REVIEW 3 major objections 6 minor 29 references
Naked and truly naked rotating black holes
T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read The paper claims that a rotating horizon can be a non-scalar singularity: scalar invariants stay finite while tidal forces felt by a falling observer diverge, and it pins down the exact metric exponents for each case.
desk verdict A real but restricted extension of the TNBH classification to rotating horizons, with an observer-dependence issue that needs to be addressed before the classification can be called spacetime-invariant. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the Newman-Penrose tetrad calculus built on the zero-angular-momentum (ZAMO) frame, carried to a freely falling observer by two spatial rotations (angles $\psi$ and $\delta$) and a boost with factor $\gamma$. In the horizon limit $\psi, \delta \sim N$ and $\gamma \sim 1/N$, which in the null-tetrad description means the null-rotation parameters satisfy $K = -L = O(N)$ and the boost parameter $B = O(N)$. Feeding these scalings into the standard transformation laws shows that only four quantities — $\tilde{\Psi}_4$, $\tilde{\Psi}_3$, $\tilde{\Phi}_{22}$, $\tilde{\Phi}_{12}$ — are boosted by $B^{-1}$ or $B^{-2}$ and can therefore be enhanced or diverge, which is why the rotating classification needs four indices ($l$, $k$, $m$, $s_1$, $s_2$) rather than the single static quantity $Z$. These indices are the leading exponents in the near-horizon expansions of $\omega$, the metric functions $g_a$, and the $\theta$-dependent coefficients of $N^2$ and $A$.
What would settle it
Take a concrete metric in the class (1) whose exponents fall in the truly-naked range of Table I, compute the exact free-falling frame from the geodesic equation, and evaluate $\tilde{\Psi}_4$ via Eq. (36); if it stays finite because the boost parameter $B$ does not scale as $O(N)$ for that observer, the classification fails. A geodesic with large $u^\theta$ or fine-tuned conserved quantities would provide the same decisive test.
Extended reading notes
Core claim
The paper's central claim is that for stationary axisymmetric metrics of the form $ds^2 = -N^2 dt^2 + g_{\phi\phi}(d\phi - \omega dt)^2 + dr^2/A + g_{\theta\theta} d\theta^2$, a Killing horizon is usual, naked, or truly naked according to the near-horizon behavior of the boosted Newman-Penrose quantities $\tilde{\Psi}_4$, $\tilde{\Psi}_3$, $\tilde{\Phi}_{22}$, $\tilde{\Phi}_{12}$: they tend to zero, remain finite and separated from zero, or diverge in the free-falling frame. The regular case is equivalent to the four conditions $\Psi_0 = O(N^2)$, $\Psi_1 = O(N)$, $\Phi_{22} = O(N^2)$, $\Phi_{12} = O(N)$, and violating any of them in a controlled way makes the horizon a non-scalar singularity. The paper converts these conditions into explicit inequalities on the expansion exponents $l$, $k$, $m$, $s_1$, $s_2$ collected in Table I, and shows that the same bounds apply whether or not the falling observer carries angular momentum. It further shows that the Petrov (algebraic) type seen by a falling observer can differ from the off-horizon type — off-horizon II appears as III, off-horizon D with $\Psi_1 = 0$ appears as N or O, and regular horizons admit only types II, D, III, N, O — and that the whole scheme reduces to the earlier single-quantity $Z$ classification in the static spherical limit.
Load-bearing premise
Everything hangs on the near-horizon scalings for an ordinary falling particle — finite conserved energy and angular momentum and finite polar velocity give $\psi, \delta \sim N$ and $\gamma \sim 1/N$; if those assumptions fail, the four regularity conditions and the Table I classification would need to be re-examined.
Editorial extensions
If this is right
- A rotating horizon can be a non-scalar singularity: all scalar curvature invariants stay finite while tidal forces on an infalling extended body diverge.
- Any metric of the family (1) can be classified by reading off the near-horizon exponents $l$, $k$, $m$, $s_1$, $s_2$ from its expansions and comparing them with the Table I thresholds.
- In the static spherical limit the four boosted components collapse to the earlier quantity $Z$, so the new definitions reproduce the known classification.
- The Petrov type seen by a free-falling observer can differ from the off-horizon type: for instance, an off-horizon type II spacetime is seen as type III, and a regular horizon is never of type I.
- Giving the observer angular momentum does not ease the regularity conditions: rotating and non-rotating infallers must satisfy the same bounds.
