Love Numbers of Covariant Loop Quantum Black Holes

Meysam Motaharfar; Parampreet Singh

REVIEW 3 major objections 4 minor 1 cited by

Love Numbers of Covariant Loop Quantum Black Holes

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Loop quantum black holes respond to tides, unlike Schwarzschild black holes.

desk verdict New TLN results for three covariant LQG black holes, but the universal Planck-suppression claim rests on an unforced ABV assumption and a numerical conversion slip; worth refereeing after correction. read the letter →

arxiv 2505.14784 v1 pith:Y2WR3NJX submitted 2025-05-20 gr-qc

Meysam Motaharfar , Parampreet Singh This is my paper

classification gr-qc MSC 83C5783C45 PACS 04.60.Pp04.70.-s

keywords tidalLovenumbersloopquantumblackholeseffectivespacetimeGreen'sfunctiontechniquePlanck-scalesuppressionlogarithmicrunninghairaxialgravitationalperturbations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Classical Schwarzschild black holes have exactly vanishing tidal Love numbers, so they do not deform under a static external tide. This paper asks whether the same is true for three effective black hole metrics coming out of covariant loop quantum gravity, and finds that it is not. The ZLMY-I, ZLMY-II, and ABV models all acquire generally nonzero TLNs whose sign depends on the spin and multipole of the perturbation, whose magnitude is suppressed by powers of the Planck mass, and whose leading order can grow logarithmically with radius. These quantum corrections are tiny for astrophysical black holes but become significant at Planck mass, and their pattern differs between the models. A positive answer here supports the idea that loop quantum black holes carry quantum hair observable through tidal response.

What carries the argument

The extraction rests on a field redefinition $\Psi(\tilde r)=\Phi(\tilde r)/\sqrt{Z(\tilde r)}$ with $Z=\sqrt{A B}/f$, which folds every quantum modification of the metric into a correction to the Schwarzschild effective potential. That correction is expanded in powers of the small dimensionless quantum parameter $\tilde{\xi}=\xi/(2M)$ (for ZLMY) or $\tilde{r}_0=r_0/(2M)$ (for ABV), and the perturbed equation is solved order by order with a Green's function built from horizon-regular hypergeometric solutions. Analytic continuation of the multipole $\ell \to \ell+\epsilon$ with $\epsilon\to 0$ separates growing and decaying modes so the TLN is read off uniquely from the constant part, including any logarithmic piece, of the integral $I[S^{(k)}]$. Three orders in the expansion are used for the lowest multipole numbers in each channel.

What would settle it

Solve the static perturbation equation (3.2) numerically on the full metrics (2.8), (2.10), and (2.14) without expanding in the quantum parameter, for a mass such as $M=10^4\,M_{\rm Pl}$; if the extracted TLNs disagree with the quoted leading-order expressions $\kappa_s^\ell=[C(s,\ell)+C_{\log}(s,\ell)\log\tilde r]\,M^{-\alpha}$ beyond the stated order of accuracy, the perturbative extraction is not capturing the true response.

Watch

Extended reading notes

Core claim

For the ZLMY-I, ZLMY-II, and ABV covariant loop quantum black hole models, the leading-order corrections to the TLNs are nonzero and scale as $\kappa_s^\ell \propto M^{-2}$ for the ZLMY models and $\kappa_s^\ell \propto M^{-1}$ for the ABV model in Planck units. Scalar TLNs are negative for $\ell \geq 1$ (vanishing for $\ell=0$ in the ZLMY models), vector TLNs change sign across multipoles and models, and tensor TLNs are positive with no leading-order running. At leading order the scalar and vector TLNs in all three models display logarithmic running in the dimensionless radial coordinate even for low multipole numbers, a feature absent from the earlier AOS-model result and from classical GR. The paper reads the sign, magnitude, and running structure as a fingerprint of the quantization ambiguity, and notes that Planck-mass black holes would be substantially more tidally deformable than their astrophysical counterparts.

Load-bearing premise

The whole calculation assumes the three effective metrics correctly describe the spacetime near the horizon within linear response, and that the small parameters used in the series expansion stay small enough that the leading-order terms actually give the true tidal Love numbers.

Editorial extensions

If this is right

Astrophysical black holes in these models have TLNs far below current and near-future detector sensitivity, while Planck-mass remnants would be strongly deformable.
The sign and running of the TLNs give a concrete way to distinguish ZLMY-I, ZLMY-II, and ABV quantizations from each other and from other modified-gravity black holes.
Nonzero TLNs mean loop quantum black holes carry a static tidal response that classical four-dimensional GR black holes do not, effectively a quantum hair visible to outside observers.
As a black hole evaporates toward Planck mass, its tidal deformability grows, potentially altering the final emission spectrum and lifetime, an extrapolation the paper itself flags as beyond the perturbative regime.
The ZLMY models share an $M^{-2}$ suppression while the ABV model is more deformable at fixed mass with $M^{-1}$, so the mass scaling itself is a model discriminator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: applying the same Green's-function machinery to the rotating versions of these models (which the paper lists as future work) would map out spin-dependent TLNs, a sharper target for gravitational-wave tests of horizon-scale quantum structure.
Beyond the paper: the leading-order logarithmic running behaves like a scale-dependent coupling, so the coefficients $C_{\log}(s,\ell)$ could be treated as a renormalization-group tell; comparing these coefficients across models may give a quantization-ambiguity diagnostic that does not require absolute amplitude calibration.
Beyond the paper: the paper's own caveat that Planck-mass predictions exceed the perturbative expansion's validity means the strong-remnant-deformability claim should be tested with a nonperturbative or numerical solution of the full effective-metric perturbation equations rather than accepted from the series alone.
Beyond the paper: since the ZLMY-I vector TLN flips sign between $\ell=1$ and $\ell=2$, a future measurement of the quadrupole-to-dipole ratio of a tidally perturbed compact object could in principle separate loop-quantum-gravity effects from environmental matter effects that typically share a common sign.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is the first calculation of tidal Love numbers for the ZLMY-I, ZLMY-II, and ABV effective black hole metrics, extending the authors' earlier AOS-model work. The Green's function technique is applied cleanly: the perturbation equation is set up, the effective potentials are expanded order by order, and the TLN coefficients are read off from the constant and logarithmic parts of the asymptotic integrals. There is no circular fitting; the ZLMY results (α = 2 Planck suppression) are solid within the model assumptions, and the tabulated sign patterns for scalars, vectors, and axial tensors are a useful fingerprint for distinguishing quantization ambiguities.

