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For a regularized BTZ black hole, the scalar quasinormal spectrum remains stable but its complex branches collide with the imaginary axis as the core size grows, splitting into purely imaginary branches and reordering overtones.

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2026-08-02 18:08 UTC pith:KV4U6XAA

load-bearing objection First QNM computation for the regular BTZ background, with a plausible bifurcation story; the numbers are probably right, but the numerical evidence is not fully certified and the Horowitz–Hubeny section has an internal inconsistency. the 3 major comments →

arxiv 2603.14872 v2 pith:KV4U6XAA submitted 2026-03-16 gr-qc hep-th

Spectral Bifurcations in Quasinormal Modes of Regular BTZ Black Holes

Kartheek Hegde , Tajron Juri\'c , A. Naveena Kumara This is my paper
classification gr-qc hep-th
keywords Quasinormal modesRegular BTZ black holesSpectral bifurcationsLovelock gravityBlack hole perturbationsContinued fraction methodHorowitz-Hubeny methodMassless scalar field
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that the massless scalar quasinormal spectrum of the regular BTZ black hole is linearly stable for all core sizes below the AdS radius: every mode has negative imaginary frequency. More strikingly, as the regularization scale grows, the standard BTZ complex branches are driven toward the imaginary axis and bifurcate into purely imaginary, overdamped branches. For the m=1 harmonic there are two such branches after a critical core size; for m=2 a second collision produces three. The frequencies are computed with Leaver's continued-fraction and Horowitz-Hubeny power-series methods, which agree to three decimal places. A sympathetic reader would care because this places regular three-dimensional black holes in the same family of bifurcating quasinormal spectra known from nearly extremal Kerr and AdS black holes, and because it ties the bifurcation to a geometric, near-horizon feature of the potential.

Core claim

For the regular BTZ black hole from an infinite tower of dimensionally regularized Lovelock corrections with c_n=1, the massless scalar quasinormal spectrum is qualitatively reshaped while remaining linearly stable. At the critical values for m=1 and m=2, the complex BTZ frequency branches reach the imaginary axis and split into purely imaginary branches; for m=2 a second collision creates a third branch. The paper identifies the bifurcation locus of the fundamental mode with the vanishing second derivative of the effective potential at the horizon, giving the closed-form threshold. It also shows that the leading near-horizon equation and the Hawking temperature are unchanged from BTZ, so th

What carries the argument

The central object is the radial Klein-Gordon equation for a massless scalar on the regular BTZ metric, whose effective potential has six singular points and is not reducible to a hypergeometric or Heun equation. The computation rests on two numerical methods: Leaver's continued fraction method, using a Frobenius ansatz and an eight-term recurrence reduced by Gaussian elimination to a three-term recurrence, and the Horowitz-Hubeny power-series method. The proposed physical mechanism is the near-horizon concavity of the effective potential: the condition V''_eff(r_h)=0 singles out the critical core size for the fundamental mode, yielding the closed-form threshold in terms of m.

Load-bearing premise

The numerical roots are assumed to be true quasinormal modes: the Frobenius series is asserted to converge for all core sizes below the AdS radius because additional singular points lie outside the unit circle, but no convergence or residual check is shown, and the Horowitz-Hubeny recurrence treats rational coefficient functions as polynomial truncations; if those roots are numerical artifacts, the bifurcation picture collapses.

