Loopy Black-Hole Remnants

Asier Alonso-Bardaji

REVIEW 3 major objections 4 minor 14 references

Loopy Black-Hole Remnants

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read In a covariant effective model of loop quantum gravity, black holes evaporate to an asymptotic minimum horizon radius, leaving a stable Planck-scale remnant of mass about 20.94 micrograms.

desk verdict A clean evaporation calculation inside an effective LQG model, but the headline remnant mass rests on an asserted identification with the area gap and Eq. (6) has an algebraic error. read the letter →

arxiv 2508.21159 v1 pith:JI6EGNXA submitted 2025-08-28 gr-qc

Asier Alonso-Bardaji This is my paper · ORCID

classification gr-qc MSC 83C5783C4583C75 PACS 04.70.Dy04.60.Pp

keywords loopquantumgravityblackholeremnantsHawkingevaporationminimumareagapBarbero-ImmirziparameterPlanck-massrelicdarkmattercandidatethirdlawofblack-holethermodynamics

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

This paper argues that in a covariant effective model of spherical loop quantum gravity, black holes cannot evaporate to zero size: their horizons asymptotically approach a minimum area set by the area gap of loop quantum gravity. It derives an evaporation law from Stefan-Boltzmann emission and shows that the final state has zero temperature and zero entropy, so by the third law of black-hole thermodynamics it is reached only after infinite time. The result is a stable Planck-scale remnant whose asymptotic mass is fixed by the Barbero-Immirzi parameter, about 20.94 micrograms. If correct, this turns the usual endpoint of Hawking evaporation into a lingering, gravitationally interacting particle that is a candidate dark-matter constituent.

What carries the argument

The load-bearing object is the one-parameter generalization of Schwarzschild (Eq. 1) in which the singularity is replaced by an asymptotic boundary T at radius r0, reached only at infinite affine parameter. Two identities carry the argument: r0/rg = λ²/(1+λ²), making quantum corrections grow as the horizon shrinks, and the area-gap identification r0² = √3 γ, which converts the model's minimum area infimum into the smallest eigenvalue of the loop-quantum-gravity area operator. The evaporation law (Eq. 5), derived from perfect black-body emission and the temperature (2), is the dynamical engine: it gives a monotone decrease of rg that asymptotes to r0, producing the infinite-time remnant.

What would settle it

Observing a remnant whose mass deviates from 20.94 μg—or an independent determination of the Barbero-Immirzi parameter that makes the predicted mass untenable—would falsify the mass formula. A more complete quantum-gravity calculation showing that the horizon reaches r0 (or crosses below it) in finite time would falsify the infinite-time remnant claim.

Watch

Extended reading notes

Core claim

The central claim is that within the effective covariant model described by the line element (1), the quantization of area in loop quantum gravity manifests as a lower bound r0 on the horizon radius of nonsingular quasi-static black holes. Hawking temperature (2) and entropy (4) both vanish as rg → r0, and the Stefan-Boltzmann evaporation law (5) makes rg decrease monotonically toward r0 without ever reaching it in finite time. Identifying r0² with the smallest nonzero eigenvalue of the loop-quantum-gravity area operator, r0² = √3 γ, fixes the remnant mass M0 = (3/2)r0 ≈ 0.9621 mP ≈ 20.94 μg. The paper presents this as proof of the intuition that quantized area forbids horizons from shrinkin

Load-bearing premise

The paper assumes that the model's minimum-area radius r0 equals the smallest nonzero area eigenvalue of loop quantum gravity via r0² = √3γ, and that the quasi-static black-body evaporation law remains valid all the way down to that radius; if either fails, the remnant mass and the infinite-time conclusion do not follow.

Editorial extensions

If this is right

Black holes in this model end not in a singularity but in a Planck-scale remnant; no naked singularity forms.
The remnant mass is fixed by the Barbero-Immirzi parameter; for γ = 0.2375 it is about 20.94 μg, and exactly one Planck mass would correspond to γ = 0.2566.
Because the final state has vanishing temperature and entropy, its formation takes infinite time—evaporation effectively stalls near the minimum radius.
Astrophysical black holes have a tiny polymerization parameter in this scheme, so quantum corrections remain negligible in low-curvature regions while the interior stays singularity-free.
The remnants are metastable at worst (decay time scales as Rg⁴) and are plausible long-lived dark-matter candidates; detecting them could measure γ.

