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

Anzhong Wang; Geeth Ongole; Parampreet Singh

REVIEW 2 major objections 5 minor 60 references

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

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Loop quantizing Schwarzschild-de Sitter with constant polymerization parameters always produces an extra black-hole-like horizon at low curvature, even though all curvature invariants stay finite.

desk verdict A clean, parameter-independent argument that constant-polymerization loop quantum black holes fail for Λ>0; the main caveat is the effective-description assumption, which the authors themselves flag. read the letter →

arxiv 2502.09718 v1 pith:O3C45T6G submitted 2025-02-13 gr-qc math-phmath.MP

Geeth Ongole , Parampreet Singh , Anzhong Wang This is my paper · ORCID

classification gr-qcmath-phmath.MP PACS 04.60.Pp04.70.-s

keywords loopquantumgravitySchwarzschild-deSitterKantowski-Sachsgaugepolymerizationcosmologicalconstantsingularityresolutioneffectivespacetimeblackholehorizon

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 asks whether loop quantization of Schwarzschild black holes survives the addition of a cosmological constant when the quantization is done in the Kantowski-Sachs gauge with polymerization parameters held constant. The answer it reaches is no for a positive cosmological constant: in the effective spacetime description a new black-hole-like horizon always forms far from the central singularity, in a region where the curvature is tiny and quantum effects would be expected to be negligible. All curvature invariants remain finite at that horizon, which appears purely because the classical connection variable grows exponentially and inevitably crosses the point $\delta_b b = \pi$ where the polymerization sine vanishes. For a negative cosmological constant, by contrast, the quantization behaves like the well-studied case without a cosmological constant: the central singularity is replaced by a regular transition surface and no spurious horizon appears. The paper concludes that constant-polymerization schemes are incompatible with the Kantowski-Sachs gauge when $\Lambda > 0$, a limitation of the same kind already known for the constant-parameter $\mu_o$ scheme in loop quantum cosmology.

What carries the argument

The central object is the polymerized effective Hamiltonian obtained by the replacements $b \to \sin(\delta_b b)/\delta_b$ and $c \to \sin(\delta_c c)/\delta_c$ in the Kantowski-Sachs Hamiltonian of Schwarzschild with a cosmological constant, with $\delta_b$ and $\delta_c$ constant along dynamical trajectories. Its lapse $N = \gamma\delta_b \sqrt{|p_c|}/\sin(\delta_b b)$ and the null expansions $\Theta_\pm = -\dot p_c \sin(\delta_b b)/(\sqrt{2}\,\gamma\delta_b p_c^{3/2})$ make the condition $\sin(\delta_b b)=0$ a horizon condition: $N^2$ diverges and $\Theta_\pm = 0$ while curvature invariants stay finite. The load-bearing identity is $\delta_b b(T_{BH}) = \pi$, which is guaranteed for $\Lambda>0$ because the classical connection in this gauge grows as $e^T$ in the asymptotic region.

What would settle it

Evolve the full loop quantum Hamiltonian constraint for the same Kantowski-Sachs model with $\Lambda>0$ and constant polymerization parameters from a sharply peaked semiclassical state, and check whether the peak follows the effective trajectory up to $\delta_b b = \pi$. If the state does not track the effective description, or if the expectation value of $\sin(\delta_b b)$ never vanishes, the predicted extra horizon would not be a genuine feature of the loop quantization.

Watch

Extended reading notes

Core claim

The central claim is that the failure is structural, not a parameter-tuning accident. In the effective dynamics, the far region of the Schwarzschild-de Sitter spacetime has $b(T) \simeq \gamma \sqrt{r_g^2 \Lambda/3}\, e^T$, so for any constant $\delta_b$ there is a finite moment $T_{BH}$ with $\delta_b b(T_{BH}) = \pi$. At that moment $\sin(\delta_b b)=0$, $N^2 \to \infty$, and the null expansions $\Theta_\pm = 0$, while all curvature scalars, energy density, and pressures stay finite, which is the signature of a black-hole-like horizon. The same mechanism operates for $0<\Lambda<\Lambda_c$ outside the cosmological horizon, at $\Lambda=\Lambda_c$ beyond the degenerate horizon, and for $\Lambda>\Lambda_c$ in the naked-singularity case, independent of the mass and of the particular values of $\delta_b,\delta_c$. Inside the classical black hole horizon the quantization is healthy: the singularity is replaced by a regular transition surface connecting a trapped region with a black hole horizon to an anti-trapped region with a white hole horizon, with negligible quantum corrections near a macroscopic horizon. The paper therefore claims the extra horizon is an artifact of the combination of Kantowski-Sachs gauge fixing with constant polymerization parameters.

