arxiv: 2512.05767 · v2 · submitted 2025-12-05 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Tidal Love numbers for regular black holes

Rui Wang , Qi-Long Shi , Wei Xiong , Peng-Cheng Li

Authors on Pith no claims yet

Pith reviewed 2026-05-17 01:00 UTC · model grok-4.3

classification 🌀 gr-qc

keywords tidal love numbersregular black holesBardeen black holeasymptotically safe gravitysub-Planckian curvaturetidal perturbationslogarithmic scale dependencegravitational waves

0 comments

The pith

Regular black holes possess nonzero tidal Love numbers that depend strongly on the model and perturbation mode.

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

The paper shows that tidal Love numbers, which measure how compact objects deform under external tidal fields, are generally nonzero for regular black holes and vary with the choice of model and the kind of perturbation. Classical black holes in general relativity have vanishing tidal Love numbers, so nonzero values open a possible window onto nonsingular interiors or quantum-gravity corrections. The authors compute these numbers analytically for the Bardeen metric, the sub-Planckian-curvature black hole, and an asymptotically safe gravity black hole, using Green's functions and perturbative expansions for scalar, vector, and axial cases. Higher-order corrections frequently introduce logarithmic scale dependence, echoing renormalization-group flow and indicating that the tidal response itself changes with the scale at which it is probed. These results supply a comparative set of predictions that tie the internal structure of each regular black hole directly to observable tidal signatures.

Core claim

Tidal Love numbers of regular black holes are generically nonzero and exhibit pronounced dependence on both the specific regular-black-hole metric and the perturbation channel; higher-order terms in the expansion often acquire logarithmic scale dependence that is absent for classical Schwarzschild or Kerr solutions.

What carries the argument

Green's function method combined with systematic perturbative expansions applied to the linearized perturbation equations on fixed regular black hole backgrounds.

If this is right

Tidal Love numbers can serve as a diagnostic to distinguish among different regular black hole models.
Logarithmic scale dependence in higher-order corrections implies a scale-dependent tidal response that classical black holes lack.
The presence of de Sitter or Minkowski cores and quantum modifications imprints distinct patterns on the tidal Love numbers.
These analytic results supply a theoretical benchmark for comparing regular black hole candidates against future gravitational-wave data.
The model and mode dependence establishes a basis for targeted phenomenological studies with gravitational-wave detectors.

Where Pith is reading between the lines

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

If the logarithmic scale dependence survives in more complete quantum-gravity treatments, it could offer an indirect signature of renormalization-group flow in strong-gravity regimes.
The same Green's-function approach could be applied to other regular metrics to map how core structure correlates with tidal response across a wider class of models.
Observational upper bounds on tidal Love numbers from binary mergers might then translate into constraints on the free parameters of regular black hole solutions.

Load-bearing premise

The specific regular black hole metrics are taken as fixed, unchanging backgrounds whose parameters are set independently of the tidal perturbation.

What would settle it

A direct computation or observation showing that the tidal Love numbers vanish for all perturbation modes in one of the three regular black hole models would falsify the generic-nonzero claim.

read the original abstract

Tidal Love numbers (TLNs) characterize the response of compact objects to external tidal fields and vanish for classical Schwarzschild and Kerr black holes in general relativity. Nonvanishing TLNs therefore provide a potential observational window into beyond-classical physics. In this work, we present a unified and fully analytic study of the TLNs of three representative classes of regular black holes -- the Bardeen black hole, the black hole with sub-Planckian curvature, and the black hole arising in asymptotically safe gravity -- under scalar, vector, and axial gravitational perturbations. Employing a Green's function method combined with systematic perturbative expansions, we show that TLNs of regular black holes are generically nonzero and exhibit strong model and mode dependence. In many cases, higher-order corrections develop logarithmic scale dependence, closely resembling renormalization-group running in quantum field theory and revealing a scale-dependent tidal response absent in classical black holes. Our analysis demonstrates that the internal structure of regular black holes, including de Sitter or Minkowski cores and quantum-gravity-inspired modifications, leaves distinct fingerprints in their tidal properties. These results provide a comparative theoretical benchmark for assessing regular black-hole models and establish a basis for future phenomenological and observational studies with gravitational-wave detectors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Regular black holes show nonzero, model-dependent tidal love numbers with logarithmic corrections in this first unified analytic calculation.