The main soft spot is the ABV sector. The stress-test note is right: the paper's 'Planck suppressed for all three models' statement depends on fixing the ABV polymerization parameter λ through the area-gap condition Δ = 4π r0², making r̃0 ∝ 1/M. That identification is an extra physical assumption, not forced by the ABV construction itself. If λ is taken as a fixed constant — which is how the model is introduced in Eq. (2.11) and in the original literature — then r̃0 is mass-independent and the leading TLNs are O(1) (or at best ~10⁻² for λ ~ 0.1), not Planck suppressed. The paper should either justify the area-gap choice as the only physically reasonable one or present the ABV results for both possibilities. As written, the 'ABV > ZLMY' comparison and the abstract's blanket claim are both contingent.

There is also a concrete numerical conversion error: from Δ = 4π r0² one gets r0/(2M) = √Δ/(4√π M), not √Δ/(16π M). The 16π factor appears to be a typo, but it propagates into Table III, so all ABV coefficients are off by roughly a factor of seven. The authors should recheck the table.

Two smaller issues. First, the abstract and Section V say results are computed 'up to third order in the perturbation series,' but scalar and vector TLNs for several multipoles stop at second order; only the tensor TLNs reach third order. This is an overstatement, not a fatal flaw. Second, there are internal typos in the displayed coefficients—e.g., Eq. (4.46) gives 95/368 for the ℓ=3 tensor TLN while the text says 95/378. These need cleanup.

The paper itself honestly notes at the end that the Planck-mass extrapolation goes beyond the perturbative regime, so the authors are aware of the limitation. That does not rescue the abstract, which sells the suppression as universal.

My take: the ZLMY part is a competent and valuable contribution; the ABV part needs a major revision that either defends the area-gap identification or reports the λ-constant scenario separately, and the numerical slip must be fixed. I would send it to peer review with a recommendation of 'major revision.' It is exactly the kind of paper that a referee can improve substantially, and the corrected version would be worth citing.

Referee Report

3 major / 4 minor

Summary. The paper computes the static tidal Love numbers (TLNs) of three covariant effective loop quantum black hole spacetimes—ZLMY-I, ZLMY-II, and ABV—in response to scalar, vector, and axial gravitational perturbations. The authors map the radial perturbation equation into a Schwarzschild-like form with a modified potential, expand in the small quantum parameters, and extract TLNs for low multipoles using the Green's function technique of Ref. [30]. They report that, unlike classical Schwarzschild black holes, these loop quantum black holes have generically nonzero TLNs, with model-dependent signs, logarithmic running, and Planck-scale suppression of the magnitude (M^{-2} for the ZLMY models and M^{-1} for the ABV model under an area-gap identification). These features are interpreted as quantum hair and as a possible way to distinguish quantization ambiguities in loop quantum black holes.

Significance. If the results hold, the paper provides a useful extension of the earlier AOS-model analysis to covariant loop quantum black hole models, with explicit analytic series and tabulated coefficients. The central qualitative message—that these effective LQG black holes acquire a nonzero tidal response—is interesting and does not appear to involve circular reasoning or fitting of the target TLN values. The ZLMY results are robust within the stated perturbative framework. The main weakness is that the claimed Planck-scale suppression for the ABV model relies on a mass-dependent identification of the polymerization parameter λ that is not forced by the model as defined; if λ is treated as a fixed constant, the ABV TLNs are mass-independent. The manuscript also contains several internal inconsistencies in the presentation of the expansion order and in the reported coefficients. These issues are fixable by qualification and correction, so the paper is suitable for major revision rather than rejection.

major comments (3)

[§II.B, §IV.C (Eq. 4.74), §V] The central claim that the ABV TLNs are Planck-scale suppressed with κ ∝ M^{-1} is not a consequence of the ABV model as defined. In §II.B, λ is introduced as a positive dimensionless constant (Eq. 2.11), which makes ṝ̃0 = λ²/(1+λ²) mass-independent; then the leading-order TLNs κ = C ṝ̃0 are O(λ²) constants and do not scale as M^{-1}. The M^{-1} scaling in Eq. (4.74) follows from the additional identification ṝ̃0 = √Δ/(16π) M^{-1} based on the area gap, which is a separate physical assumption and is not forced by the ABV construction. The abstract's 'Planck scale suppressed for all three models' and the ABV-versus-ZLMY comparison therefore depend on this unexamined choice. Please state this assumption explicitly and, ideally, provide the constant-λ results as a limiting case so that the reader can see how the conclusions depend on it.
[Abstract and §IV.C] The abstract and §V state that the TLNs exhibit logarithmic running at leading order in response to scalar and vector perturbations, but this is not true of all computed cases. In §IV.C.2 the ABV vector TLNs have zero running at leading order (κ¹₁ = 1/2, κ¹₂ = 9/160, κ¹₃ = 1/175), and in §IV.C.1 the ABV scalar ℓ = 0 TLN has no running. The tensor TLNs for all three models also have no leading-order running (Tables I–III). The narrative should be qualified to 'for several multipoles and models' or otherwise made consistent with the detailed results.
[Abstract, §IV.C, §V] The statement that, for the same black hole mass, the ABV model has larger TLNs than the ZLMY models is not true over the full mass range. Because the ABV TLNs scale as M^{-1} while the ZLMY TLNs scale as M^{-2}, there is a mass-dependent crossover. For example, for the scalar ℓ = 1 case, the ABV coefficient in Table III is 0.0868 M^{-1} while the ZLMY-I coefficient in Table I is 2.5847 M^{-2}, so ABV dominates only for M ≳ 30 M_Pl. The comparison should be qualified with the mass range in which it holds, especially since the paper emphasizes Planck-mass black holes.

