Loop quantum black hole

Leonardo Modesto

REVIEW

Loop quantum black hole

Leonardo Modesto

General Relativity and Quantum Cosmology · Submitted 20 Sep 2005 · 16 references

Pith's verdict: A real but modest computation — explicit Ashtekar-variable spectra for the Schwarzschild interior — wrapped around a dynamical claim the paper never actually demonstrates.

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arXiv IDgr-qc/0509078 (version 2)

Pith IDV3IKDWX7, Pith's permanent record of this paper and its review

CitationLeonardo Modesto. "Loop quantum black hole." arXiv:gr-qc/0509078 (2005). https://pith.science/paper/V3IKDWX7

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Worth a look for the computation, not for the conclusion it advertises. Modesto takes the Kantowski-Sachs (Schwarzschild interior) minisuperspace, sets it up in Ashtekar variables, and writes down the inverse-volume spectrum (34), the 1/sqrt|E| spectrum (37), and the Hamiltonian constraint as a difference equation (44). That is genuine, checkable work, and it is the first time this model is done in connection variables rather than ADM. The physical conclusion, though, is the same one he already published twice in [3] and [4]; the new formalism is the contribution, not the singularity resolution. He says so himself in the introduction, which I appreciate. The Thiemann regularization in section 1.4 is applied correctly and the resulting bounded spectra are what you'd expect. Nothing is invented, and the algebra is at a level a specialist can verify in an afternoon. Now the soft spots. The abstract claims the dynamics can be extended past the classical singularity (abstract:s4), but (44) is never solved, no boundary condition is imposed, and no physical state is constructed. The paper admits this — 'imposing a boundary condition we can obtain the wave function' — so it's an acknowledged gap, not a hidden one, but it means the headline claim is kinematical. The 'psi(0,0) decouples' line is also too narrow: from (45), beta(0, muE1) vanishes for every muE1, so the whole muE=0 slice drops out of the recurrence, and whether that lets you propagate through or instead forces fresh data on each side is exactly the question that needs answering. Second, eq (37) as printed does not reduce to the stated classical limit (38); the prefactor looks inverted. That's a checkable slip and it suggests the spectra weren't cross-checked against their own limits. Third, the curvature invariant is claimed finite by analogy to 1/sqrt|E| rather than computed, and the scheme dependence of the inverse-volume operator is not discussed at all. Who should read it: people working on loop quantization of black hole interiors, mainly as a stepping stone and as a comparison point for the later Ashtekar-Bojowald work. I would not cite it myself; the results are superseded and the errors make it a shaky reference. But it deserves referees — the computation is real and the errors are fixable.

Referee report

Major objections (4)