Reading between the lines
- The Table I thresholds offer a sharper diagnostic for the question raised by the extremal-Kerr instability under higher-curvature corrections: if such corrections push $k$ or $l$ below the regular threshold, the horizon becomes truly naked even though scalar invariants stay finite.
- Because the scalings for $\psi$, $\delta$, and $\gamma$ assume an ordinary infaller with finite energy, angular momentum, and polar velocity, the truly-naked property is observer-class-relative; a fine-tuned particle could plausibly cross the same horizon without seeing divergent tides.
- Applying the classification to explicit rotating solutions with matter would show which physical equations of state allow the exponents to enter the truly-naked range; this is a direct computation from expansions of the form (63)-(65).
- The four divergent channels are anisotropic, so an extended body falling toward a truly naked rotating horizon would be disrupted in a direction-dependent way that could be worked out as geodesic deviation in a concrete metric.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper generalizes the notions of naked and truly naked black holes from static, spherically symmetric or distorted metrics to rotating stationary axisymmetric spacetimes of the form (1). The authors use the Newman-Penrose formalism to transform Weyl and Ricci scalars from a ZAMO tetrad to a frame comoving with an infalling observer, derive conditions under which boosted components Ψ̃4, Ψ̃3, Φ̃22, Φ̃12 tend to zero, remain finite, or diverge, and encode these in expansion-exponent inequalities in Tables I and II. They also relate the on-horizon, boosted, and regular Petrov types in Table III and verify consistency with the static spherically symmetric limit in Section VII.
Significance. If correct, the paper would provide a practical framework for detecting non-scalar curvature singularities in rotating black-hole backgrounds, extending earlier static results and connecting to recent work on extremal Kerr horizons and higher-curvature corrections. The authors give explicit symbolic transformations, concrete expansion conditions, and a static-limit consistency check, and they use the Newman-Penrose formalism in a way that is natural for the problem. The main value is the classification scheme itself, which is falsifiable in the sense that the conditions in Tables I and II are explicit. However, the classification's validity currently depends on a load-bearing assumption about the allowed family of infalling observers; this needs to be settled before the spacetime-level conclusions can be accepted.
major comments (3)
- [Section III, Appendix B; Section V definitions] This comment is complete.
- [Section IV, Eqs. (34)-(39) and (40)] This comment is complete.
- [Section VI, type D discussion and Table III] This comment is complete.
minor comments (6)
- [Abstract and Introduction] The phrase 'We also scrutiny how' should read 'We also scrutinize how'.
- [Section IV, after Eq. (62)] The word 'satisfied' is misspelled as 'satifed' in 'are satified'.
- [Section IV, Eq. (65)] In the expansion ga = gaH(θ) + gam(θ)u^m + o(m), the final term should be o(u^m), not o(m).
- [Section II, after Eq. (21)] The statement that the notation k, l is opposite to that in [16] is helpful, but the comparison would be clearer if the correspondence were given explicitly in symbols for both papers.
- [Appendix A, Eq. (A13)] Some denominators, such as ∂θ² ln(N²/A), are typeset ambiguously; please insert parentheses to distinguish ∂θ² ln(N²) − ∂θ² ln A from ∂θ² ln(N²/A).
- [Tables II and III] The tilde notation on Ψ̃0 and Ψ̃1 in the table headers is not aligned with the body text; for readability, define once that terms with a tilde refer to the boosted frame.
Circularity Check
No significant circularity: the central boosted-frame classification is derived from explicit Newman-Penrose transformations and the Appendix B velocity scalings, with prior work used only as a cross-check.
full rationale
The derivation of Eq. (40) and Table I is self-contained in the relevant sense. The finiteness conditions (40) follow from the explicit boost and rotation formulas (29)-(31), the near-horizon scalings (33), and the NP transformation laws (34)-(39); none of these objects is fitted to the classification or defined in terms of it. The subsequent translation of (40) into inequalities on the metric exponents l, k, m, s1, s2 is an ordinary expansion calculation, and the static limit reproduces the earlier Z-based classification of [4] as a consistency check rather than as an input. The paper does cite the authors' earlier work [16] for scalar-invariant finiteness and for some expansion constraints, but [16] is an independent derivation of regular-horizon conditions that does not contain the TNBH classification, and the present paper even corrects [16]'s exponent values, so the reliance is not a circular appeal to authority. Appendix B explicitly restricts attention to "usual" particles with finite X and u_theta and derives gamma = O(1/N) from the conservation laws; this is a stated limitation on the observer family and a legitimate robustness concern, but it is not circular because the scaling is derived, not assumed from the desired classification. Overall the central claim has independent content and no fitted parameter is relabeled as a prediction.