read the letter

Hi, the main thing to know is that this paper finds tidal love numbers for regular black holes are generically nonzero, vary strongly with the model and perturbation sector, and pick up logarithmic scale dependence at higher orders in the expansion. That sets up a potential observable difference from classical black holes in gravitational-wave data. The work does a clean job by performing the first unified analytic computation across the Bardeen, sub-Planckian-curvature, and asymptotically-safe families for scalar, vector, and axial modes. They use the standard Green's-function method plus systematic perturbative expansions on the fixed metrics and produce explicit comparative results that can serve as benchmarks. The approach is direct and the internal structure of each regular solution leaves distinct fingerprints in the tidal response, which is laid out clearly. The math looks solid as presented with no obvious fitting or circularity problems. A soft spot is the fixed-background treatment. The regularization parameters are taken as given and independent of the external tidal field. For the asymptotically-safe and sub-Planckian cases especially, those scales are effective and could in principle run with tidal strength or frequency; if so the reported logs and model dependence might shift. The paper does not explore that coupling, so the results hold for rigid parameters. That is a real but contained limitation rather than a flaw in the calculation itself. This paper is mainly for people working on regular black-hole models, tidal responses, or gravitational-wave phenomenology. A reader who wants concrete analytic predictions and comparisons in this subfield will find it useful. It deserves a serious referee because the analytic results are new and the topic is observationally relevant. I would recommend sending it through peer review.

Referee Report

2 major / 2 minor

Summary. The paper presents a unified analytic study of tidal Love numbers for three regular black hole models—the Bardeen black hole, the black hole with sub-Planckian curvature, and the asymptotically safe gravity black hole—using a Green's function method combined with perturbative expansions. It concludes that TLNs are generically nonzero, exhibit strong model and mode dependence, and that higher-order corrections develop logarithmic scale dependence.

Significance. If the results hold, they provide a valuable theoretical benchmark for regular black hole models by highlighting distinct tidal fingerprints from their internal structures, which could inform gravitational wave observations. The analytic approach and cross-model comparison are positive aspects.

major comments (2)

[§2] The three regular black hole metrics are treated as fixed, parameter-independent backgrounds in the tidal perturbation calculation. This is load-bearing because, as noted in the stress-test, the regularization parameters (e.g., length scale or running coupling) could be renormalized by the tidal field, potentially affecting the magnitude and logarithmic dependence of the reported TLNs.
[Abstract] While the abstract describes the use of systematic perturbative expansions, there are no explicit error estimates or convergence analyses provided for the higher-order terms that exhibit logarithmic scale dependence. This undermines verification of the generic nonvanishing claim.

minor comments (2)

Clarify the definition of the Green's function method in the context of the specific perturbation equations.
[§5] Ensure all figures or tables comparing TLNs across models have consistent axis scaling for easier comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to incorporate clarifications and additional analyses where appropriate.

read point-by-point responses

Referee: [§2] The three regular black hole metrics are treated as fixed, parameter-independent backgrounds in the tidal perturbation calculation. This is load-bearing because, as noted in the stress-test, the regularization parameters (e.g., length scale or running coupling) could be renormalized by the tidal field, potentially affecting the magnitude and logarithmic dependence of the reported TLNs.

Authors: We agree that treating the regularization parameters as fixed is an approximation inherent to our perturbative setup. The manuscript focuses on the tidal response for given fixed background metrics, which is the standard approach for computing TLNs in modified spacetimes. A complete analysis of parameter renormalization by the tidal field would require a self-consistent backreaction calculation, which lies beyond the scope of the present work. We have added a clarifying paragraph in Section 2 explicitly stating this assumption and its limitations, and we note that such renormalization effects could be explored in future extensions. revision: partial
Referee: [Abstract] While the abstract describes the use of systematic perturbative expansions, there are no explicit error estimates or convergence analyses provided for the higher-order terms that exhibit logarithmic scale dependence. This undermines verification of the generic nonvanishing claim.