[§2.1, Eqs. (34)–(38)] The stated classical limits do not follow from the spectra as written. In the limit |µE|,|µE1| ≫ µ0, |µE1+µ0|^{1/2}−|µE1−µ0|^{1/2} ≈ µ0|µE1|^{-1/2} and |µE+µ0|^{1/4}−|µE−µ0|^{1/4} ≈ (µ0/2)|µE|^{-3/4}; inserting these into (37) with its prefactor √2/(µ0^2 lP |µE|^{1/4}|µE1|^{1/2}) gives an eigenvalue scaling as |µE|^{-1}|µE1|^{-1}, not the |µE|^{-1/2} of (38). The same check applied to (34)→(35) gives 4√2 lP |µE|^{-1/4}|µE1|^{-1/2} rather than 2√2 lP |µE|^{-1/4}|µE1|^{-1/2}, and as written (35) carries dimensions of length whereas 1/√V̄ should carry inverse length. Since the classical limit is the only check that these operators really represent 1/√V̄ and 1/√|E|, the prefactors and exponents in (34) and (37) need to be rechecked and corrected before conclusion 2 can be maintained.
[§1.4, Eq. (26); Conclusions, item 2] There is a mismatch between the operator computed and the observable claimed. §1.4 states that the quantity relevant to the curvature invariant is 1/r^2 = 1/b^2 = 1/|E|, but Eq. (26) and its quantization (36)–(37) compute sgn(E)sgn(E1)/√|E| = 1/b, not 1/|E|. Conclusion 2 then asserts that R_{µνρσ}R^{µνρσ} ~ 1/b^6 is singularity-free. Since 1/b^6 = 1/|E|^3, this requires the sixth power of the operator, which is not computed. The implication would follow only if the operator is diagonal on the basis (27) and the functional calculus applies, which the paper does not state. The paper should either compute the operator corresponding to 1/|E|^3 or explicitly argue why boundedness of 1/√|E| implies boundedness of the curvature invariant.
[§2.2, Eq. (44); Abstract, sentence s4] The central dynamical claim is asserted rather than shown. Equation (44) is written down, but no boundary condition is imposed, no solution is constructed, and no argument is given that the recurrence propagates data from µE>0 to µE<0. The statement that 'the state ψ(0,0) decouples from the dynamics' is also imprecise: from (45), β(0,µE1)=0 for every µE1, so the entire µE=0 slice drops out of the recurrence. That means the constraint does not determine the µE=0 data and does not by itself establish continuation through the singularity; whether the recurrence terminates and requires fresh data on each side is not analyzed. The paper should either construct a physical state and exhibit the continuation, or state the extendability claim as conditional.
[§1.3, Eqs. (21) and (39)] Only the Euclidean part H_E of the Hamiltonian constraint is quantized; the Lorentzian term, the second term of Eq. (4), is dropped without discussion. For the Kantowski–Sachs reduction this term is not obviously zero or proportional to the Gauss constraint, and the paper does not show that it is. Unless it vanishes identically for this ansatz, the states annihilated by (44) are not solutions of the full Hamiltonian constraint, and the dynamical conclusions drawn from (44) do not follow. The paper should either justify the omission or include the Lorentzian part.

Minor objections (12)

[§2.2, Eqs. (40) and (45)] The symbol β is used both for the constant µ0^2/2 in (40) and for the function β(µE,µE1) in (45); one of them should be renamed.
[Text following Eq. (44)] The sentence defining the functions reads 'α(µE,µE1) and α(µE,µE1)'; the second should be β.
[Throughout] There are numerous typographical errors: 'Kanthowski' in the abstract, 'Schwarzsc hild', 'sapec-time', 'Hiilbert', 'dinamics', 'well defines' for 'well defined', 'limite' for 'limit', and 'extend further the classical singularity' in the abstract and conclusions.
[Eq. (7)] The brackets in the expression for H_E are unbalanced and the equation cannot be parsed as printed.
[References] Reference [7] 'gr-qc/041112' appears truncated; reference [9] gives the page range '1747-1745', which is reversed; 'Igemar Bengtsson' should read 'Ingemar Bengtsson'; and 'quant - ph/01 10102' has stray spacing.
[§1.2–§2.1] The parameter µ0 enters the spectra (34), (37) and the recurrence (44), but its value and the µ0-independence of the physical conclusions are not discussed; the classical limit is taken at fixed µ0, which should be stated explicitly.
[§1.2, Eq. (12)] The choice R = lP fixes the symplectic structure ({A,E} = κlP) and hence the spectra; the R-dependence of the physical results should be stated.
[§2.2] The interpretation of ψ(µE,µE1) as the wave function of E1 'at the time E' is asserted without a deparametrization or a Dirac observable; this should be softened or justified.
[Conclusions] The claim that the 'other side' of the singularity corresponds to a domain where the triad reverses its orientation is asserted without derivation.
[§2.1, below Eq. (34)] The statement that the inverse-volume spectrum is 'upper bounded near the classical singularity' is weaker than what (34) shows: the eigenvalue vanishes at µE=0. This stronger and cleaner statement should be made directly.
[§1.2, Eq. (9)] The relation between the general Kantowski–Sachs metric (9) and the Schwarzschild interior (the specific relation between a(t) and b(t)) should be stated explicitly.
[§1.1–§2] The Immirzi parameter γ does not appear in the final spectra; the paper should state where it enters and why the results are γ-independent.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Quantizing the Schwarzschild interior in loop variables replaces the black hole singularity with a finite, extendable quantum evolution.

