Scalar, electromagnetic, and Dirac perturbations of a regular black hole supported by primordial dark matter

Abstract: We study massless scalar, electromagnetic, and Dirac perturbations of the exact asymptotically flat regular black hole supported by a phantom Dirac--Born--Infeld scalar. Using the Padé-improved WKB method, with a time-domain Prony check for the scalar fundamental mode, we compute representative quasinormal frequencies and find that larger regularity shifts the spectrum toward smaller oscillation frequencies and damping rates, whereas the quality factor changes only weakly. The spectral shifts remain well above the estimated numerical uncertainty, demonstrating that the DBI regularity scale leaves a robust spin-dependent imprint on ringdown.

• The paper analyzes massless scalar, electromagnetic, and Dirac perturbations of a DBI-regularized black hole, finding that the DBI regularity scale shifts quasinormal mode frequencies by about 10% for increasing $a$ values, with a monotonic decrease in both real and imaginary parts.
• The study employs the sixth-order Padé improved WKB method, cross-verified by a time-domain method, which confirms the shift in oscillation and damping frequencies is a significant physical effect rather than a numerical artifact.
• The effective barrier potentials qualitatively systematically affected by increasing the regularity scale $a$: the magnitude and position of the scalar, electromagnetic, and Dirac barriers are modified, providing a robust observational signature of primordial dark matter in black hole systems.

Background and motivation

This paper by Lütfüoğlu analyzes massless scalar, electromagnetic, and Dirac test-field perturbations of the exact asymptotically flat regular black-hole solution recently constructed by Parvez and Shankaranarayanan from a phantom Dirac–Born–Infeld (DBI) scalar (Parvez et al., 18 Nov 2025). The motivation is twofold. First, the background is not an ad hoc deformation of Schwarzschild: it follows from a definite matter action with scalar hair, and it interpolates between black-hole, extremal-remnant, and horizonless configurations depending on parameters. Second, comparing several spin sectors within one exact geometry allows one to separate universal spectral features from field-dependent ones, in line with the observation that strong-field observables often depend on only a few effective metric parameters (Konoplya et al., 2020).

The geometry is static and spherically symmetric, with $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right)$$ and areal radius $R^2(r)=r^2+a^2$, where $M$ is the ADM mass and $a$ sets the regularity scale. The areal radius never drops below $a$, so the center is replaced by a minimal two-sphere rather than a curvature singularity; at large distances $f(r)=1-2M/r+O(r^{-3})$. The limit $a\to 0$ recovers Schwarzschild.

Master equations and effective potentials

All three sectors reduce to a Schrödinger-like master equation $\frac{d^2\Psi}{dr_*^2}+(\omega^2-V)\Psi=0$ in the tortoise coordinate, with potentials vanishing at both asymptotic ends — hence standard quasinormal boundary conditions of purely ingoing waves at the horizon and outgoing waves at infinity.

• Scalar: $$V_s=f\left[\ell(\ell+1)/(r^2+a^2)+rf'/(r^2+a^2)+a^2f/(r^2+a^2)^2\right]$$, containing both centrifugal and geometric terms.
• Electromagnetic: $V_{em}=f\,\ell(\ell+1)/(r^2+a^2)$, purely centrifugal; both parities share this potential.
• Dirac: obtained via supersymmetric factorization with superpotential $R^2(r)=r^2+a^2$0, giving isospectral partner potentials $R^2(r)=r^2+a^2$1.

Qualitatively, for the same low multipole the scalar barrier is highest, the electromagnetic barrier is lower, and the Dirac barrier peaks closer to the horizon. Increasing $R^2(r)=r^2+a^2$2 lowers and broadens all barriers mildly but systematically — precisely the deformation expected to shift quasinormal frequencies away from the Schwarzschild values.

Methods

The spectrum is computed primarily with the Padé-improved WKB method at 16th order (approximant $R^2(r)=r^2+a^2$3), cross-checked against 14th order ($R^2(r)=r^2+a^2$4) as an internal accuracy estimate (Matyjasek et al., 2017). The single-barrier structure of all potentials makes high-order WKB applicable whenever $R^2(r)=r^2+a^2$5. Two time-domain verifications using the Gundlach–Price–Pullin characteristic scheme [gr-qc/9307009] with Prony fits confirm the dominant modes: for the scalar $R^2(r)=r^2+a^2$6, $R^2(r)=r^2+a^2$7 mode the Prony fit agrees with WKB to better than $R^2(r)=r^2+a^2$8, and for the electromagnetic $R^2(r)=r^2+a^2$9, $M$0 mode to better than $M$1.