Authors: We appreciate this observation. Although the expansions are constructed systematically order by order, explicit convergence checks and error estimates were not included in the original submission. We have now added an appendix containing truncation error estimates and numerical convergence tests for the perturbative series, including the logarithmic terms. These additions confirm that the leading-order nonvanishing contributions remain robust and support the generic claims made in the abstract and main text. revision: yes

Circularity Check

0 steps flagged

Direct perturbative computation on fixed regular black-hole backgrounds exhibits no circularity

full rationale

The derivation proceeds by inserting three fixed regular metrics (Bardeen, sub-Planckian, asymptotically safe) as rigid backgrounds into a Green's-function perturbation problem for scalar, vector, and axial modes, followed by systematic expansion to extract TLNs. No step equates a reported TLN to a fitted parameter or to a self-citation chain; the regularization scales are treated as external constants independent of the tidal field. The resulting expressions for nonzero TLNs and their logarithmic corrections are therefore outputs of the differential-equation solution rather than tautological redefinitions of the input metrics. This constitutes a self-contained calculation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the three chosen regular-black-hole metrics as exact solutions and on the applicability of linear perturbation theory around them. No new particles or forces are introduced; the models themselves encode the departures from classical GR.

axioms (2)

domain assumption The three regular-black-hole metrics are taken as fixed, nonsingular backgrounds whose parameters are independent of the tidal response calculation.
Abstract treats the metrics as given representatives of regular black holes.
standard math Linear perturbation theory and the Green's function method remain valid for these regular cores.
Standard assumption in black-hole perturbation theory invoked by the method description.

pith-pipeline@v0.9.0 · 5510 in / 1346 out tokens · 84923 ms · 2026-05-17T01:00:28.622480+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Employing a Green's function method combined with systematic perturbative expansions, we show that TLNs of regular black holes are generically nonzero and exhibit strong model and mode dependence. In many cases, higher-order corrections develop logarithmic scale dependence, closely resembling renormalization-group running
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The metrics for all three can be uniformly expressed... f(r) = 1−2M/r²/(r²+q²)^{3/2} ... f(r) = 1−2M/r exp(−α₀ M x/r^c) ... f(r) = 1−r²/3ξ log(1+6Mξ/r³)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Proca-Maxwell System in an Infinite Tower of Higher-Derivative Gravity
gr-qc 2026-04 unverdicted novelty 7.0

Higher-order terms in an infinite tower of higher-derivative gravity regularize a 5D Proca-Maxwell system, creating frozen regular cores that mimic extremal black holes and satisfy all energy conditions.

Reference graph

Works this paper leans on

88 extracted references · 88 canonical work pages · cited by 1 Pith paper · 28 internal anchors

[1]

Well-known examples include the Hay- ward BH [60] and the Frolov BH [61], in addition to the Bardeen BH

showed that, within GR, any static spherically sym- metric BH with a regular center and matter obeying the weak energy condition must approach a de Sitter core at the center. Well-known examples include the Hay- ward BH [60] and the Frolov BH [61], in addition to the Bardeen BH. To study the gravitational perturbations of all three RBHs within a unified f...

work page
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In contrast, the effective potential in [71] was obtained specifically for the case of vanishing background radial pressure

1 V(r) = l(l+ 1) r2 − f ′(r) r +f ′′(r).(17) Next, we consider the following expansions for the effec- tive potential and the master function: U(r) = X k≥0 ηk U (k)(r),Φ(r) = X k≥0 ηk Φ(k)(r),(18) 1 It is important to note that the effective potential derived in [70] applies to an anisotropic fluid with an arbitrary non-zero radial pressure. In contrast, ...

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The lowest multipole number starts froml= 0

Scalar field response In this section, we compute the TLN of Bardeen BH under scalar perturbation. The lowest multipole number starts froml= 0. Forl= 0, the zeroth-orderO(q 0) growing and decaying modes are given by Eqs. (20) and (21), and take the following form: Φ(0) + (r) = Φ(0)(r) = Φhor-reg(r) =r,(38) Φ(0) − (r) =−rlog 1− 1 r .(39) We should point ou...

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[4]

Vector field response Now we compute the TLNs of the Bardeen BH under vector perturbation. Similar to the steps present in the previous subsection, fors= 1 andl= 1, the TLNs can be obtained from the following expressions: I[S (1)](r) = 1 2 − r 4 − r2 4 −2 logr,(54) I[S (2)](r) =− 411 16 − π2 2 + 3 8r2 + 217 8r − 7r 4 + π2r 4 − r2 16 + π2r2 4 − 21 logr 4 +...

work page
[5]

We carry the calculations up to second order inq 4

Axial gravitational field response Here, we compute the TLNs of the Bardeen BH under axial gravitational perturbations forl= 2 andl= 3. We carry the calculations up to second order inq 4. Forl= 2 the expansion of the integration (29) inq 4 reads I[S (1)](r) = 103 40 − 21r 2 10 − r3 4 − 9r 4 40 ,(60) I[S (2)](r) = 3541 128 − 2163 320r 2 − 13081 640r + 837r...