minor comments (4)

[§IV opening and §V] The paper says TLNs are computed 'up to the third order in the perturbation series expansion', but for several cases the results are given only through second order: ZLMY-I scalar ℓ = 1,2 and vector ℓ = 1,2,3, ZLMY-II scalar and vector cases, and ABV scalar ℓ = 1,2. Please revise the order claims or extend the computations.
[§II.B and Eq. (4.74)] The displayed conversion ṝ̃0 = √Δ/(16π) M^{-1} is arithmetically wrong: from Δ = 4π r₀² one obtains ṝ̃0 = √Δ/(4√π M). The numerical values in Table III and the leading coefficients in §IV.C are consistent with the correct factor, so the text should be corrected.
[Eq. (4.45) vs Eq. (4.46) and Table I] For the ZLMY-I tensor ℓ = 3 case, the constant term in Eq. (4.45) is 95/378, while Eq. (4.46) uses 95/368 and Table I gives C = 0.3336, which is consistent with 95/368 once the Δ/4 factor is included. Please correct the inconsistent fraction.
[Eq. (4.92)] In the coefficient of the O(ṝ̃0³) term in Eq. (4.92) there is a dangling plus sign: '−4021/6912 +' should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the TLN coefficients are computed from the effective metrics via an external Green's-function method, and the ABV mass scaling is an explicit input assumption rather than a fitted prediction.

full rationale

The derivation chain is self-contained. The TLNs are obtained by inserting the three effective metrics (2.8), (2.10), and (2.14) into the standard static perturbation equation (3.2), transforming to the Schwarzschild-like form (3.9), expanding the effective potential in the stated quantum parameter, and applying the external Green's-function matching (3.24)-(3.26). None of the extracted coefficients C(s,ell) or C_log(s,ell) is fitted to a target TLN; the Green's function is fixed by the zeroth-order Schwarzschild problem, whose vanishing TLN is an independent classical result. For the ZLMY models, the mass scaling kappa ~ M^{-2} follows from the LQG-fixed parameter xi = sqrt(Delta) entering only through xi-tilde = xi/(2M); this is a dimensional consequence of the model input, not a hidden fit. For the ABV model, the M^{-1} scaling is not concealed: Section II B explicitly assumes the area-gap identification r0-tilde = r0/(2M) proportional to M^{-1}, and the leading-order result kappa = C r0-tilde then inherits that scaling by simple algebra. This is an explicit physical assumption about the model rather than a circular prediction. The printed conversion r0/(2M) = sqrt(Delta)/(16 pi) M^{-1} in Eq. (4.74) is numerically inconsistent with Delta = 4 pi r0^2 (the correct value would be sqrt(Delta)/(4 sqrt(pi)) M^{-1}, which Table III appears to use), but this is a numerical/typographical issue, not a circularity. The only same-author citation, Ref. [65], is used for comparison with the AOS model and is not a premise of the calculation. The uniqueness claim comes from the analytic continuation of the multipole number, an external technique, not from a self-citation. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central results depend only on the two model parameters inherited from LQG and the assumption that the effective metrics are the correct spacetimes. No ad hoc parameters are fitted to the TLN results.

free parameters (2)

xi (ZLMY regularization parameter) = sqrt(Delta), Delta=4 sqrt(3) pi gamma l_Pl^2
Inherited from LQG area gap via Refs [61,62]; enters the metrics (2.8) and (2.10) and controls the expansion parameter xi-tilde=xi/(2M). Not fitted to data in this paper.
lambda (ABV polymerization parameter) = lambda^2/(1+lambda^2)=r0/(2M)=sqrt(Delta)/(4 sqrt(pi))M^{-1} (paper has a typo with 16pi)
Fixed by area gap in Ref [63]; mass-dependent; controls the small parameter r0-tilde. Not fitted to data in this paper.

assumptions (4)

domain assumption Effective dynamics from polymerization describe the correct spacetime geometry of loop quantum black holes.
Section II adopts the effective metrics (2.8), (2.10), and (2.14) as the physical backgrounds; the whole TLN computation rests on these, not on a full quantum gravity derivation.
domain assumption The perturbation equation (3.2) with effective potential (3.3) governs scalar, vector, and axial gravitational tidal responses.
Quoted from Refs [68,80,81]; the paper relies on this master equation for spin-s fields in spherical symmetry without deriving it.
standard math The Green's function perturbative scheme with analytic continuation in l yields the unique TLN.
Method of Ref [30] adopted wholesale; the paper applies it but assumes its validity and uniqueness for the modified potentials.
domain assumption The quantum parameters xi-tilde and r0-tilde are small for macroscopic black holes, so truncating the perturbation series is valid.
Assumed in Sections III and IV to justify the perturbative extraction; the paper explicitly notes the Planck-mass extrapolation is outside this validity.

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Cite this review

Pith. "Pith review of Love Numbers of Covariant Loop Quantum Black Holes." pith.science (2026). https://pith.science/paper/Y2WR3NJX

@misc{pith2026250514784,
  author       = {Pith},
  title        = {Pith review of: Love Numbers of Covariant Loop Quantum Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2WR3NJX}},
  note         = {Machine review of arXiv:2505.14784}
}

read the original abstract

We investigate the linear static response of three covariant loop quantum black holes, namely, the two models proposed by Zhang, Lewandowski, Ma, and Yang (ZLMY) and the Alonso-Bardaji, Brizuela, and Vera (ABV) model, to an external tidal field. Using effective spacetime description, we uniquely extract the tidal Love numbers (TLNs) using perturbative solutions derived from the Green's function technique. Our findings reveal that, in contrast to the classical Schwarzschild black hole, the TLNs for loop quantum black holes are generally nonzero. The sign and magnitude of the TLNs depend on the spin of the external tidal field, the multipole number, and the details of the loop quantized model. The magnitude of the TLNs is found to be Planck scale suppressed for all three models, implying a stronger tidal deformability for black holes with Planckian mass. Additionally, for the same black hole mass, the magnitude of the TLNs for the ABV model is larger than the ZLMY models. We also find that the TLNs exhibit logarithmic running behavior at the leading order, even for low multipole numbers, in response to scalar and vector field perturbations. These distinct features of the TLNs can serve as a potential tool to differentiate between various quantization ambiguities arising in loop quantum black holes. We briefly discuss the potential phenomenological and theoretical implications of nonzero TLNs for black hole physics.