Load-bearing premise

The load-bearing assumption is that the effective Hamiltonian faithfully describes the full loop quantum dynamics all the way to the low-curvature moment $\delta_b b = \pi$; if the true quantum evolution departs from the effective trajectory before that moment, the spurious horizon is an approximation artifact rather than a property of the quantization.

Editorial extensions

If this is right

For any positive cosmological constant, no adjustment of the constant values of $\delta_b$ and $\delta_c$ removes the extra black-hole-like horizon; the effect follows from constancy itself.
The quantization of the black hole interior remains singularity-free: a regular transition surface replaces the classical singularity, and for macroscopic black holes the quantum corrections near the horizon are small.
The failure is of the same type as the $\mu_o$-scheme recollapse of loop quantum cosmology with $\Lambda>0$, so it points to a general incompatibility between constant connection polymerization and de Sitter-like asymptotic regions.
For $\Lambda<0$ the same quantization scheme passes the test: the interior structure mirrors the $\Lambda=0$ case with a white-hole horizon and no additional horizon.

Reading between the lines

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

If the effective description is reliable, the result suggests that any symmetry-reduced loop quantization of Schwarzschild-de Sitter with connection polymerization must either let the polymerization parameters vary along trajectories or use a gauge in which the relevant connection component does not grow monotonically; the paper leaves this as an open question.
A natural next test is to repeat the analysis in a different interior gauge, such as a Gullstrand-Painlevé-type slicing, with constant polymerization parameters; finding no spurious horizon there would confirm the paper's diagnosis that the Kantowski-Sachs gauge choice is the trigger.
The same $\delta_b b=\pi$ mechanism could reappear in charged or rotating extensions whose homogeneous interiors have a connection component growing without bound; checking whether the extra horizon tracks the polymer scale would show how generic the obstruction is.

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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

Two things worth knowing about this one. First, the central result is genuinely new and, as far as I can tell, correct within its own assumptions: for Λ>0, any constant-δb, connection-polymerized effective model in the Kantowski-Sachs gauge produces an extra black hole-like horizon at low curvature, because b(T) tracks the classical exponential growth long enough that δb b hits π. The analytic argument in Eqs. (4.1)–(4.3) is simple, robust, and independent of the specific values of δb and δc, and the numerics back it up with the Hamiltonian constraint at 10^-13. Second, the paper is honest: it states in Section I that the effective spacetime description is assumed, and in Section IV it lists checking that assumption as an open problem. That is exactly the right place to put the caveat.

What the paper does well: it extends the constant-polymerization analysis to all Kantowski-Sachs regions of Schwarzschild–(A)dS, including the region outside the cosmological horizon and the Λ≥Λc naked-singularity cases, which earlier interior-focused studies like Ref. [45] did not cover. The contrast with Λ<0, where no such horizon appears, makes the Λ>0 pathology sharp and suggests it is a genuine feature of the scheme rather than an artifact of a particular parameter choice. The parallel with the failure of the µo scheme in LQC is drawn carefully.

The soft spots are real but not fatal. The biggest one is the one the authors flag: the inevitability of δbb=π is shown in the effective Hamiltonian, and if the quantum state delocalizes before the polymer pole, the prediction may not survive in the full quantum theory. I would have liked a more thorough discussion of why the effective description should be trusted all the way to that point, especially near the Brillouin-zone boundary. But given that this is the same approximation used throughout the LQC black-hole literature, and that the failure mode is exactly the µo-scheme failure mode, I don't think this prevents publication; it just means the strong headline should be phrased as “incompatibility at the level of effective dynamics.” The lack of error bars on the numerical horizon locations is minor, since the analytic argument carries the conclusion.

Who’s it for: anyone working on loop quantization of black holes, effective dynamics, or cosmological-constant extensions of LQBH models. It deserves a serious referee. I’d send it to review.