Reading between the lines

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

The identification r0² = √3 γ is a fragile step: a different quantization or a different mapping from the model's infimum to the area gap would shift the remnant mass, so the predicted 20.94 μg is as much a test of that identification as of the evaporation model.
The paper stops at the effective model's validity boundary (λ diverges as rg → r0); the infinite-time conclusion is a statement about the effective dynamics, not about the full quantum-gravity regime that would govern the last instants.
If remnants exist with this mass and interact only gravitationally, their abundance could be constrained by microlensing or gravitational-wave searches, providing a concrete way to probe the dark-matter scenario.
The same mechanism of an asymptotic minimum radius might apply to charged or rotating black holes, where an analogous r0 would modify the inner horizon and the final evaporation stage.

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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

Notable paper with a real new calculation and an overreach in the punchline. The evaporation analysis is sound: given the temperature (2) and Stefan-Boltzmann, the flow (5) integrates to an infinite time as rg→r0, so the model's minimal radius is not reached in finite time. That part is a clean, reproducible result. The paper is also honest about the quasi-static approximation and the λ→∞ boundary, and the entropy match to spin-network results in the large-mass limit at least shows the model connects to LQG in a known regime.

But the quantitative remnant mass does not stand on the same footing. The chain from the model to M0 depends entirely on the identification r0²=√3γ, which is asserted, not derived: the model's minimum-area boundary is a continuous parameter of an effective metric, and the area operator's smallest eigenvalue is a kinematical quantum geometry result. Equating them is a physical assumption. If you don't buy it, the rest of the numerology, including the 20.94 μg, is free. The paper should say so explicitly.

There's also an algebraic slip in Eq. (6): the displayed expression sqrt(243/16 γ²) mP does not equal 0.9621 mP for γ=0.2375; it gives 0.9256. The correct formula is M0=(3/2)(√3γ)^{1/2}mP. That's a typo-level error, but it matters because the number is the paper's headline.

Two smaller things. The abstract calls the remnants "stable" while the text admits they are metastable at worst, with a cited decay half-life. And the entropy constant S0 is set to zero by hand; the minimum entropy argument needs that choice. None of these are load-bearing for the infinite-evaporation-time claim, but they soften the dark-matter conclusion.

Who gets value? People working on effective LQG black holes and remnants will want to read the evaporation analysis; the mass estimate is a target for future models, not a robust prediction. With the identification made explicit as a conjecture and Eq. (6) fixed, I'd be happy to cite it as an example of how remnant scenarios can emerge in covariant LQG. As it stands, it deserves a serious referee—the core calculation is checkable and the assumption is exactly what a referee should probe.

Referee Report

3 major / 4 minor

Summary. This essay studies the end state of black-hole evaporation in the covariant effective-LQG model of Ref. [8]. The spherically symmetric metric (1) contains a Killing horizon at r=rg and a non-singular asymptotic boundary at r=r0. The author computes the Hawking temperature (2), the Misner–Sharp energy (3), and the entropy (4), and identifies the boundary area 4πr0² with the smallest LQG area eigenvalue, obtaining r0²=√3γ. Assuming black-body emission and the quasi-static approximation, he derives the evaporation law (5), integrates it analytically, and shows that the horizon radius approaches r0 only as τ→∞. The corresponding 'remnant' has zero temperature, zero entropy (after choosing S0=0), and asymptotic mass M0=3r0/2≈20.94 μg for γ=0.2375. The paper concludes that stable, metastable-at-worst Planck-scale remnants form and are a candidate for dark matter, and that their observation would measure γ.

Significance. The paper's analytic demonstration is a useful contribution: starting from a concrete covariant polymerization, it shows explicitly that the horizon can shrink to a finite minimal area and that the evaporation time diverges there. The entropy formula interpolating between the classical area law and full spin-network results is also suggestive. The integration leading to the infinite evaporation time is correct and can be reproduced. The quantitative remnant mass, however, is not a parameter-free prediction: it rests on the assumed equality between the model's r0 and the LQG area gap, and on a particular choice of mass variable. These caveats are acknowledged in parts of the text, but the abstract and conclusion state the 20.94 μg value without them. If the identification is accepted, the paper gives a clean and potentially testable relation between remnant mass and γ.

major comments (3)