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Forward citations

Cited by 1 Pith paper

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

Echoes of Love Beyond the Horizon: A Bridge to Recovering Information from Black Holes
gr-qc 2025-05 conditional novelty 5.0 of 10

Black holes may acquire Planck-suppressed tidal Love numbers from quantum gravity, which the authors argue could carry information about the initial collapse and help resolve the information loss paradox.

Reference graph

Works this paper leans on

95 extracted references · 61 canonical work pages · cited by 1 Pith paper

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(3.17) and (3.18), take the following form Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r, (4.6) Φ (0) − (˜r) = −˜r log ( 1 − 1 ˜r )

Scalar ﬁeld response For the scalar ﬁeld perturbation, s = 0, and the lowest multipole number, ℓ = 0, the growing and decaying modes at the zeroth order, O(˜ξ0), which are given by Eqs. (3.17) and (3.18), take the following form Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r, (4.6) Φ (0) − (˜r) = −˜r log ( 1 − 1 ˜r ) . (4.7) We should point out that the ﬁr...
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Vector ﬁeld response Now we compute the vector TLNs for the ZLMY-I model. For s = 1 and ℓ = 1, the zeroth order growing and decaying solutions are given by Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r2, (4.27) Φ (0) − (˜r) = −3 [ ˜r2 log ( 1 − 1 ˜r ) + ˜r + 1 2 ] . (4.28) The ﬁrst order correction to the TLN can also be obtained from the following integr...
[3]

For s = 2 and ℓ = 2, we have Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r3, (4.39) Φ (0) − (˜r) = −5 [ ˜r3 log ( 1 − 1 ˜r ) + ˜r2 + ˜r 2 + 1 4˜r + 1 3 ]

Axial gravitational ﬁeld response Here, we extract the TLNs in response to the axial gravitatio nal ﬁeld perturbation for the ZLMY-I model. For s = 2 and ℓ = 2, we have Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r3, (4.39) Φ (0) − (˜r) = −5 [ ˜r3 log ( 1 − 1 ˜r ) + ˜r2 + ˜r 2 + 1 4˜r + 1 3 ] . (4.40) Hence, the integration for extracting the ﬁrst order c...

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Scalar ﬁeld response Here, we compute the TLNs in response to a scalar ﬁeld perturb ation for the ZLMY-II model. Having the ﬁrst order eﬀective potential (4.54) and the zerot h order solutions (4.6) and (4.7) for ℓ = 0, the ﬁrst order correction to the TLN in response to a scala r ﬁeld is obtained from the following integration I[S (1)(˜r)] = − ∫ ˜r 1 Φ (...
[5]

Vector ﬁeld response Now we calculate the TLNs in response to a vector ﬁeld perturb ation, s = 1. For ℓ = 1, the zeroth order solutions are given by (4.27) and (4.28), and th e ﬁrst order correction to the TLN can then be derived from the following integration I[S (1)(˜r)] = − 1 3 ∫ ˜r 1 Φ (0) + (˜r′)S (1)(˜r′)d˜r′ = 5 4˜r − 17 12 + log(˜r) + ˜r 6 . (4.62...

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For ℓ = 2, the zeroth order solutions are given by Eqs

Axial gravitational ﬁeld response In this section, we compute the TLNs for the axial gravitatio nal ﬁeld response for the ZLMY-II model. For ℓ = 2, the zeroth order solutions are given by Eqs. (4.39) and (4 .40), and we can compute the following integration I[S (1)(˜r)] = − 1 5 ∫ ˜r 1 Φ (0) + (˜r′)S (1)(˜r′)d˜r′ = ˜r3 10 + 7˜r2 20 − 21˜r 20 + 3 5 . (4.68)...
[7]

Scalar ﬁeld response Here, we compute the TLNs in response to the scalar ﬁeld pertu rbation for the ABV model. For s = 0 and ℓ = 0, one can compute the ﬁrst order correction to the TLN from t he following integration I[S (1)(˜r)] = − ∫ ˜r 1 Φ (0) + (˜r′)S (1)(˜r′)d˜r′ = − 1 4 + 1 4˜r , (4.79) implying that κ0(1) 0 = −1/4. This TLN is negative and has no l...

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Tidal Love Numbers of Ker r Black Holes,

A. Le Tiec, M. Casals and E. Franzin, “Tidal Love Numbers of Ker r Black Holes,” Phys. Rev. D 103, no.8, 084021 (2021); H. S. Chia, “Tidal deformation and dissipatio n of rotating black holes,” Phys. Rev. D 104, no.2, 024013 (2021); P. Charalambous, S. Dubovsky and M. M. Iv an...