Quasinormal spectra

The fundamental ($M$2) frequencies were tabulated for $M$3 (scalar), $M$4 (electromagnetic), and $M$5 (Dirac), plus electromagnetic overtones up to $M$6 at $M$7. Representative values:

Sector	$M$8	$M$9
Scalar, $a$0	$a$1	$a$2
Electromagnetic, $a$3	$a$4	$a$5
Dirac, $a$6	$a$7	$a$8

The central result is that increasing $a$9 shifts every mode monotonically toward the origin of the complex-frequency plane: both $a$0 and $a$1 decrease. The effect is not marginal — roughly 11% in $a$2 and 10% in $a$3 for the representative scalar mode over $a$4, with very similar magnitudes (~10.6–11.3%) in the electromagnetic and Dirac sectors. The quality factor $a$5 changes only weakly: it decreases slightly in the scalar sector (from ~1.499 to ~1.482) while increasing mildly in the electromagnetic (~1.342→1.349) and Dirac (~0.939→0.945) sectors. Thus the DBI regularity scale alters the damping/oscillation balance only marginally while producing robust absolute spectral shifts.

The significance assessment rests on the comparison between physical shift and numerical uncertainty. For fundamental modes with $a$6 or $a$7, the WKB16/WKB14 discrepancy is typically zero in quoted digits or at most $a$8–$a$9; even higher electromagnetic overtones stay below $f(r)=1-2M/r+O(r^{-3})$0. The least favorable cases are the Dirac $f(r)=1-2M/r+O(r^{-3})$1 mode (~0.13% internal spread) and the scalar monopole (~0.25%), yet even there the cumulative shift across the table (~11% and ~5.2%/3.5% respectively) exceeds the numerical error by at least one order of magnitude. The paper therefore concludes that the dependence on $f(r)=1-2M/r+O(r^{-3})$2 is a genuine physical effect rather than a numerical artifact.

As a secondary application, the paper notes that for single-barrier potentials the dominant quasinormal frequencies can be used to reconstruct grey-body factors semi-analytically via the correspondence developed in (Konoplya et al., 2024, Konoplya et al., 2024), without solving a separate scattering problem.

Limitations and open questions

Several caveats are stated explicitly. For scalar perturbations with $f(r)=1-2M/r+O(r^{-3})$3 and $f(r)=1-2M/r+O(r^{-3})$4 at $f(r)=1-2M/r+O(r^{-3})$5, the effective potential has no barrier maximum, so the WKB method is inapplicable there; the corresponding spectra remain uncomputed within this framework. The grey-body-factor correspondence inherits the single-barrier assumption and would deteriorate if the potential developed a double-well profile or if higher-curvature corrections triggered eikonal-type instabilities. All results pertain to test fields only: gravitational perturbations — essential for direct observational contact with ringdown data — are not included. The analysis also covers only the black-hole branch of the DBI solution, leaving the extremal-remnant and horizonless configurations unexplored. Finally, the time-domain verification covers only two representative bosonic modes; a systematic time-domain study of higher overtones and late-time tails is left open.

Conclusion

The paper establishes that the DBI regularity scale leaves a consistent, spin-dependent imprint on the quasinormal spectrum of its regular black hole: larger $f(r)=1-2M/r+O(r^{-3})$6 lowers and broadens all effective barriers and shifts the fundamental modes toward smaller oscillation frequencies and damping rates, by roughly 10% over the studied range, while the quality factor changes only weakly. Because these shifts exceed the estimated numerical uncertainty by one to several orders of magnitude — and are confirmed by time-domain Prony fits at the $f(r)=1-2M/r+O(r^{-3})$7 level — they constitute a robust spectroscopic signature of the DBI regularity scale. Extending the analysis to gravitational perturbations, to the non-black-hole branches of the solution, and to systematic overtone and tail studies remains the natural continuation of this work.