[After Eq. (4)] The identification r0² = √3γ is an additional input, not a consequence of the model. Nothing in the metric (1) or in the construction of Ref. [8] forces the classical infimum area 4πr0² to equal the smallest LQG area eigenvalue 4π√3γ; r0 is a continuous parameter of an effective metric, while the area gap belongs to the full quantum kinematics. All of the paper's quantitative results – Eq. (6), the 20.94 μg mass, and the value γ=0.2566 for a one-Planck-mass remnant – depend on this step. The manuscript should either derive this identification or explicitly label it as a physical assumption and show how the remnant mass changes if it is relaxed.
[Eq. (5) and following] The proof that the remnant cannot be realized in finite time assumes that Eq. (5), with A=4πrg² and T from (2), remains valid for every rg>r0. But the paper itself states that as rg→r0 the polymerization parameter λ² = r0/(rg−r0) diverges and 'marks the boundary of validity for the model.' Thus the infinite-time conclusion is an extrapolation of the effective model beyond its controlled regime. This does not invalidate the result as a conditional statement, but the abstract's word 'prove' and the claim that the horizon 'attains' the LQG area gap should be softened to 'approaches in the formal model'.
[After Eq. (3)] The remnant mass M0=3r0/2 follows from choosing the asymptotic value of the Misner–Sharp energy, E→rg/2+r0, as 'the mass.' The paper acknowledges that Einstein's equations are not satisfied, so the usual equivalence between mass notions is lost. For metric (1), g_tt→1−(rg−r0)/r while g_rr→1+(rg+2r0)/r, so other natural asymptotic quantities (e.g., the coefficient of g_tt) would give a different remnant mass, namely (rg−r0)/2, which vanishes at rg=r0. Since the numerical headline depends on this convention, the choice should be justified physically and its ambiguity acknowledged.

minor comments (4)

[Eq. (6)] The fourth-root notation in Eq. (6) is easy to misread as a square root; please ensure the typesetting clearly shows ∛[4]{(243/16)γ²}. The numerical value 0.9621 is correct for the fourth root with γ=0.2375.
[Eq. (4)] The zero-entropy claim relies on setting the integration constant S0=0. Please state explicitly that this is a convention and that the entropy is determined up to this constant; the mass prediction is unaffected.
[Before Eq. (5)] The word 'prove' is stronger than what is shown. 'We show under the stated assumptions' would be more accurate, given the quasi-static and black-body assumptions and the model's validity boundary.
[Penultimate paragraph] The statement 'the estimated half-life for such processes is Rg^4' appears without derivation; please clarify that this is the estimate from Ref. [11] if that is the source.

Circularity Check

1 steps flagged · score 6.0 of 10

Remnant mass is the assumed area-gap identification restated in mass units; evaporation-to-r0 itself is model-derived.

self definitional [After Eq. (4) ('Once this connection is established...') through Eq. (6)]
"By identifying this area gap with our infimum, we find r2 0 = √ 3γ. In turn, the polymerization parameter becomes a running constant, varying over the space of solutions (like the mass does in Schwarzschild) as λ2 = r0/(rg − r0). ... The mass of black-hole remnants (those with rg = r0), as measured by asymptotic observers, is thusM0 := 3 2 r0. Exploiting the relation between r0 and loop quantum gravity found above, we obtain M0 = 4 r 243 16 γ2 mP ≈ 0.9621mP = 20.94µg"

In metric (1), r0 is a free length parameter inherited from the polymerization scheme; its boundary area 4πr0² is not forced by LQG kinematics. The paper stipulates 'By identifying this area gap with our infimum', i.e. sets r0²=√3γ as an input. Then M0=(3/2)r0 and Eq. (6) are just this input converted to mass units. The advertised 20.94 μg therefore reduces by construction to the assumed identification plus the externally chosen γ; it is not an independent prediction. The evaporation asymptotics to rg→r0 (Eqs. (2),(5)) do not depend on this identification, so the circularity is confined to the numerical remnant-mass claim.

full rationale

The main evaporation chain is self-contained: temperature (2) follows from surface gravity of metric (1), entropy (4) from the first law, and the evaporation law (5) from Stefan-Boltzmann; integrating (5) gives the asymptotic approach to r0 and the infinite-time conclusion without any appeal to the LQG area gap. Self-citations to [8,9] provide the effective model but the evaporation analysis is new, so they are not load-bearing circularity. The one genuine reduction-by-construction is the mass estimate: r0²=√3γ is an asserted identification, not derived, and M0=20.94 μg is a direct rescaling of that assertion. This makes the quantitative mass 'prediction' circular even though the remnant-formation claim has independent content. Score 6 reflects that one central numerical prediction is forced by its input, while the core dynamical statement is not.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The model has a free length scale r0 that is then identified with the LQG area gap, making the mass prediction dependent on this identification and on the Barbero-Immirzi parameter. The entropy constant S0 is set to zero by hand. No new fundamental entities are invented beyond the remnant itself, which lacks independent evidence.