Referee Report

2 major / 5 minor

Summary. The paper analyzes the effective loop quantization of Schwarzschild black holes with a cosmological constant in the Kantowski-Sachs gauge, using constant polymerization parameters δb and δc. The authors solve the effective Hamiltonian equations numerically for Λ>0 (subcritical, critical, and supercritical with respect to Λc=1/(9m^2)) and for Λ<0. They find that the classical singularity is replaced by a regular transition surface in all cases, but for Λ>0 an additional black-hole-like horizon forms in a low-curvature region, far from the central singularity, at the moment δb b reaches π, where the lapse diverges and the null expansions vanish while curvature invariants remain finite. The paper interprets this as evidence that constant-polymerization schemes are incompatible with the Kantowski-Sachs gauge in the presence of a positive cosmological constant, in analogy with the μo-scheme problems in loop quantum cosmology, whereas the Λ<0 case is free of this feature.

Significance. If the effective-dynamics argument is accepted, the paper provides a sharp, largely parameter-independent obstruction: for any Λ>0 and any fixed δb>0, the classical growth b ∝ e^T drives δb b to π, producing an unphysical-looking additional horizon at low curvature. The analytic derivation in Eqs. (4.1)-(4.3) is simple and robust, and the numerical solutions are carefully checked by monitoring the Hamiltonian constraint to order 10^-13. The systematic tables for different masses and values of Λ are useful, and the negative-Λ result gives a clean contrast that sharpens the lesson. The main caveat is that the central claim is formulated within the effective spacetime description, whose validity up to the pole at sin(δb b)=0 is assumed rather than demonstrated.

major comments (2)

[Section I, Section IV, Eqs. (4.1)-(4.3)] The central claim that constant-polymerization, connection-polymerized loop quantization is incompatible with the Kantowski-Sachs gauge for Λ>0 is established only inside the effective Hamiltonian dynamics, whose validity up to the pole at sin(δb b)=0 is assumed (Section I: 'In this manuscript we assume the validity of the effective spacetime description') and is explicitly left unchecked in Section IV. The cited numerical support [46,47] concerns effective dynamics in LQC models and does not cover the present model near the Brillouin-zone boundary with Λ>0. Since the predicted horizon occurs exactly at δb b=π, the possibility that the full quantum dynamics ceases to be sharply peaked before this point means the extra horizon could be an artifact of the approximation rather than a consequence of the quantization scheme. The paper should either provide evidence from full loop quantum dynamics (for example, sharply peaked states following the effective trajectory up to the pole) or qualify the abstract and conclusion to 'within the effective spacetime description'.
[Section IV, Eq. (4.2)] The analytic inevitability argument assumes b(T) ≃ b_GR(T) up to the moment δb b = π. This is not automatic: the effective equation (3.4) contains terms proportional to 1/sin(δb b), which diverge at the pole, so the approximation b ≈ b_GR must be justified precisely in the regime where quantum corrections are no longer small. The numerical solutions support the claim, but the paper does not provide a quantitative criterion (for example, a bound on |b_eff - b_GR| or a statement about the domain of validity of the expansion) before invoking Eq. (4.2) to conclude that the horizon is 'always' formed.

minor comments (5)

[Figure 1 caption] The caption states that the blue dashed curve in panel (a) represents Λ < Λc and has no horizons; from the text and from the figure this should be Λ > Λc, since for Λ < Λc two horizons exist.
[Eq. (2.17)] The symbol co in the first integration constant is not defined in the surrounding text, and it does not appear in the subsequent expressions in Eq. (2.18); this appears to be a typographical error that should be corrected.
[Tables I-VI] The horizon locations are quoted to four to six decimal places without numerical error estimates. Since the solutions approach poles where N^2 diverges, a brief statement of the ODE solver tolerances and the resulting uncertainty in TBH would help readers assess the precision of the tables.
[Section III and Section IV] The paper claims that any other choice of constant polymerization parameters will give the same results, but the numerical tables use only two parameter sets. The analytic argument in Eqs. (4.1)-(4.3) supports the claim for δb>0, but the text should make explicit that the numerical demonstration is illustrative and that the universality is analytic.
[Section III.B.2 and Section IV] The phrase 'not physical and must be discarded' is a methodological judgment that goes beyond what the effective spacetime description can say. Within the effective description, the additional horizon is a well-defined prediction; consider phrasing such as 'a pathology of the scheme' to separate the formalism's output from the interpretive conclusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the extra-horizon result follows from the effective equations and the gauge choice; self-citations to AOS and effective-dynamics numerics are not load-bearing because the conclusion is explicitly independent of the cited parameter values.

