $M = r_{\mathrm{SCH}}/2$ (green solid line). The horizontal black dashed line represents an arbitrary positive mass $M > 0$. For any such $M$, the intersection with the blue curve occurs at $r_0(M)$ before that with the red curve at $r_h(M)$, ensuring $r_0(M) < r_h(M)$ and thus the absence of naked singularities.