free parameters (3)

r0 (minimum-area length scale) = r0 = (√3 γ)^{1/2} l_P ≈ 0.641 l_P for γ=0.2375
The length scale r0 in the metric (1) is a free parameter of the effective model; it is set to the LQG area gap via the identification r0² = √3 γ.
S0 (entropy integration constant) = 0
The integration constant in Eq. (4) is set to zero by hand to satisfy the third law.
γ (Barbero-Immirzi parameter) = 0.2375
Taken from Ref. [12] as a fixed constant; not fitted here, but the remnant mass depends directly on it.

assumptions (4)

domain assumption The effective LQG model of Ref. [8] correctly describes nonsingular spherical black holes
The metric (1) is taken as the starting point; its derivation is in prior work by the same author.
ad hoc to paper The infimum area 4π r0² equals the smallest positive eigenvalue of the LQG area operator
Stated after Eq. (4): 'By identifying this area gap with our infimum, we find r0² = √3 γ.' This identification is not derived within the paper.
domain assumption The quasi-static approximation and Stefan-Boltzmann law remain valid until rg → r0
Acknowledged in the text: 'Assuming that the quasi-static approximation remains valid, thermal emission produces stable objects...'
domain assumption γ = 0.2375 from entropy matching is correct
Uses the widely accepted value from Ref. [12].

invented entities (1)

Black-hole remnant (minimum-area final state)
purpose: Stable end-state of evaporation, dark-matter candidate
The paper predicts these objects but provides no observational handle beyond the mass estimate; detection proposals are cited from other work.

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Cite this review

Pith. "Pith review of Loopy Black-Hole Remnants." pith.science (2026). https://pith.science/paper/JI6EGNXA

@misc{pith2026250821159,
  author       = {Pith},
  title        = {Pith review of: Loopy Black-Hole Remnants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JI6EGNXA}},
  note         = {Machine review of arXiv:2508.21159}
}

abstract

The quantized area predicted by loop quantum gravity suggests the existence of a lower bound for black-hole horizons. We prove this intuition within a covariant effective model for spherical loop quantum gravity, where nonsingular quasi-static black holes evaporate until their horizons attain the smallest positive eigenvalue of the area operator. Consistent with the third law of black-hole thermodynamics, this final state -- characterized by vanishing temperature and entropy -- cannot be realized in finite time. The process thus leads to the formation of stable remnants, whose estimated masses are approximately 20.94$\mu$g, lying in the Planck regime.

Figures

Figures reproduced from arXiv: 2508.21159 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

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Bianchi, M

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [8]

Ashtekar, J

A. Ashtekar, J. Olmedo and P. Singh, Quantum extension of the Kruskal spacetime. Phys. Rev. D 98, 126003 (2018)

work page 2018

[2] [1]

classical

More precisely (and removing the zero-energy null geodesics lying on H), the interval s ∈ R is the preimage of r ∈ (r0, ∞) (for the 2 FIG. 1: Conformal diagram of the maximally extended spacetime [8, 9]. The horizon H is drawn in red, and the novel boundaries T, which replace the singularity, are depicted by the dashed purple lines. Dashed gray lines repr...

[3] [2]

J. H. MacGibbon, Can Planck-mass relics of evaporating black holes close the universe? Nature 329 308-309 (1987)

work page 1987

[4] [3]

S. B. Giddings, Black holes and massive remnants. Phys. Rev. D 46, 1347-1352 (1992)

work page 1992

[5] [4]

Amadei and A

L. Amadei and A. Perez, Planckian discreteness as seeds for cosmic structure. Phys. Rev. D 106, 063528 (2022)

work page 2022

[6] [5]

Perez, C

A. Perez, C. Rovelli and M. Christodoulou, Detecting Gravitationally Interacting Dark Matter with Quantum Interference. Phys. Rev. Lett. 133, 111001 (2024)

work page 2024

[7] [6]

Bianchi, M

E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. M. Haggard and C. Rovelli, White Holes as Remnants: A Surprising Scenario for the End of a Black Hole. Class. Quant. Grav. 35, 225003 (2018)

work page 2018

[8] [7]

Rovelli and F

C. Rovelli and F. Vidotto, Planck stars, White Holes, Remnants and Planck-mass quasi- 8 particles. The quantum gravity phase in black holes’ evolution and its manifestations. arXiv:2407.09584 [gr-qc] (2024)

arXiv 2024