The reading

Inside a black hole horizon the Schwarzschild solution looks like a homogeneous but anisotropic cosmology of the Kantowski-Sachs type, and that is the model this paper quantizes. The author sets the model up in Ashtekar connection variables, builds holonomies along the three preferred directions, and rewrites the volume, the inverse volume, and the classically divergent quantity $1/

Original abstract

In this paper we consider the Kantowski-Sachs space-time in Ashtekar variables and the quantization of this space-time starting from the complete loop quantum gravity theory. The Kanthowski-Sachs space-time coincides with the Schwarzschild black hole solution inside the horizon. By studying this model we can obtain information about the black hole singularity and about the dynamics across the point r=0. We studied this space-time in ADM variables in two previous papers where we showed that the classical black hole singularity disappears in quantum theory. In this work we study the same model in Ashtekar variables and we obtain a regular space-time inside the horizon region and that the dynamics can be extend further the classical singularity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,252 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

In this paper we consider the Kantowski-Sachs space-time in Ashtekar variables and the quantization of this space-time starting from the complete loop quantum gravity theory. The Kanthowski-Sachs space-time coincides with the Schwarzschild black hole solution inside the horizon. By studying this model we can obtain information about the black hole singularity and about the dynamics across the point r=0. We studied this space-time in ADM variables in two previous papers where we showed that the classical black hole singularity disappears in quantum theory. In this work we study the same model in Ashtekar variables and we obtain a regular space-time inside the horizon region and that the dynamics can be extend further the classical singularity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we consider the Kantowski-Sachs space-time in Ashtekar variables and the quantization of this space-time starting from the complete loop quantum gravity theory.	contribution 0.84	0.84	0.04	asserted 0.72
s1	The Kanthowski-Sachs space-time coincides with the Schwarzschild black hole solution inside the horizon.	background 0.99	0.09	0.04	asserted 1.00
s2	By studying this model we can obtain information about the black hole singularity and about the dynamics across the point r=0.	interpretation 0.55	0.34	0.11	hedged possibility 0.81
s3	We studied this space-time in ADM variables in two previous papers where we showed that the classical black hole singularity disappears in quantum theory.	background 0.99	0.62	0.05	asserted 1.00
s4	In this work we study the same model in Ashtekar variables and we obtain a regular space-time inside the horizon region and that the dynamics can be extend further the classical singularity.	result 0.79	0.95	0.88	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (16)

[3]

Leonardo Modesto, Disappearance of the black hole singularity in loop quantum gravity, Phys. Rev. D 70 (2004) 124009, gr-qc/0407097

View on Pith Pith review arXiv 2004
[4]

Leonardo Modesto, The kantowski-Sachs space-time in loop quantum gravity , gr-qc/0411032

View on Pith Pith review arXiv
[1]

Abhay Ashtekar, New Hamiltonian formulation of general relativity , Phys. Rev. D 36 1587-1602

View on Pith
[2]

Carlo Rovelli, Quantum Gravity, (Cambridge University Press, Cambridge, 2004)

View on Pith 2004
[5]

M. Bojowald, Inverse scale factor in isotropic quantum geometry , Phys. Rev. D64 084018 (2001); M. Bojowald, Loop Quantum Cosmology IV: discrete time evolution , Class. Quant. Grav. 18, 1071 (2001); A. Ashtekar, M. Bojowald and J. Lewandowski, Mathematica structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7 (2003) 233-268, gr-qc/0304074

View on Pith Pith review arXiv 2001
[6]

R. Kantowski and R. K. Sachs, J. Math. Phys. 7 (3) (1966)

View on Pith 1966
[7]