What would settle it

Recompute the modes with a pseudospectral or other discretization that does not rely on the rational-to-polynomial truncation, and check whether (i) both numerical methods still agree, (ii) the real part crosses zero at approximately 0.433 for m=1 and 0.655 for m=2, and (iii) the three m=2 branches persist at higher resolution. A discrepancy in any of these three checks would falsify the bifurcation claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Scalar perturbations of these regular BTZ black holes remain linearly stable for all core sizes below the AdS radius.
  • Regularization reduces the decay rate of the least damped mode, lengthening ringdown, while also creating overdamped branches.
  • The overtone ladder is reordered as the core size grows, so the mode that dominates late-time decay changes.
  • The bifurcation threshold saturates at large harmonic index, suggesting a high-frequency geometric mechanism independent of the core details.
  • The leading near-horizon conformal structure is BTZ-universal, so the global spectral change arises from subleading near-horizon corrections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Since the bifurcation threshold saturates as m grows, an analytic eikonal derivation of the threshold from the potential should be possible; the paper does not provide one.
  • The m=2 secondary bifurcation resembles exceptional-point collision dynamics; if the same mechanism holds for other fields, the pattern may be generic to regularized cores.
  • Because the Hawking temperature is unchanged even though the QNM pole structure shifts, holographic two-point functions might carry a core signature not visible in thermodynamics.
  • If such regular cores exist, ringdown templates that assume a single damped sinusoid may need to include purely imaginary branches above the critical core size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This paper studies massless scalar quasinormal modes of a family of regular BTZ black holes obtained from an infinite tower of dimensionally regularized Lovelock corrections, focusing on the representative case c_n=1. The authors formulate the radial perturbation problem, note the presence of multiple singularities, and compute QNM frequencies using Leaver's continued-fraction method and the Horowitz–Hubeny power-series method. They report agreement to three decimal places and describe a sequence of spectral bifurcations as the regularization scale ℓ increases: for m=0 the purely imaginary BTZ mode splits into two branches; for m=1 the complex BTZ-like branch reaches the imaginary axis at ℓ_crit≈0.433 and splits into two purely imaginary branches; for m=2 a second collision produces a third purely imaginary branch. The paper claims linear stability (ω_I<0) throughout and connects the branch collisions to a change in the concavity of the effective potential at the horizon, deriving ℓ_crit(m)=√3 m/(2√(m^2+3)).

Significance. If the numerical results are correct, the paper provides a new and controllable three-dimensional example of QNM spectral bifurcation in a regular black hole, complementing earlier examples in nearly extremal Kerr and Maxwell perturbations of AdS black holes. The authors include several positive elements: a benchmark of Leaver's method against the exact BTZ spectrum (Table II), two independent numerical schemes, and an analytic condition for the bifurcation threshold based on V''_eff(r_h)=0. However, the central claim rests on numerical eigenvalues whose certification is incomplete; the Horowitz–Hubeny implementation as written is internally inconsistent, and the Leaver convergence statement omits the singular point at spatial infinity. These issues must be resolved before the bifurcation pattern can be considered established.

major comments (3)
  1. [Sec. V, Eqs. (34)–(37), (40)] The text states that s(z), t(z), u(z) in Eq. (34) are 'third-order polynomials in z', but the displayed forms are rational functions with denominator [ℓ²(L²M z²−1)+L²] and its square. The Taylor expansions (35)–(37) truncated at third order therefore discard an infinite tail, and the recurrence (40) is not an exact Horowitz–Hubeny recurrence for the actual ODE (33). As a result, the HH results are not a valid independent verification of the Leaver spectrum. The authors should either compute the exact Taylor coefficients of the rational functions to all orders and use them in a consistent recurrence, or replace the implementation with a clearly convergent spectral method, and show that the results converge as the truncation order is increased.
  2. [Sec. IV, Eq. (22) and convergence discussion] The Leaver convergence argument states that all singular points of the transformed radial equation lie outside the unit circle in z=(r−r_h)/r. This is incomplete: spatial infinity r=∞ maps to z=1, which lies on the unit circle and is a singular point of the ODE. The prefactor (r_h/r)^{3/2} removes the leading branch point, but because the indicial exponents at z=1 differ by an integer, a logarithmic term may be present in the decaying solution, and convergence of Σ a_n z^n at z=1 is not guaranteed by the location of the other singularities. The paper provides no partial-sum convergence test near z=1, no residual check, and no N-truncation study for the continued-fraction roots. I request a concrete convergence study of the Frobenius series at z=1 for representative (m,ℓ) values, including the behavior of |a_n| and the residual of Eq. (19).
  3. [Sec. VI, text above Eq. (16) and ℓ_crit formula] The paper asserts that the condition V''_eff(r_h)=0 singles out the bifurcation point of the fundamental mode and leads to ℓ_crit(m)=√3 m/(2√(m^2+3)). This connection is stated without derivation and is load-bearing for the interpretation of the numerically observed branch collisions. Please either derive the relation from the perturbation equation (e.g., by analyzing the behavior of the QNM condition near ω_R=0), or explicitly compare the analytic ℓ_crit with the numerical value of ℓ at which ω_R crosses zero for each m and overtone in Table I. As written, the analytic formula appears to be transferred from Refs. [102,122] without a direct proof for the present potential.
minor comments (3)
  1. [Sec. VI, Fig. 2 and Table I (m=0)] For m=0 at ℓ=0, Table I lists a single purely imaginary value per overtone, while for ℓ>0 it lists two. The text says the BTZ value 'splits' as soon as ℓ is turned on. It would be helpful to clarify whether the two branches emanate from a degenerate point at ℓ=0 (and how the degeneracy is resolved for infinitesimal ℓ) in the figure and table.
  2. [General numerical reporting] No error estimates, convergence criteria, or code are provided. A statement of the Leaver truncation order N, the number of continued-fraction iterations, and the observed convergence as N grows (even for a few representative cases) would greatly increase confidence in the three-decimal agreement claimed in Table I.
  3. [Fig. 4 caption] The caption states the plot shows 'fundamental scalar quasinormal frequencies' for m=1 and m=2, while the main text (Sec. VI) says the fundamental and the first two overtones are plotted. Please make the caption consistent.

