Massive scalar quasinormal modes of an asymptotically flat regular black hole supported by a phantom Dirac--Born--Infeld field

Abstract: We study the quasinormal spectrum of a massive test scalar field in the exact asymptotically flat regular black-hole geometry supported by a phantom Dirac--Born--Infeld scalar. Using high-order WKB approximation improved by Padé resummation, together with characteristic time-domain integration and Prony extraction, we compute the fundamental mode and the first two overtones for representative values of the regularity parameter and the field mass. We show that increasing the field mass raises the oscillation frequency and reduces the damping rate, while increasing the regularity scale generally makes the ringing softer and longer lived. The time-domain profiles are in very good agreement with the WKB--Padé results and confirm the robustness of the spectrum. For sufficiently large field mass, the damping tends to zero, indicating the onset of quasiresonant behavior, although in the time domain these modes are eventually masked by oscillatory late-time tails. Our results show that massive scalar ringing provides a sensitive probe of this DBI-supported regular geometry.

• The paper computes massive scalar quasinormal modes using 16th-order Padé-improved WKB methods and time-domain evolution, with frequencies agreeing to better than 0.01% in representative tests.
• The spectrum shows that heavier fields oscillate faster and decay more slowly, while increasing the regularity scale lowers both frequency and damping, making the black hole’s internal structure spectrally measurable.
• The analysis indicates a possible quasiresonant threshold near μc≈0.844 for the fundamental ℓ=1 mode, but oscillatory power-law tails can mask ultra-long-lived ringing and require convergent methods for confirmation.

Background and motivation

This paper computes the quasinormal mode (QNM) spectrum of a massive test scalar field on the exact, asymptotically flat, nonsingular black-hole solution recently derived by Parvez and Shankaranarayanan from Einstein gravity coupled to a phantom Dirac–Born–Infeld (DBI) scalar (Parvez et al., 18 Nov 2025). The geometry is not a phenomenological ansatz: it follows from the DBI matter action, carries scalar hair, replaces the central singularity with a regular two-sphere of finite areal radius $R^2(r)=r^2+a^2$, and admits black-hole, extremal-remnant, and horizonless branches depending on parameters. Because the same metric can arise in effective-field-theory or asymptotically safe frameworks, its QNM spectrum is also relevant for distinguishing microscopic interpretations of one spacetime.

The motivation for considering a massive perturbing field is twofold. First, effective masses arise generically in brane-world, massive-gravity, and magnetized backgrounds, and they qualitatively alter both the spectrum and late-time relaxation, replacing power-law tails by oscillatory ones. Second, massive fields can exhibit quasiresonances—arbitrarily long-lived modes at critical field masses—which have been observed across many spins and geometries but are known not to be universal. Whether such long-lived modes develop in this DBI-supported geometry is therefore a nontrivial question that the paper addresses directly.

Geometry and wave equation

The background is determined by imposing the regular areal radius $R^2(r)=r^2+a^2$; one gravitational field equation becomes independent of the scalar profile, allowing the metric to be fixed by regularity and asymptotic flatness alone:

$$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$

with $f(r)=1-2M/r+\mathcal{O}(r^{-3})$ at large $r$. A test scalar of mass $\mu$ obeys $(\Box-\mu^2)\Phi=0$, which reduces in the tortoise coordinate to a Schrödinger-like problem with potential

$$V(r)=f(r)\left(\mu^2+\frac{\ell(\ell+1)}{R^2(r)}\right)+\frac{1}{R(r)}\frac{d^2 R}{dr_*^2}.$$

The potential vanishes at the horizon and approaches $\mu^2$ at infinity, so increasing $\mu$ raises the asymptotic plateau and flattens the barrier—a structural feature that underlies all spectral trends reported below.

Methods

Two complementary techniques are used. On the frequency side, the WKB expansion around the peak of the potential is carried to 16th order with Padé approximants ($R^2(r)=r^2+a^2$0), cross-checked against 14th order ($R^2(r)=r^2+a^2$1). On the time side, the wave equation is evolved via the Gundlach–Price–Pullin characteristic scheme, and the dominant ringing is extracted with the Prony method applied to the intermediate-time window between prompt response and late-time tail. Representative comparisons show excellent agreement: for $R^2(r)=r^2+a^2$2, $R^2(r)=r^2+a^2$3, $R^2(r)=r^2+a^2$4, Prony gives $R^2(r)=r^2+a^2$5 versus WKB $R^2(r)=r^2+a^2$6 (agreement within 0.13% and 0.23%); for $R^2(r)=r^2+a^2$7, $R^2(r)=r^2+a^2$8, $R^2(r)=r^2+a^2$9, agreement reaches 0.0069% and 0.00012%.