Ioannis Raptis, Finitary-Algebraic ‘Resolution’ of the Inner Schwarzschi ld Singularity , gr-qc/0408045; Anastasios Mallios and Ioannis Raptis, Smooth Singularities Exposed: Chimeras of the Diﬀerential Spacetime Manifold , gr-qc/041112; Ioannis Raptis, ‘Iconoclastic’, Categorical Quantum Gravity, gr-qc/0509089

View on Pith Pith review arXiv
[8]

Igemar Bengtsson, Note on Ashtekar’s variables in the spherically symmetric c ase, Class. Quant. Grav. 5 (1988) L139-L142; Igemar Bengtsson, A new phase for general relativity? , Class. Quant. Grav. 7 (1990) 27-39; H.A. Kastrup & T. Thiemann, Spherically symmetric gravity as a complete integrable system, Nucl. Phys. B 425 (1994) 665-686, gr-qc/9401032; M....

View on Pith Pith review arXiv 1988
[9]

Luca Bombelli & R. J. Torrence Perfect ﬂuids and Ashtekar variables, with application to Kantowski-Sachs models Class. Quant. Grav. 7 (1990) 1747-1745

View on Pith 1990
[10]

A. Ashtekar, S. Fairhurst and J. Willis, Quantum gravity, shadow states, and quantum mechanics , Class. Quant. Grav. 20 1031-1062 (2003)

View on Pith 2003
[11]

C Rovelli, “Partial observables”, Phy Rev D65 (2002) 12 4013; gr-qc/0110035

View on Pith Pith review arXiv 2002
[12]

Viqar Husain and Oliver Winkler, On singularity resolution in quantum gravity ; gr-qc/0312094

View on Pith Pith review arXiv
[13]

H. Halvorson, Complementary of representations in quantum mechanics , Studies in History and Philosophy of Modern Physics 35, 45-56 (2004), quant - ph/01 10102

View on Pith 2004
[14]

T. Thiemann, Quantum Spin Dynamics , Class. Quant. Grav. 15, 839 (1998)

View on Pith 1998
[15]

T. Thiemann, Introduction to Modern Canonical Quantum General Relativi ty ; gr -qc/0110034; Lectures on Loop Quantum Gravity , gr-qc/0210094

View on Pith Pith review arXiv
[16]

Abhay Ashtekar & Martin Bojowald, Black hole evaporation : A paradigm Class. Quant. Grav. 22 (2005) 3349-3362, gr-qc/0504029 12

View on Pith Pith review arXiv 2005

Forward citations

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How the literature uses it

93 arXiv papers cite this in 145 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 78 sentences without a confident label

Contests it 2 papers

As in the early quantization of cosmological models in LQC, the quantizations of black hole spacetime given in Refs. [ref] suffer from lack of independence of the the auxiliary structure, such as the `size' of the fiducial cell, needed for the Hamiltonian description.

Loop quantization of the Schwarzschild interior revisited · Introduction · contest 0.87
This idea was follow up on in several papers already, however one often encountered physically insensible results [ref] or had to deviate from the effective Hamiltonians typically arising in loop quantum cosmology [ref] .

(b,v)-type variables for black to white hole transitions in effective loop quantum gravity · Introduction · contest 0.82

Uses its method or tool 2 papers

In order to derive the canonical equations, one specifies the lapse function as [ref] [equation] and denotes the corresponding time variable as $T$.

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Following [ref] , we take the classical limit of the polymer Hamiltonian ([ref]) by keeping the polymerization scale $μ$ fixed and making the replacement $U_μ→e^iμp$, where $p$ is the classical momentum.

Polymer quantization, singularity resolution and the 1/r^2 potential · Semiclassical polymer theory · 0.94

Compares with it 5 papers

This appears in contrast to treatments of the interior [ref] of the Schwarzschild space-time and may shed light on the treatment of the complete space-time where the issue of uniqueness is still not settled [ref] .

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While in both quantizations the singularity is replaced by a bounce, the former leads to a single-horizon solution whereas the latter exhibits a cyclic behavior with multiple horizons, reminiscent of the results of [ref] .