Circularity Check

0 steps flagged

No circular reduction: QNMs are not fitted inputs, the bifurcation condition is analytic, and self-citations are peripheral.

full rationale

The central QNM calculation is not circular. M, L, m, and ℓ are fixed physical inputs, and no eigenfrequency is used as a fitting parameter. The reported branch points ℓcrit(m)=√3m/(2√(m²+3)) are obtained analytically from the concavity condition V''_eff(r_h)=0 (Sec. VI), applied to the independently derived effective potential (16), and then compared with the numerically computed spectrum rather than imposed on it. The ℓ=0 limit is validated against the exact BTZ analytic spectrum in Table II, and the two numerical methods are independent implementations that share only the background and boundary conditions. The self-citations to the authors' earlier noncommutative BTZ papers (Refs. [71,72]) appear only in a broad survey list and are not load-bearing for any argument. The numerical convergence concerns in Secs. IV and V (e.g., the singular point z=1 on the unit circle, and the apparent inconsistency between the polynomial truncation and the rational forms in Eq. (34)) are correctness or robustness risks, not circular reductions: they do not make the output equal to an input by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central claim rests on the regular BTZ background of Ref. [48], standard AdS QNM boundary conditions, numerical convergence of the spectral methods, and the potential-concavity criterion for bifurcations. No new particles or fields are introduced, and there are no fitted parameters beyond the hand-chosen c_n = 1 tower.

free parameters (1)
  • Lovelock coefficients c_n = 1 for all n, with c_1 = 1
    The closed-form metric Eq. (6) and all numerical results use this hand-chosen tower; the paper does not explore other c_n, so the claimed 'family' behavior is demonstrated only for this choice.
axioms (5)
  • domain assumption The metric f(r) = r^2 f_BTZ / (r^2 - ℓ^2 f_BTZ) is a valid background solution of the infinite-tower Lovelock field equations.
    The entire QNM calculation uses this geometry, which is taken from Ref. [48] and not re-derived here.
  • domain assumption Standard AdS QNM boundary conditions apply: purely ingoing at the horizon and vanishing at spatial infinity.
    These boundary conditions are imposed in Secs. IV and V; the paper does not justify them beyond standard AdS/CFT practice.
  • domain assumption The Leaver Frobenius series (22) converges for all ℓ < 1 because the singular points r±_ℓ lie outside the unit circle.
    Asserted in Sec. IV with no convergence proof, truncation test, or Nollert-tail validation; the central numerical results depend on this.
  • domain assumption The V''_eff(r_h) = 0 condition identifies the fundamental-mode bifurcation point.
    Borrowed from Refs. [102,122] and used to produce the analytic ℓ_crit(m) formula, without a direct numerical verification that the complex branch collides at exactly that value.
  • ad hoc to paper The results for c_n = 1 are representative of the broader family of regular BTZ black holes.
    Only the simplest tower is solved; the paper itself lists varying c_n as future work, so this representativeness is assumed, not shown.

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read the original 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 ($\omega_I < 0$) 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

Figures reproduced from arXiv: 2603.14872 by A. Naveena Kumara, Kartheek Hegde, Tajron Juri\'c.

Figure 1
Figure 1. Figure 1: FIG. 1. Effective potential for the massless scalar field for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. QNM frequencies for the first three harmonics [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fundamental mode ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The trajectory plot of the fundamental scalar quasi [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Left: Near-horizon behavior of the effective potential for [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

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