2021
[27]

Tes ting strong-ﬁeld gravity with tidal Love numbers,

V. Cardoso, E. Franzin, A. Maselli, P. Pani and G. Raposo, “Tes ting strong-ﬁeld gravity with tidal Love numbers,” Phys. Rev. D 95, no.8, 084014 (2017)

2017
[28]

Love numbers and magnetic s usceptibility of charged black holes,

D. Pere˜ niguez and V. Cardoso, “Love numbers and magnetic s usceptibility of charged black holes,” Phys. Rev. D 105, no.4, 044026 (2022)

2022
[29]

Ladder symmetries and Love numbers o f Reissner-Nordstr¨ om black holes,

M. Rai and L. Santoni, “Ladder symmetries and Love numbers o f Reissner-Nordstr¨ om black holes,” JHEP 07, 098 (2024)

2024
[31]

and by eﬀective ﬁeld theory of gravity [28, 30], as each fr amework predicts a distinct TLN behavior. V. SUMMARY AND CONCLUSIONS Classical GR predicts that black holes are the most resilien t objects in the universe with pre- cisely vanishing TLNs. This implies that classical ...
[32]

Vanishing of n onlinear tidal Love numbers of Schwarzschild black holes,

M. M. Riva, L. Santoni, N. Savi´ c and F. Vernizzi, “Vanishing of n onlinear tidal Love numbers of Schwarzschild black holes,” Phys. Lett. B 854, 138710 (2024); S. Iteanu, M. M. Riva, L. Santoni, N. Savi´ c and F. Vernizzi, “Vanishing of Quadratic Love Numbers of Schwarzschild...

2024 arXiv
[33]

Symmetries of Vanishing Nonlinear Love Numbers of Schwarzschild Black Holes,

O. Combaluzier-Szteinsznaider, L. Hui, L. Santoni, A. R. Solomo n and S. S. C. Wong, “Symmetries of Vanishing Nonlinear Love Numbers of Schwarzschild Black Holes,” [arX iv:2410.10952 [gr-qc]]

arXiv
[34]

The Vanishing of the No n-linear Static Love Number of Kerr Black Holes and the Role of Symmetries,

L. R. Gounis, A. Kehagias and A. Riotto, “The Vanishing of the No n-linear Static Love Number of Kerr Black Holes and the Role of Symmetries,” [arXiv:2412.08249 [gr-qc ]]

arXiv
[35]

Hidden Symm etry of Vanishing Love Numbers,

P. Charalambous, S. Dubovsky and M. M. Ivanov, “Hidden Symm etry of Vanishing Love Numbers,” Phys. Rev. Lett. 127, no.10, 101101 (2021); L. Hui, A. Joyce, R. Penco, L. Santoni an d A. R. Solomon, “Ladder symmetries of black holes. Implications for love numbers an d no-hair the...

2021
[36]

Gravitational wave s and higher dimensions: Love numbers and Kaluza-Klein excitations,

V. Cardoso, L. Gualtieri and C. J. Moore, “Gravitational wave s and higher dimensions: Love numbers and Kaluza-Klein excitations,” Phys. Rev. D 100, no.12, 124037 (2019); M. J. Rodriguez, L. Santoni, A. R. Solomon and L. F. Temoche, “Love numbers for rotating black holes in hi...

2019 arXiv
[37]

Asymptotically de Sitt er black holes have nonzero tidal Love numbers,

S. Nair, S. Chakraborty and S. Sarkar, “Asymptotically de Sitt er black holes have nonzero tidal Love numbers,” Phys. Rev. D 109, no.6, 6 (2024)

2024
[38]

Tidal Love number s of static black holes in anti-de Sitter,

E. Franzin, A. M. Frassino and J. V. Rocha, “Tidal Love number s of static black holes in anti-de Sitter,” [arXiv:2410.23545 [hep-th]]

arXiv
[39]

Black Holes in an Eﬀective Field Theory Extension of General Relativity,

V. Cardoso, M. Kimura, A. Maselli and L. Senatore, “Black Holes in an Eﬀective Field Theory Extension of General Relativity,” Phys. Rev. Lett. 121, no.25, 251105 (2018) [erratum: Phys. Rev. Lett. 131, no.10, 109903 (2023)]

2018
[40]

Implications of the we ak gravity conjecture for tidal Love numbers of black holes,

V. De Luca, J. Khoury and S. S. C. Wong, “Implications of the we ak gravity conjecture for tidal Love numbers of black holes,” Phys. Rev. D 108, no.4, 4 (2023)

2023
[41]

Tidal response beyond vacuum General Relativity with a canonical deﬁnition,

T. Katagiri, V. Cardoso, T. Ikeda and K. Yagi, “Tidal response beyond vacuum General Relativity with a canonical deﬁnition,” [arXiv:2410.02531 [gr-qc]]

arXiv
[42]

Tidal Love Numbers o f Neutron Stars in Horndeski Theories,

R. F. Diedrichs, S. Tsujikawa and K. Yagi, “Tidal Love Numbers o f Neutron Stars in Horndeski Theories,” [arXiv:2501.07998 [gr-qc]]

arXiv
[43]

Environmental eﬀects in gravitatio nal-wave physics: Tidal deformability of black holes immersed in matter,

V. Cardoso and F. Duque, “Environmental eﬀects in gravitatio nal-wave physics: Tidal deformability of black holes immersed in matter,” Phys. Rev. D 101, no.6, 064028 (2020)

2020
[44]

Tidal deformability of dressed black hole s and tests of ultralight bosons in extended mass ranges,

V. De Luca and P. Pani, “Tidal deformability of dressed black hole s and tests of ultralight bosons in extended mass ranges,” JCAP 08, 032 (2021)

2021
[45]

modeling frequency-depend ent tidal deformability for environ- mental black hole mergers,

V. De Luca, A. Maselli and P. Pani, “modeling frequency-depend ent tidal deformability for environ- mental black hole mergers,” Phys. Rev. D 107, no.4, 044058 (2023)

2023
[46]

Stability of relativistic tida l response against small potential modiﬁcation,

T. Katagiri, H. Nakano and K. Omukai, “Stability of relativistic tida l response against small potential modiﬁcation,” Phys. Rev. D 108, no.8, 084049 (2023)

2023
[47]

Tidal deformability of bla ck holes surrounded by thin accretion disks,

E. Cannizzaro, V. De Luca and P. Pani, “Tidal deformability of bla ck holes surrounded by thin accretion disks,” Phys. Rev. D 110, no.12, 123004 (2024)

2024
[48]

Loop Quantum Cosmology: A Status Report,

A. Ashtekar and P. Singh, “Loop Quantum Cosmology: A Status Report,” Class. Quant. Grav. 28, 213001 (2011)

2011
[49]