full rationale

The central derivation is self-contained. The effective Hamiltonian (3.2)-(3.7) is solved with initial data matched to GR (3.8)-(3.9), and the extra horizon is derived from the asymptotic classical solution b(T) ~ γ r_g sqrt(Λ/3) e^T (Eq. 4.1), which forces δb b(TBH)=π (Eq. 4.2) for any fixed positive δb. No parameter is fitted to the predicted horizon; the AOS values (δb,δc) are used only for representative numerics and the paper states that any constant choice gives the same result. Citations [21,22,46,47] are self-citations (P. Singh and prior AOS work), but they are not the logical load: the paper explicitly assumes validity of the effective description ('In this manuscript we assume the validity of the effective spacetime description') and even lists as an open question whether that assumption breaks down for Λ>0. That is an acknowledged limitation, not a circular step. The only 'by construction' element — N²→∞ at sin(δb b)=0 via the lapse choice (3.3) — is the very feature under investigation, namely a property of constant-polymerization Kantowski-Sachs schemes, not a hidden reuse of the conclusion. Score 2 reflects minor non-load-bearing self-citations; no circular reduction is present.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the effective dynamics approximation and on the constant-polymerization map; both are stated openly. No new entities are introduced. The only numerical inputs are the AOS polymerization parameters, whose specific values the paper argues are irrelevant to the main conclusion.

free parameters (3)

δb (polymerization parameter for b) = 0.1172 for m=10^4, 0.0252406 for m=10^6
Constant polymerization parameter for the connection component b, fixed using the AOS procedure [21,22]. The paper argues the central result is independent of its value.
δc (polymerization parameter for c) = 0.0126 for m=10^4, 0.0027187 for m=10^6
Constant polymerization parameter for the connection component c, fixed using the AOS procedure [21,22]. The paper argues the central result is independent of its value.
rg for Λ>Λc = rg=2m (chosen by hand)
When no classical black hole horizon exists, rg is a numerical constant used to set initial data; authors state the choice does not affect qualitative results.

assumptions (6)

domain assumption The effective spacetime description accurately captures the underlying loop quantum dynamics in the symmetry-reduced model.
Invoked in Section I: 'In this manuscript we assume the validity of the effective spacetime description.' The central claim about the extra horizon is a property of this effective description.
domain assumption Polymerization replacement b→sin(δbb)/δb and c→sin(δcc)/δc with constant δb,δc is the correct quantization map.
Eq. (3.1) is the basis of the effective Hamiltonian (3.2); the paper studies exactly this class of schemes.
domain assumption The Kantowski-Sachs metric (2.1) faithfully represents all the Schwarzschild-(A)dS regions considered (τ<τBH, τ>τCH, τ>τDH, and the naked singularity region).
Section II states that in all regions where A(τ)>0 the spacetime can be cast in KS form; the quantization is applied region by region.
domain assumption The lapse choice (3.3) and gauge fixing (2.4), (2.14) do not change physical conclusions.
The paper uses gauge freedom to fix lapse and integration constants; the divergence of N² at sin(δbb)=0 is interpreted as a horizon.
domain assumption A viable loop quantization must recover general relativity in the infra-red or low-curvature limit.
Used in Section IV to classify the low-curvature extra horizon as unphysical; this is a standard consistency criterion from LQC.
domain assumption Barbero-Immirzi parameter γ≈0.2375 from black hole thermodynamics in LQG.
Section II sets γ≈0.2375 following black hole thermodynamics in LQG; not fitted in this paper.

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Pith. "Pith review of Lessons from gauge fixing and polymerization of loop quantum black holes with a cosmological constant." pith.science (2026). https://pith.science/paper/O3C45T6G

@misc{pith2026250209718,
  author       = {Pith},
  title        = {Pith review of: Lessons from gauge fixing and polymerization of loop quantum black holes with a cosmological constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3C45T6G}},
  note         = {Machine review of arXiv:2502.09718}
}

abstract

Loop quantization of Schwarzschild black holes with a cosmological constant for polymerization parameters which are constant is studied in the effective spacetime description. We show that for the positive cosmological constant there can be an appearance of large quantum effects at small spacetime curvatures. These effects can manifest as an additional black hole horizon. While the central singularity is resolved in all the cases, these limitations demonstrate incompatibility of the Kantowski-Sachs gauge and schemes with fixed polymerization parameters in the presence of a positive cosmological constant. In contrast, the case of a negative cosmological constant is free of such problematic features. Noted limitations are similar to those in the $\mu_o$ scheme for the loop quantization of cosmological models.

