Quasinormal modes and excitation factors of a regular black hole with zero-point length

Abstract: We study the ringdown of the regular Jusufi-Singleton black hole, whose nonsingular core is controlled by a zero-point length arising from a non-local, T-duality-inspired gravitational model. Scalar, electromagnetic and Dirac perturbations are considered. The zero-point-length parameter raises the effective scattering barrier and produces a systematic increase of the oscillation frequencies, while also making the damping faster over most of the parameter range. High-order WKB results are checked against time-domain integration and show very good agreement for the dominant modes. We also compute excitation factors, which characterize the source-independent strength of the quasinormal-mode poles and show a smooth dependence on the new length scale.

• The paper computes quasinormal mode (QNM) spectra for a regular black hole geometry, yielding frequency and damping highly dependent on a zero-point length ($l_0$).
• It shows that increasing $l_0$ causes both the real and imaginary parts of QNM frequencies, for scalar, electromagnetic, and Dirac perturbations, to monotonically increase or saturate, respectively. For example, the scalar monopole fundamental increases from $0.016093-0.015245i$ at $l_0=0.05$ to $0.088221-0.069215i$ at $l_0=0.65$.
• The results are consistent across numerical methods, including high-order WKB and time-domain computations. Excitation factors show instrumental stability, resolving complementarty grey-body and Hawking enterprises for black hole perturbation.

Background and motivation

This paper by Skvortsova (2606.15785) computes the quasinormal mode (QNM) spectra and excitation factors of the neutral, nonsingular black hole proposed by Jusufi and Singleton (Jusufi et al., 10 Sep 2025). The distinguishing feature of this geometry is that the regularizing scale is not an electric charge or a quantum-gravity cutoff in the usual sense, but a zero-point length $l_0$ arising from a non-local, T-duality-inspired description of the source. In that construction, the point source is replaced by an extended distribution whose finite gravitational self-energy is folded back into the effective mass, producing an Ayón-Beato–García-type metric with ADM mass $M_{\rm ADM}=1+3\pi/(32l_0)$ (in units where $M=1$). While grey-body factors and Hawking radiation for this spacetime had been studied previously, no QNM analysis existed; the present work fills that gap.

The motivation is twofold. First, ringdown frequencies are fixed by the geometry rather than by the initial perturbation, so they provide a clean diagnostic of how a short-distance non-local scale deforms the exterior potential relative to Schwarzschild. Second, excitation factors — the residues of the frequency-domain Green function at the QNM poles — carry complementary, source-independent information about the intrinsic strength of each pole, which has rarely been computed for regular black holes.

Geometry and perturbation equations

The metric is static and spherically symmetric, $ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$, with a metric function containing both the $(l_0^2+r^2)$-regularized mass distribution and terms proportional to $\tan^{-1}(r/l_0)$ coming from the regularized self-energy. At large $r$ the geometry approaches Schwarzschild with subleading corrections including a $1/r^2$ term $(1+3l_0^2/8)/r^2$; near the center the even dependence on $r$ renders the core smooth.

Scalar ($M_{\rm ADM}=1+3\pi/(32l_0)$0), electromagnetic ($M_{\rm ADM}=1+3\pi/(32l_0)$1), and massless Dirac perturbations are treated as test fields on this fixed background, neglecting backreaction. Each reduces to a Schrödinger-type wave equation on the tortoise coordinate, with effective potentials of the standard Regge–Wheeler form for bosonic spins and supersymmetric partner potentials $M_{\rm ADM}=1+3\pi/(32l_0)$2 for Dirac fields. The two Dirac potentials are related by a Darboux transformation and are isospectral, so only $M_{\rm ADM}=1+3\pi/(32l_0)$3 is used numerically. For all three sectors the potential is a single positive barrier vanishing at the horizon and at infinity, which justifies WKB treatment: increasing $M_{\rm ADM}=1+3\pi/(32l_0)$4 raises the barrier height and modifies its curvature near the maximum, a fact that directly explains the spectral trends below.

Numerical methods

Two complementary methods are employed. The primary results come from high-order WKB expansion (up to 16th order) with Padé resummation, using symmetric approximants $M_{\rm ADM}=1+3\pi/(32l_0)$5 at orders 14 and 16; the difference between these two estimates serves as an internal accuracy check. As an independent validation, the wave equation is evolved on a characteristic null grid using the Gundlach–Price–Pullin scheme, and complex frequencies are extracted from the ringing stage via Prony fitting, with stability checked against variations of the fitting window and exponent number.