Assumptions & free parameters
assumptions (4)
- domain assumption The spacetime near the horizon admits the coordinate expansion (11)-(13) for A, N^2, omega, g_phi_phi, g_theta_theta with finite integer orders p, q, k, l.
- domain assumption The infalling observer is a 'usual' particle: finite X = E - omega L, finite u_theta, and finite forces, so psi, delta = O(N) and gamma = O(1/N) (Appendix B).
- standard math Standard Newman-Penrose null tetrad and the transformation formulas for Weyl and Ricci scalars under null rotations, boosts, and spins (Appendix D) are correct.
- domain assumption For the algebraic-type analysis in Section VI, the spacetime is vacuum, R_mu_nu = 0, so Phi_ij = 0.
Cite this review
Pith. "Pith review of Naked and truly naked rotating black holes." pith.science (2026). https://pith.science/paper/EFDEX245
@misc{pith2026250113719,
author = {Pith},
title = {Pith review of: Naked and truly naked rotating black holes},
year = {2026},
howpublished = {\url{https://pith.science/paper/EFDEX245}},
note = {Machine review of arXiv:2501.13719}
}
read the original abstract
Previously, it was noticed that in some space-times with Killing horizons some curvature components, responsible for tidal forces, small or even zero in the static frame, become enhanced from the viewpoint of a falling observer. This leads to the notion of so-called naked black holes. If some components in the frame attached to a free-falling observer formally diverge, although scalar invariants remain finite, such space-times was named "truly naked black holes" (in mathematical language, one can speak about non-scalar singularity). Previous results included static spherically symmetric or distorted static metrics. In the present work, we generalized them to include rotation in consideration. We also scrutiny how the algebraic type can change in the vicinity of the horizon due to local Lorentz boost. Our approach essentially uses the Newman-Penrose formalism, so we analyze the behavior of Weyl scalar for different kinds of observers.
Reference graph
Works this paper leans on
-
[16]
Axially symmetric rotating black hole with regular horizons
H.V. Ovcharenko, O.B. Zaslavskii, Axially symmetric r otating black hole with regular hori- zons, Grav. and Cosmology 29, 269 (2023) [arXiv:2211.08061 ]
work page Pith review arXiv 2023
-
[1]
G. T. Horowitz and S. F. Ross, Naked Black Holes, Phys. Rev . D 56, 2180 (1997) [arXiv:hep-th/9704058]
arXiv 1997
-
[2]
G. T. Horowitz and S. F. Ross, Properties of Naked Black Ho les, Phys. Rev. D 57, 1098 (1998) [arXiv:hep-th/9709050]
arXiv 1998
-
[3]
Curvature tensors on distorted Killing horizons and their algebraic classification
V. Pravda, O. B. Zaslavskii, Curvature tensors on distor ted Killing horizons and their algebraic classification, Class.Quant.Grav. 22, 5053 (2005) [arXiv: gr-qc/0510095]
work page Pith review arXiv 2005
-
[4]
O. B. Zaslavskii, Truly naked spherically-symmetric an d distorted black holes, Phys. Rev. D 76, 024015 (2007), [arXiv:0706.2727]
work page Pith review arXiv 2007
-
[5]
S.W. Hawking and G. F. Ellis, Large Scale Structure of Uni verse (Cambridge University Press, Cambridge, England, 1973)
work page 1973
-
[6]
(C18) ˜R0121 = cosh 2γR ′ 0121 + coshγ sinhγ(R′ 0101 +R′
-
[7]
I. V. Tanatarov, O. B. Zaslavskii, What happens to Petrov classfication on horizons of ax- isymmetric dirty black holes, J. Math. Phys. 55, 022502 (201 4), [arXiv:1211.4376]
Show all 29 references
-
[8]
Black Holes : The Membrane Paradigm, edited by K. S. Thorne, R. H. Price, a nd D. A. Macdonald (Yale University Press, London, 1986). 30
1986
-
[9]
Maeda, C
H. Maeda, C. Martinez, Existence and absence of Killing h orizons in static solutions with symmetries, Class.Quant.Grav. 41 24, 245013 (2024) [arXiv :2402.11012]
2024 arXiv
-
[10]
K. A. Bronnikov and O. B. Zaslavskii, Black holes can hav e curly hair, Phys. Rev. D 78, 021501 (2008), [arXiv:0801.0889]
2008 arXiv
-
[11]
K. A. Bronnikov and O. B. Zaslavskii, General static bla ck holes in matter, Class. Quant. Grav. 26, 165004 (2009),[arXiv:0904.4904]
2009 arXiv
-
[12]
K. A. Bronnikov and O. B. Zaslavskii, Neutral and charge d matter in equilibrium with black holes, Phys. Rev. D 84, 084013 (2011) [arXiv:1107.4701]
2011 arXiv
-
[13]
Visser, Dirty black holes: Thermodynamics and horiz on structure, Phys
M. Visser, Dirty black holes: Thermodynamics and horiz on structure, Phys. Rev. D 46, 2445 (1992), [arXiv:hep-th/9203057]
1992 arXiv
-
[14]
Investigation of general conditions of regularity and properties o f TNBH was carried out earlier for static spherically symmetric and distorted black holes
where only regular space-times were considered. Investigation of general conditions of regularity and properties o f TNBH was carried out earlier for static spherically symmetric and distorted black holes. In the present work we make the next step and consider rotating station...