Figures

Figures reproduced from arXiv: 2502.09718 by the authors.

**Figure 1.** (b), and the spacetime inside this horizon can also be cast in the Kantowski-Sachs form (2.1) using the gauge freedom. As a result, loop quantization of black holes can be applied to all the regions in which the corresponding metric can be written in the Kantowski-Sachs form (2.1). In particular, these include the regions τ > τCH in the cases 0 < Λ ≤ Λc. Loop quantization of the Schwarzschild spacetime with a cosmol… view at source ↗

**Figure 2.** FIG. 2. The effective dynamics of the phase space variables and metric components for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Curvature invariants such as the Kretschmann scalar [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. Plots of the phase space variables [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. The dynamics of the phase space variables in the black hole interior ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. Curvature invariants such as the Kretschmann scalar ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. The dynamics of the phase space variables in the region [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Curvature invariants such as the Kretschmann scalar ( [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9. The dynamics of the phase space variables in the region [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. Curvature invariants such as the Kretschmann scalar ( [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: FIG. 11. The dynamics of the phase space variables in the region [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: FIG. 12. Curvature invariants such as the Kretschmann scalar ( [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: FIG. 13. The dynamics of the phase space variables in the region [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

**Figure 14.** Figure 14: FIG. 14. Curvature invariants such as the Kretschmann scalar ( [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

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

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Regular black holes and their relationship to polymerized models and mimetic gravity

K. Giesel, H. Liu, P. Singh, and S. A. Weigl, (2024), arXiv:2405.03554 [gr-qc]

work page Pith review arXiv 2024

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Using the gauge freedom (2.4), without loss of the gen- erality, we can always set TBH(τBH) = 0, a condition that we shall adopt in this subsection

Spacetime in the Region T < TBH Classically, the black hole horizon is given by Eq.(2.20). Using the gauge freedom (2.4), without loss of the gen- erality, we can always set TBH(τBH) = 0, a condition that we shall adopt in this subsection. We numerically solve the system of Eqs.(3.4) - (3.7) for various choices of the mass parameter m and the cosmological...

work page

[3] [2]

On the other hand, in the region T < TT , b(T ) re- mains almost constant for T ≳ TWH (≃ −22.1844), but suddenly drops to zero at T ≃ TWH

Then, as T continuously increases, there will exist a moment TBH such that δbb(TBH) = π, at which the lapse function N (∝ 1/ sin(δbb) diverges, whereby a black hole horizon is formed. On the other hand, in the region T < TT , b(T ) re- mains almost constant for T ≳ TWH (≃ −22.1844), but suddenly drops to zero at T ≃ TWH. Note that as T → TWH, we have N 2(...

work page

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Numerical stud- ies of the spacetime with a choice of Ti, where Ti > TCH are carried out and the results for m = 104, Λ/Λc = 0.9 are shown in Figs

Spacetime in the Region T > TCH In this subsection, we study quantum geometric effects on the spacetime in the region T > TCH. Numerical stud- ies of the spacetime with a choice of Ti, where Ti > TCH are carried out and the results for m = 104, Λ/Λc = 0.9 are shown in Figs. 7 and 8. From Fig. 7 we can see the dynamics of the phase-space variables to the r...

work page

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Quantum Geom- etry and Black Holes,

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9 and 10

Spacetime in the Region T < TDH For comparison with the cases studied earlier, we also choose m = 10 4 in this case and present the results in Figs. 9 and 10. From these figures, we can see that the properties of the corresponding spacetime are very similar to the case Λ < Λc in the region inside the black hole horizon. In particular, the central singular...

work page

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11 and 12

Spacetime in the Region T > TDH In this region, we show the physical quantities for m = 10 4 in Figs. 11 and 12. From these figures we can see that the properties are similar in nature to the T > TCH in the Λ < Λc case. In particular, in addition to the presence of a degenerate horizon, a black hole-like horizon is always developed at TBH = 4 .7267, at wh...

work page

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