Quasinormal spectra

The central result is a systematic, monotonic dependence of the spectrum on the zero-point length across all three spin sectors:

• Real part: $M_{\rm ADM}=1+3\pi/(32l_0)$6 increases monotonically with $M_{\rm ADM}=1+3\pi/(32l_0)$7, tracking the growth of the effective barrier. For example, the scalar monopole fundamental rises from $M_{\rm ADM}=1+3\pi/(32l_0)$8 at $M_{\rm ADM}=1+3\pi/(32l_0)$9 to $M=1$0 at $M=1$1.
• Imaginary part: damping is faster over most of the parameter range, with $M=1$2 saturating around $M=1$3–$M=1$4 at large $M=1$5. All modes remain stable ($M=1$6).
• Overtones: electromagnetic $M=1$7 overtones ($M=1$8) follow the same qualitative trend with substantially larger damping rates, as expected.

A representative sample illustrates the scale of the effect:

Mode	$M=1$9	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$0
Scalar $ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$1	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$2	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$3
Electromagnetic $ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$4	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$5	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$6
Dirac $ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$7	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$8	$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$9

The internal consistency between WKB16 and WKB14 is strong: for multipoles $(l_0^2+r^2)$0 the relative differences are zero at displayed precision, while the worst cases — the scalar monopole and Dirac $(l_0^2+r^2)$1 fundamental — stay at or below roughly $(l_0^2+r^2)$2. Time-domain checks at $(l_0^2+r^2)$3 confirm this independently: Prony extraction agrees with WKB16 to $(l_0^2+r^2)$4 for the scalar monopole, $(l_0^2+r^2)$5 for the lowest Dirac mode, and to parts in $(l_0^2+r^2)$6–$(l_0^2+r^2)$7 percent for higher multipoles. Because the tabulated mass parameter is not the ADM mass, the paper also presents ADM-rescaled frequencies via $(l_0^2+r^2)$8; in the small-$(l_0^2+r^2)$9 limit these approach the expected Schwarzschild values (e.g., $\tan^{-1}(r/l_0)$0 for the scalar monopole at $\tan^{-1}(r/l_0)$1), providing a normalization check.

The implication is that the zero-point length leaves a clean, monotonic spectroscopic signature without introducing new branches or instabilities: it shifts the usual black-hole ringing frequencies upward and shortens damping times in a controlled way.

Excitation factors

Excitation factors are computed as Green-function residues,

$\tan^{-1}(r/l_0)$2

with the horizon-regular solution matched at finite radius $\tan^{-1}(r/l_0)$3 to asymptotic series truncated at $\tan^{-1}(r/l_0)$4, and derivatives obtained by finite differencing. The phase convention is fixed by $\tan^{-1}(r/l_0)$5 at infinity. The results vary smoothly with $\tan^{-1}(r/l_0)$6: the scalar monopole residue grows in magnitude (from $\tan^{-1}(r/l_0)$7 to $\tan^{-1}(r/l_0)$8), while the electromagnetic and Dirac residues mainly rotate in the complex plane at comparable magnitude. The paper explicitly cautions that negative real or imaginary parts of $\tan^{-1}(r/l_0)$9 reflect only the tortoise-coordinate phase convention and must not be read as instability indicators; rephasing the basis rotates the residue by an overall phase. It also stresses that $r$0 characterizes intrinsic pole strength, not waveform amplitude, which additionally depends on the source integral.

Limitations and open questions

Several caveats are stated plainly in the paper. All perturbations are treated as test fields with neglected backreaction, and gravitational perturbations are not considered, so the astrophysically dominant axial/polar gravitational ringdown remains uncomputed for this background. The WKB method is least reliable for the lowest barriers — the scalar monopole and Dirac $r$1 modes — where uncertainties reach a few tenths of a percent, and the time-domain validation covers only the largest $r$2 value studied. The excitation-factor calculation depends on the additive convention for $r$3 and on matching radius and series truncation, with the scalar monopole again the most sensitive case. Finally, the analysis is restricted to $r$4; behavior closer to any extremal configuration is unexplored. A concrete open question raised by the author is whether the recently established correspondence between QNMs and grey-body factors can be quantitatively verified for this background, and what excitation coefficients arise for explicit sources or initial data.

Conclusion

The paper establishes that the zero-point-length parameter of the Jusufi–Singleton regular black hole produces a robust and systematic imprint on its ringdown: oscillation frequencies increase and damping generally accelerates as $r$5 grows, uniformly across scalar, electromagnetic, and Dirac sectors, with no qualitative restructuring of the spectrum. High-order WKB–Padé results are validated against time-domain integration at the sub-percent level even for the most difficult low-multipole cases, and the newly computed excitation factors vary smoothly with the non-local scale while correctly reducing to the Schwarzschild limit after ADM rescaling. The work provides the spectroscopic baseline needed to connect this T-duality-inspired regularization to observable ringdown phenomenology, pending extension to gravitational perturbations and sourced waveforms.