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For the casel= 0, The zeroth-order (O(α 0 0)) growing and decaying modes, corresponding to Eqs

Scalar field response First of all, we compute the TLNs for the RBH with sub-Planckian curvature under scalar field perturbations. For the casel= 0, The zeroth-order (O(α 0 0)) growing and decaying modes, corresponding to Eqs. (20) and (21), are given explicitly by Φ(0) + (r) = Φ(0)(r) = Φhor-reg(r) =r,(74) Φ(0) − (r) =−rlog 1− 1 r .(75) In the cases= 0 a...

work page
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is derived from Eq. (25) as S(1)(r) =U (1)(r) Φ(0)(r) = 9 4r5 − 1 2r4 − 1 4r3 .(76) Applying the Green’s function formalism, the first- order solution and the second-order source term (O(α 2 0)) are expressed as Φ(1)(r) = Z ∞ 1 G(r, r′)S (1)(r′)dr ′ =− 1 4 − 1 4r2 − 1 4r , (77) S(2)(r) =U (2)(r) Φ(0)(r) +U (1)(r) Φ(1)(r) =− 9 8r8 + 15 16r7 + 1 2r6 + 15 8r...

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Vector field response Next, we compute the TLNs for vector perturbations (s= 1). Similarly, for the casel= 1, the TLNs can be obtained from the following integrations I[S (1)](r) =− 13 12 + 5 4r − r 12 − r2 12 ,(91) I[S (2)](r) =− 35 48 − 7 16r4 + 7 48r3 + 1 8r2 + 25 24r − r 12 − r2 16 , (92) I[S (3)](r) =− 4687 8064 + 65 896r7 − 103 1152r6 − 317 5760r5 −...

work page
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Axial gravitational field response In this subsection, we compute the TLNs of the RBH with sub-Planckian curvature under axial gravitational perturbations forl= 2 andl= 3. For the case ofl= 2, expanding the integral (29) inα 0, the leading-order and next-to-leading-order terms are given by I[S (1)](r) = 337 120 − 51r 20 − r2 10 − r3 12 − 3r 4 40 ,(98) I[S...

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in ASG under scalar, vector, and axial gravitational perturbations. Similarly, settingr h = 1 we can express the BH massMas a series expansion in terms of the deviation parameterξ, M(ξ) = 1 2 + 3ξ 4 + 3ξ2 4 + 9ξ3 16 +O(ξ 4).(104) Up to the same order in the parameterξ, the metric function can be expanded as f(r) = 1− 1 r − 3(−1 +r 3)ξ 2r4 − 3(−2 +r 3)(−1 ...

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For the case l= 0, the zeroth-order (O(ξ 0) modes, representing the growing and decaying solutions in Eqs

Scalar field response We first compute the TLNs of the RBH in ASG under scalar perturbations forl= 0 andl= 1. For the case l= 0, the zeroth-order (O(ξ 0) modes, representing the growing and decaying solutions in Eqs. (20) and (21), take the explicit forms Φ(0) + (r) = Φ(0)(r) = Φhor-reg(r) =r,(112) Φ(0) − (r) =−rlog 1− 1 r .(113) Whens= 0 andl= 0, the fir...

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Vector field response Next, we compute the TLNs of the RBH in ASG under vector perturbations, forl= 1 andl= 2. Forl= 1, the TLNs can be obtained from the following expressions: I[S (1)](r) =− 13 4 + 15 4r − r 4 − r2 4 ,(127) I[S (2)](r) =− 35 8 − 123 16r 4 + 13 8r 3 + 23 16r 2 + 155 16r − 7r 16 − r2 4 ,(128) I[S (3)](r) =− 4917 896 + 18063 896r 7 − 801 12...

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fingerprint

Axial gravitational field response Finally, we compute the TLNs of the RBH under axial gravitational perturbations within the framework of ASG 14 forl= 2 andl= 3. For the case ofl= 2, the expansion of the integral (29) inξis given by I[S (1)](r) = 9− 171r 20 − 3r2 20 − 3r3 20 − 3r4 20 ,(134) I[S (2)](r) = 159 2 − 513 16r3 − 927 40r2 + 177 80r − 411r 16 − ...

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