Regular black holes from L oop Quantum Gravity,

A. Ashtekar, J. Olmedo and P. Singh, “Regular black holes from L oop Quantum Gravity,” [arXiv:2301.01309 [gr-qc]]

arXiv
[50]

Quantum nature of t he big bang,

A. Ashtekar, T. Pawlowski and P. Singh, “Quantum nature of t he big bang,” Phys. Rev. Lett. 96, 141301 (2006)

2006
[51]

Quantum Nature of t he Big Bang: Improved dynamics,

A. Ashtekar, T. Pawlowski and P. Singh, “Quantum Nature of t he Big Bang: Improved dynamics,” Phys. Rev. D 74, 084003 (2006)

2006
[52]

Robustness of key fea tures of loop quantum cosmology,

A. Ashtekar, A. Corichi and P. Singh, “Robustness of key fea tures of loop quantum cosmology,” Phys. Rev. D 77, 024046 (2008). 33

2008
[53]

Numerical simulations of a loop q uantum cosmos: robustness of the quantum bounce and the validity of eﬀective dynamics,

P. Diener, B. Gupt and P. Singh, “Numerical simulations of a loop q uantum cosmos: robustness of the quantum bounce and the validity of eﬀective dynamics,” Class. Quant . Grav. 31, 105015 (2014)

2014
[54]

Numerical simulat ions of loop quantum Bianchi-I spacetimes,

P. Diener, A. Joe, M. Megevand and P. Singh, “Numerical simulat ions of loop quantum Bianchi-I spacetimes,” Class. Quant. Grav. 34, 094004 (2017)

2017
[55]

Glimpses of Space-Time Beyond the Singularities Using S upercomputers,

P. Singh, “Glimpses of Space-Time Beyond the Singularities Using S upercomputers,” Comput. Sci. Eng. 20, no.4, 26-38 (2018)

2018
[56]

Are loop quantum cosmos never singular?,

P. Singh, “Are loop quantum cosmos never singular?,” Class. Qua nt. Grav. 26, 125005 (2009); P. Singh, “Loop quantum cosmology and the fate of cosmological singularities ,” Bull. Astron. Soc. India 42, 121 (2014); S. Saini and P. Singh, “Resolution of strong singularitie s and...

2009
[57]

Quantum geometry and the Sch warzschild singularity,

A. Ashtekar and M. Bojowald, “Quantum geometry and the Sch warzschild singularity,” Class. Quant. Grav. 23, 391-411 (2006)

2006
[58]

Loop quantum black hole,

L. Modesto, “Loop quantum black hole,” Class. Quant. Grav. 23, 5587-5602 (2006)

2006
[59]

Phenomenological loop quantum geometry of the S chwarzschild black hole,

D. W. Chiou, “Phenomenological loop quantum geometry of the S chwarzschild black hole,” Phys. Rev. D 78, 064040 (2008)

2008
[60]

Loop quantization of the Schwarzsch ild interior revisited,

A. Corichi and P. Singh, “Loop quantization of the Schwarzsch ild interior revisited,” Class. Quant. Grav. 33, no.5, 055006 (2016)

2016
[61]

Emergence of the product of constant curvature spaces in loop quantum cosmology,

N. Dadhich, A. Joe and P. Singh, “Emergence of the product of constant curvature spaces in loop quantum cosmology,” Class. Quant. Grav. 32, no.18, 185006 (2015)

2015
[62]

From black holes to white holes: a quantum gravitational, symmetric bounce,

J. Olmedo, S. Saini and P. Singh, “From black holes to white holes: a quantum gravitational, symmetric bounce,” Class. Quant. Grav. 34, no.22, 225011 (2017)

2017
[63]

Quantum Transﬁguratio n of Kruskal Black Holes,

A. Ashtekar, J. Olmedo and P. Singh, “Quantum Transﬁguratio n of Kruskal Black Holes,” Phys. Rev. Lett. 121, no.24, 241301 (2018)

2018
[64]

Quantum extension of th e Kruskal spacetime,

A. Ashtekar, J. Olmedo and P. Singh, “Quantum extension of th e Kruskal spacetime,” Phys. Rev. D 98, no.12, 126003 (2018)

2018
[65]

(b,v)-type variab les for black to white hole transitions in eﬀective loop quantum gravity,

N. Bodendorfer, F. M. Mele and J. M¨ unch, “(b,v)-type variab les for black to white hole transitions in eﬀective loop quantum gravity,” Phys. Lett. B 819, 136390 (2021)

2021
[66]

Loop quantum Schwar zschild interior and black hole rem- nant,

C. Zhang, Y. Ma, S. Song and X. Zhang, “Loop quantum Schwar zschild interior and black hole rem- nant,” Phys. Rev. D 102, no.4, 041502
[67]

Hamiltonian formulation and loop quantization of a recent extension of the Kruskal spacetime,

B. Elizaga Navascu´ es, A. Garc ´ ıa-Quismondo and G. A. Mena Marug´ an, “Hamiltonian formulation and loop quantization of a recent extension of the Kruskal spacetime,” Phys. Rev. D 106, no.4, 043531 (2022)

2022
[68]

Revisiting quantum black holes f rom eﬀective loop quantum gravity,

G. Ongole, P. Singh and A. Wang, “Revisiting quantum black holes f rom eﬀective loop quantum gravity,” Phys. Rev. D 109, no.2, 026015 (2024)

2024
[69]

Lessons from gauge ﬁxing and polymerization of loop quantum black holes with a cosmological constant,

G. Ongole, P. Singh and A. Wang, “Lessons from gauge ﬁxing and polymerization of loop quantum black holes with a cosmological constant,” Phys. Rev. D 111, no.8, 086019 (2025)

2025
[70]

Loop quantization of the Schwarzschild black hole,

R. Gambini and J. Pullin, “Loop quantization of the Schwarzschild black hole,” Phys. Rev. Lett. 110, no.21, 211301 (2013); V. Husain, J. G. Kelly, R. Santacruz and E. Wilson-Ewing, “Quantum Gravity of Dust Collapse: Shock Waves from Black Holes,” Phys. Rev. Lett. 128, no.12, 1...