Spectral results

The central quantitative finding concerns the hierarchy between numerical uncertainty and physical parameter dependence. For the fundamental $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$0 mode at $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$1, varying the regularity scale $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$2 from 0.2 to 1 shifts $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$3 by about 1.7% and $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$4 by about 5.2%, while the internal WKB–Padé spread stays below $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$5. For $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$6 the WKB consistency check typically lies in the range $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$7%–$$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$8%, so the dependence on $$f(r)=1+\frac{3M}{a}\left(\frac{r}{a}-\frac{a^2+r^2}{a^2}\arctan\frac{a}{r}\right), \qquad R^2(r)=a^2+r^2,$$9 is a genuine physical effect rather than a numerical artifact. The notable exception is the $f(r)=1-2M/r+\mathcal{O}(r^{-3})$0 sector near the largest masses considered, where the WKB spread reaches several percent (up to 8.33% at $f(r)=1-2M/r+\mathcal{O}(r^{-3})$1, $f(r)=1-2M/r+\mathcal{O}(r^{-3})$2); those entries are flagged as requiring caution.

The qualitative trends are consistent across multipoles and overtones:

• Field mass: increasing $f(r)=1-2M/r+\mathcal{O}(r^{-3})$3 raises $f(r)=1-2M/r+\mathcal{O}(r^{-3})$4 and lowers $f(r)=1-2M/r+\mathcal{O}(r^{-3})$5—heavier fields ring faster but decay more slowly.
• Regularity scale: increasing $f(r)=1-2M/r+\mathcal{O}(r^{-3})$6 generally lowers both frequency and damping, making the ringing softer and longer lived.
• Multipole number: $f(r)=1-2M/r+\mathcal{O}(r^{-3})$7 raises the oscillation frequency through the centrifugal term.
• Overtone number: $f(r)=1-2M/r+\mathcal{O}(r^{-3})$8 primarily controls damping; first and second overtones decay much faster than the fundamental even when their real parts are comparable.

Quasiresonances and their time-domain masking

The monotonic decrease of the damping rate with $f(r)=1-2M/r+\mathcal{O}(r^{-3})$9 indicates an approach to quasiresonance. For the fundamental $r$0 mode at $r$1, a quadratic extrapolation of the WKB data crosses zero damping at $r$2. The author is explicit that this is an extrapolated estimate, not an exact threshold; moreover, accessing the true quasiresonant regime would require a convergent method such as Leaver's, which demands casting the metric into rational parametrized form—not done here.

A further structural limitation is identified and analyzed rather than glossed over: because quasinormal modes do not form a complete set for massive fields, and the asymptotic signal is governed by oscillatory tails with power-law envelope, the waveform is overtaken by the tail before an ultra-long-lived QNM can dominate as a clean exponentially damped stage. Time-domain profiles at large $r$3 confirm this: the intermediate and asymptotic tails take over at relatively early times. Thus the absence of a visibly extended ultra-long-lived ringdown does not contradict the frequency-domain indication of quasiresonance—an important caveat for any observational interpretation.

Grey-body factors

As a byproduct, the paper notes that the computed fundamental mode and first overtone suffice to estimate grey-body factors through the established correspondence between QNMs and transmission probabilities, valid provided the effective potential has a single-barrier shape. This correspondence may break down for double-well potentials or certain higher-curvature corrections that induce eikonal instabilities; neither pathology occurs here, so the estimate is applicable, though no explicit grey-body numbers are tabulated.

Limitations and open questions

Several limitations are stated plainly in the paper. The quasiresonant threshold $r$4 rests on quadratic extrapolation of WKB data rather than a convergent eigenvalue computation. The $r$5 sector at large $r$6 has WKB uncertainties comparable to physical effects, limiting conclusions there. Only the black-hole branch of the phantom DBI solution is treated; the extremal-remnant and horizonless configurations remain unexamined, as do electromagnetic, Dirac, and gravitational perturbations. The interplay between long-lived modes and oscillatory tails in the quasiresonant regime—including dedicated scattering and tail analyses—is left open.

Conclusion

The paper establishes that massive scalar ringing is a sensitive probe of the regularity scale in the phantom DBI-supported regular black hole: the dependence of the spectrum on $r$7 exceeds numerical uncertainty by orders of magnitude for $r$8, heavier fields approach a quasiresonant regime with vanishing damping, and time-domain evolution independently validates the dominant frequencies while explaining why ultra-long-lived modes are masked by power-law-enveloped oscillatory tails. The work extends the QNM program to a new exactly solvable regular geometry and identifies the remaining sectors of the solution whose spectra would complete the perturbative picture.