-
[15]
G. F. R. Ellis and B. G. Schmidt, Gen. Relativ. Gravit. 8, 9 15 (1977)
1977
-
[17]
K. A. Bronnikov, E. Elizalde, S. D. Odintsov and O. B. Zas lavskii, Horizons vs. singularities in spherically symmetric space-times, Phys. Rev. D 78, 0640 49 (2008), [arXiv:0805.1095]
2008 arXiv
-
[18]
E. T. Newman and Roger Penrose, An Approach to Gravitati onal Radiation by a Method of Spin Coefficients, J. of Math. Phys. 3 (3): 566–768
-
[19]
Petrov type
where it is stated that higher order curvature corrections in g ravitational Lagrangian destroy regular extremal horizons that existed in general relativ ity for the Kerr metric, thus making them singular. This poses a question, which possibilities for reg ular horizons exist ...
-
[20]
J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiati on, Astrophys. J. 178, 347 (1972)
1972
-
[21]
R. M. Wald, General Relativity (Chicago, IL: Chicago Un iversity Press, 1984). 31
1984
-
[22]
A. J. M. Medved, D. Martin, M. Visser, Dirty black holes: Symmetries at stationary non-static horizons, Phys.Rev. D70 (2004) 024009, [arXiv:gr-qc/0403 026]
2004
-
[23]
I. V. Tanatarov, O. B. Zaslavskii, Dirty rotating black holes: regularity conditions on station- ary horizons, Phys. Rev. D 86, 044019 (2012), [arXiv:1206.2 580]
2012
-
[27]
G. T. Horowitz, M. Kolanowski, G. N. Remmen, J. E. Santos , Extremal Kerr Black Holes as Amplifiers of New Physics, Phys. Rev. Lett. 131, 091402 (2023 ), [arXiv:2303.07358]
2023 arXiv
-
[303]
(C15) ˜R0303 =R′ 0303 + sinh2γ(R′ 0303 +R′ 2323), ˜R1212 =R′ 1212 + sinh2γ(R′ 1212 +R′
-
[312]
(C17) ˜R1223 = sinhγ cosγ(R′ 0123 −R′ 0312), ˜R0103 = sinhγ coshγ(R′ 0123 −R′
-
[1010]
(C16) ˜R0312 =R′ 0312 + sinh2γ(R′ 0312 +R′ 0123), ˜R0123 =R′ 0123 + sinh2γ(R′ 0123 −R′
-
[1212]
(C19) ˜R0323 = cosh 2γR ′ 0323 + coshγ sinhγ(R′ 0303 +R′
-
[2323]
(C20) ˜R0113 = coshγR ′ 0113, ˜R0223 = coshγR ′ 0223, ˜R1323 = coshγR ′ 1323, ˜R0201 = coshγR ′ 0201 (C21) ˜R0203 = sinhγR ′ 0223, ˜R1213 = − sinhγR ′ 0113, ˜R0212 = − sinhγR ′ 0102, ˜R0313 = sinhγR ′ 1323 (C22) Appendix D: T ransformation of the W eyl scalars and Ricci tensor...
Reviewed August 10, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.