2013
[71]

Black Holes and C ovariance in Eﬀective Quantum 34 Gravity,

C. Zhang, J. Lewandowski, Y. Ma and J. Yang, “Black Holes and C ovariance in Eﬀective Quantum 34 Gravity,” [arXiv:2407.10168 [gr-qc]]
[72]

Black holes and c ovariance in eﬀective quantum gravity: A solution without Cauchy horizons,

C. Zhang, J. Lewandowski, Y. Ma and J. Yang, “Black holes and c ovariance in eﬀective quantum gravity: A solution without Cauchy horizons,” [arXiv:2412.02487 [gr-q c]]
[73]

An eﬀective model for the quantum Schwarzschild black hole,

A. Alonso-Bardaji, D. Brizuela and R. Vera, “An eﬀective model for the quantum Schwarzschild black hole,” Phys. Lett. B 829, 137075 (2022)

2022
[74]

Planck stars, White Holes, Remnants and Planck-mass quasi-particles. The quantum gravity phase in black holes’ evolution and its manifestation s,

C. Rovelli and F. Vidotto, “Planck stars, White Holes, Remnants and Planck-mass quasi-particles. The quantum gravity phase in black holes’ evolution and its manifestation s,” [arXiv:2407.09584 [gr-qc]]

arXiv
[75]

Loop Quantum Gravitational Sign atures via Love Numbers,

M. Motaharfar and P. Singh, “Loop Quantum Gravitational Sign atures via Love Numbers,” [arXiv:2501.09151 [gr-qc]]

arXiv
[76]

Quantum corrections to tidal Love numbe r for Schwarzschild black holes,

J. W. Kim and M. Shim, “Quantum corrections to tidal Love numbe r for Schwarzschild black holes,” Phys. Rev. D 104, no.4, 046022 (2021)

2021
[77]

Love in Extrema Ratio,

P. Pani and A. Maselli, “Love in Extrema Ratio,” Int. J. Mod. Phys . D 28, no.14, 1944001 (2019)

2019
[78]

Probing the Eﬀective Quant um Gravity via Quasinormal Modes and Shadows of Black Holes,

R. A. Konoplya and O. S. Stashko, “Probing the Eﬀective Quant um Gravity via Quasinormal Modes and Shadows of Black Holes,” [arXiv:2408.02578 [gr-qc]]

arXiv
[79]

Light rings and shadows of static blac k holes in eﬀective quantum gravity,

W. Liu, D. Wu and J. Wang, “Light rings and shadows of static blac k holes in eﬀective quantum gravity,” Phys. Lett. B 858, 139052 (2024)

2024
[80]

Gravitation al lensing eﬀect of black holes in eﬀective quantum gravity,

H. Liu, M. Y. Lai, X. Y. Pan, H. Huang and D. C. Zou, “Gravitation al lensing eﬀect of black holes in eﬀective quantum gravity,” Phys. Rev. D 110, no.10, 104039 (2024)

2024
[81]

Strong G ravitational Lensing by Static Black Holes in Eﬀective Quantum Gravity,

Y. Wang, A. Vachher, Q. Wu, T. Zhu and S. G. Ghosh, “Strong G ravitational Lensing by Static Black Holes in Eﬀective Quantum Gravity,” [arXiv:2410.12382 [astro-ph.CO]]

arXiv
[82]

Shadows of rotating black holes in eﬀective quantum gravity,

Z. Ban, J. Chen and J. Yang, “Shadows of rotating black holes in eﬀective quantum gravity,” [arXiv:2411.09374 [gr-qc]]

arXiv
[83]

Quasinormal modes of a holonomy corrected Schwarzschild black hole,

Z. S. Moreira, H. C. D. Lima, Junior, L. C. B. Crispino and C. A. R. Herdeiro, “Quasinormal modes of a holonomy corrected Schwarzschild black hole,” Phys. Rev. D 107, no.10, 104016 (2023)

2023
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Quasinormal modes of a nonsingular spherically s ymmetric black hole eﬀective model with holonomy corrections,

D. M. Gingrich, “Quasinormal modes of a nonsingular spherically s ymmetric black hole eﬀective model with holonomy corrections,” Phys. Rev. D 110, no.8, 084045 (2024)

2024
[85]

Long-lived quasinormal modes and overtones ’ behavior of holonomy-corrected black holes,

S. V. Bolokhov, “Long-lived quasinormal modes and overtones ’ behavior of holonomy-corrected black holes,” Phys. Rev. D 110, no.2, 024010 (2024)

2024
[86]

Peculiar prop erties in quasinormal spectra from loop quantum gravity eﬀect,

G. Fu, D. Zhang, P. Liu, X. M. Kuang and J. P. Wu, “Peculiar prop erties in quasinormal spectra from loop quantum gravity eﬀect,” Phys. Rev. D 109, no.2, 026010 (2024)

2024
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Gravitational lens eﬀect of a holonomy corrected Schwarzschild black hole,

E. L. B. Junior, F. S. N. Lobo, M. E. Rodrigues and H. A. Vieira, “ Gravitational lens eﬀect of a holonomy corrected Schwarzschild black hole,” Phys. Rev. D 109, no.2, 2 (2024)

2024
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Holonomy corrected Schwarzschild black hole lensing,

A. R. Soares, C. F. S. Pereira, R. L. L. Vit´ oria and E. M. Rocha , “Holonomy corrected Schwarzschild black hole lensing,” Phys. Rev. D 108, no.12, 124024 (2023)

2023
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Formation of nonsingular spherical black h oles with holonomy corrections,

A. Alonso-Bardaji, “Formation of nonsingular spherical black h oles with holonomy corrections,” [arXiv:2410.20529 [gr-qc]]

arXiv
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Haw king radiation by spherically- symmetric static black holes for all spins: Teukolsky equations and p otentials,