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Furthermore, we can compare to $μ_o$-schemes [ref] and generalised $μ_o$-scheme [ref] .

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Cites it as support 1 paper

This viewpoint has been reinforced by a general result obtained in an independent way by several quantum gravity modifications of black hole metrics [ref] : even at the terminal stage of the evaporation the probe of the shortest scales in the vicinity of the curvature singularities would not be possible for the formation of zero temperature black hole remnants, i.e., extremal configurations occurring also in the case of non-rotating, neutral black holes [ref] .

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Cites it as background 48 papers

The pioneer work that embarked on this task was Ref. [ref] , and several other works followed [ref] .

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In one, called the $μ_0$ scheme, $μ$ parameters are considered to be constants [ref] .

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Group quantization of the black hole minisuperspace · Introduction · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [3]

Leonardo Modesto, Disappearance of the black hole singularity in loop quantum gravity, Phys. Rev. D 70 (2004) 124009, gr-qc/0407097

View on Pith Pith review arXiv 2004

[2] [4]

Leonardo Modesto, The kantowski-Sachs space-time in loop quantum gravity , gr-qc/0411032

View on Pith Pith review arXiv

[3] [1]

Abhay Ashtekar, New Hamiltonian formulation of general relativity , Phys. Rev. D 36 1587-1602

View on Pith

[4] [2]

Carlo Rovelli, Quantum Gravity, (Cambridge University Press, Cambridge, 2004)

View on Pith 2004

[5] [5]

M. Bojowald, Inverse scale factor in isotropic quantum geometry , Phys. Rev. D64 084018 (2001); M. Bojowald, Loop Quantum Cosmology IV: discrete time evolution , Class. Quant. Grav. 18, 1071 (2001); A. Ashtekar, M. Bojowald and J. Lewandowski, Mathematica structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7 (2003) 233-268, gr-qc/0304074

View on Pith Pith review arXiv 2001

[6] [6]

R. Kantowski and R. K. Sachs, J. Math. Phys. 7 (3) (1966)

View on Pith 1966

[7] [7]

Ioannis Raptis, Finitary-Algebraic ‘Resolution’ of the Inner Schwarzschi ld Singularity , gr-qc/0408045; Anastasios Mallios and Ioannis Raptis, Smooth Singularities Exposed: Chimeras of the Diﬀerential Spacetime Manifold , gr-qc/041112; Ioannis Raptis, ‘Iconoclastic’, Categorical Quantum Gravity, gr-qc/0509089

View on Pith Pith review arXiv

[8] [8]

Igemar Bengtsson, Note on Ashtekar’s variables in the spherically symmetric c ase, Class. Quant. Grav. 5 (1988) L139-L142; Igemar Bengtsson, A new phase for general relativity? , Class. Quant. Grav. 7 (1990) 27-39; H.A. Kastrup & T. Thiemann, Spherically symmetric gravity as a complete integrable system, Nucl. Phys. B 425 (1994) 665-686, gr-qc/9401032; M....

View on Pith Pith review arXiv 1988

[9] [9]

Luca Bombelli & R. J. Torrence Perfect ﬂuids and Ashtekar variables, with application to Kantowski-Sachs models Class. Quant. Grav. 7 (1990) 1747-1745

View on Pith 1990

[10] [10]

A. Ashtekar, S. Fairhurst and J. Willis, Quantum gravity, shadow states, and quantum mechanics , Class. Quant. Grav. 20 1031-1062 (2003)

View on Pith 2003

[11] [11]

C Rovelli, “Partial observables”, Phy Rev D65 (2002) 12 4013; gr-qc/0110035

View on Pith Pith review arXiv 2002

[12] [12]

Viqar Husain and Oliver Winkler, On singularity resolution in quantum gravity ; gr-qc/0312094

View on Pith Pith review arXiv

[13] [13]

H. Halvorson, Complementary of representations in quantum mechanics , Studies in History and Philosophy of Modern Physics 35, 45-56 (2004), quant - ph/01 10102

View on Pith 2004

[14] [14]

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