Spectral Bifurcations in Quasinormal Modes of Regular BTZ Black Holes

Abstract

We study the quasinormal spectrum of massless scalar fields propagating on a family of regular BTZ black holes arising from an infinite tower of dimensionally regularized Lovelock corrections. These geometries are asymptotically AdS, reduce to the standard BTZ solution in the limit $\ell \to 0$, and resolve the central singularity by introducing a smooth core controlled by the new length scale $\ell$. The scalar quasinormal modes are computed using both Leaver's continued-fraction method and the Horowitz-Hubeny power-series method; the two approaches agree to high accuracy across the parameter space. We find that the regularization preserves linear stability ($ω_I < 0$) while qualitatively reshaping the spectrum: as $\ell$ increases, BTZ-like complex branches collide with the imaginary axis and undergo a hierarchy of bifurcations into multiple purely imaginary branches, leading to mode switching and a nontrivial reordering of overtones as functions of $\ell$ and the harmonic index $m$. Our results place regular BTZ black holes within the emerging family of bifurcating quasinormal spectra known from nearly extremal and asymptotically AdS black holes, and highlight these $(2+1)$-dimensional geometries as a controlled arena for exploring geometric mechanisms behind spectral branching and late-time ringdown in regular black hole spacetimes.

Figures (5)

• Figure 1: Effective potential for the massless scalar field for different values of the regularization parameter $\ell$, shown for the fundamental mode $m=0$. Parameters are fixed as $M = 1$, $L = 1$.
• Figure 2: QNM frequencies for the first three harmonics $m =0,1,2$ in the range $0 \leq \ell < 1$. The left panel shows the dependence of the real part $\omega_R$ on $\ell$, while the right panel shows the corresponding imaginary part $\omega_I$. For $m = 0$, the BTZ black hole has a vanishing real part $\omega_R = 0$, so the splitting of $\omega_I$ into two branches, the upper (more damped) and the lower (less damped), occurs already at $\ell = 0$. For $m = 1$, the BTZ value is $\omega_R = 1$; as $\ell$ increases, $\omega_R$ decreases and crosses zero at $\ell = \ell_{\text{crit}} = 0.433013$, where $\omega_I$ splits into two branches. For $m = 2$, the BTZ value is $\omega_R = 2$; $\omega_R$ decreases with $\ell$ and becomes zero at $\ell = \ell_{\text{crit}} = 0.654654$, and $\omega_I$ again splits into two branches. In this case $\omega_R$ also exhibits a small jump, leading to a short interval in which a nonzero $\omega_R$ branch coexists with the zero branch; when this jump merges back into the zero branch, the corresponding $\omega_I$ spectrum develops a third branch.
• Figure 3: Fundamental mode $(n=0)$ and the first two overtones $(n=1,2)$ of the scalar QNM frequencies as functions of the regularization parameter in the range $0 \leq \ell < 1$. The left panels display the dependence of the real part $\omega_R$ on $\ell$, while the right panels show the corresponding imaginary part $\omega_I$. For $m = 0$ all modes are purely imaginary and the regular BTZ geometry splits the BTZ value at $\ell = 0$ into two branches with different damping rates, the upper (more damped) and the lower (less damped). For $m = 1$ and $m = 2$, the BTZ-like complex branches move towards the imaginary axis as $\ell$ increases and, at critical values of $\ell$, bifurcate into multiple purely imaginary branches. In the $m = 2$ case, this bifurcation pattern is particularly pronounced and leads to a nontrivial reordering of the fundamental mode and its first overtones, as evidenced by the crossing of the corresponding curves.
• Figure 4: The trajectory plot of the fundamental scalar quasinormal frequencies in the $(\omega_R,-\omega_I)$ plane for $m = 1$ and $m = 2$ as the regularization scale $\ell$ is varied. Solid circles ($\bullet$) denote the $m = 1$ modes, while open circles ($\circ$) correspond to $m = 2$.
• Figure 5: Left: Near-horizon behavior of the effective potential for $M=L=1$ and $m=2$. The plot clearly shows that the concavity of the curve changes as $\ell$ is varied. Right: Contour plot of $V_{\text{eff}}"(r_\text{h})$ in the $(m , \ell)$–plane, showing the regions where the effective potential at the event horizon is concave up, concave down, or has a vanishing second derivative. The black line corresponds to the critical values $\ell_{\text{crit}}$ for the fundamental mode.