A. Arbey, J. Auﬃnger, M. Geiller, E. R. Livine and F. Sartini, “Haw king radiation by spherically- symmetric static black holes for all spins: Teukolsky equations and p otentials,” Phys. Rev. D 103, no.10, 104010 (2021)

2021
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Parametrized Love num bers of nonrotating black holes,

T. Katagiri, T. Ikeda and V. Cardoso, “Parametrized Love num bers of nonrotating black holes,” Phys. Rev. D 109, no.4, 044067 (2024)

2024
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Tidal Gravitational Radiation,

B. Mashhoon, “Tidal Gravitational Radiation,” Astrophysical J ournal, 185, 83-86 (1973)

1973
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Detecting Planck-Sc ale Dark Matter with Quantum In- terference,

M. Christodoulou, A. Perez and C. Rovelli, “Detecting Planck-Sc ale Dark Matter with Quantum In- terference,” Phys. Rev. Lett. 133, no.11, 111001 (2024)

2024
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Echoes of Love Beyond the Horiz on: A Bridge to Recovering Information from Black Holes,

M. Motaharfar and P. Singh, “Echoes of Love Beyond the Horiz on: A Bridge to Recovering Information from Black Holes,” Essay written for the Gravity Research Foundat ion 2025 Awards for Essays on Gravitation (Honorable Mention)

2025
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Shadow of qu antum extended Kruskal black hole and its super-radiance property,

S. Devi, A. N. S., S. Chakrabarti and B. R. Majhi, “Shadow of qu antum extended Kruskal black hole and its super-radiance property,” Phys. Dark Univ. 39, 101173 (2023)

2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [30]

Tidal Love numbers from EFT of black hole perturbations with timelik e scalar proﬁle,

C. G. A. Barura, H. Kobayashi, S. Mukohyama, N. Oshita, K. Ta kahashi and V. Yingcharoenrat, “Tidal Love numbers from EFT of black hole perturbations with timelik e scalar proﬁle,” JCAP 09, 001 (2024)

work page 2024

[2] [1]

(3.17) and (3.18), take the following form Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r, (4.6) Φ (0) − (˜r) = −˜r log ( 1 − 1 ˜r )

Scalar ﬁeld response For the scalar ﬁeld perturbation, s = 0, and the lowest multipole number, ℓ = 0, the growing and decaying modes at the zeroth order, O(˜ξ0), which are given by Eqs. (3.17) and (3.18), take the following form Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r, (4.6) Φ (0) − (˜r) = −˜r log ( 1 − 1 ˜r ) . (4.7) We should point out that the ﬁr...

[3] [2]

Vector ﬁeld response Now we compute the vector TLNs for the ZLMY-I model. For s = 1 and ℓ = 1, the zeroth order growing and decaying solutions are given by Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r2, (4.27) Φ (0) − (˜r) = −3 [ ˜r2 log ( 1 − 1 ˜r ) + ˜r + 1 2 ] . (4.28) The ﬁrst order correction to the TLN can also be obtained from the following integr...

[4] [3]

For s = 2 and ℓ = 2, we have Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r3, (4.39) Φ (0) − (˜r) = −5 [ ˜r3 log ( 1 − 1 ˜r ) + ˜r2 + ˜r 2 + 1 4˜r + 1 3 ]

Axial gravitational ﬁeld response Here, we extract the TLNs in response to the axial gravitatio nal ﬁeld perturbation for the ZLMY-I model. For s = 2 and ℓ = 2, we have Φ (0) + (˜r) = Φ (0)(˜r) = Φ hor-reg(˜r) = ˜r3, (4.39) Φ (0) − (˜r) = −5 [ ˜r3 log ( 1 − 1 ˜r ) + ˜r2 + ˜r 2 + 1 4˜r + 1 3 ] . (4.40) Hence, the integration for extracting the ﬁrst order c...

1960

[5] [4]

Scalar ﬁeld response Here, we compute the TLNs in response to a scalar ﬁeld perturb ation for the ZLMY-II model. Having the ﬁrst order eﬀective potential (4.54) and the zerot h order solutions (4.6) and (4.7) for ℓ = 0, the ﬁrst order correction to the TLN in response to a scala r ﬁeld is obtained from the following integration I[S (1)(˜r)] = − ∫ ˜r 1 Φ (...

[6] [5]

Vector ﬁeld response Now we calculate the TLNs in response to a vector ﬁeld perturb ation, s = 1. For ℓ = 1, the zeroth order solutions are given by (4.27) and (4.28), and th e ﬁrst order correction to the TLN can then be derived from the following integration I[S (1)(˜r)] = − 1 3 ∫ ˜r 1 Φ (0) + (˜r′)S (1)(˜r′)d˜r′ = 5 4˜r − 17 12 + log(˜r) + ˜r 6 . (4.62...

1920

[7] [6]

For ℓ = 2, the zeroth order solutions are given by Eqs

Axial gravitational ﬁeld response In this section, we compute the TLNs for the axial gravitatio nal ﬁeld response for the ZLMY-II model. For ℓ = 2, the zeroth order solutions are given by Eqs. (4.39) and (4 .40), and we can compute the following integration I[S (1)(˜r)] = − 1 5 ∫ ˜r 1 Φ (0) + (˜r′)S (1)(˜r′)d˜r′ = ˜r3 10 + 7˜r2 20 − 21˜r 20 + 3 5 . (4.68)...

[8] [7]

Scalar ﬁeld response Here, we compute the TLNs in response to the scalar ﬁeld pertu rbation for the ABV model. For s = 0 and ℓ = 0, one can compute the ﬁrst order correction to the TLN from t he following integration I[S (1)(˜r)] = − ∫ ˜r 1 Φ (0) + (˜r′)S (1)(˜r′)d˜r′ = − 1 4 + 1 4˜r , (4.79) implying that κ0(1) 0 = −1/4. This TLN is negative and has no l...