On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

Norihiro Tanahashi; Tsuyoshi Houri; Yukinori Yasui

REVIEW 2 major objections 4 minor 1 cited by

On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives the commuting symmetry operators that separate Maxwell equations on Kerr-NUT-(A)dS from Killing tensors on an Eisenhart-Duval lifted spacetime, and shows they match earlier coordinate operators up to first-order terms.

desk verdict A solid covariant construction of the LFKK commuting symmetry operators and a new Teukolsky symmetry operator; the main result holds up, with the only real soft spot being compressed algebra in Appendix C. read the letter →

arxiv 1908.10250 v2 pith:RQCBIAPO submitted 2019-08-27 gr-qc

Tsuyoshi Houri , Norihiro Tanahashi , Yukinori Yasui This is my paper · ORCID

classification gr-qc MSC 83C5083C5783C15

keywords Kerr-NUT-(A)dSspacetimeEisenhart-DuvalliftMaxwellequationcommutingsymmetryoperatorsseparationofvariablesKillingtensorsTeukolskyhiddensymmetries

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

Maxwell's equations around rotating black holes are usually solved by separation of variables, but the deeper reason the separation works has often been tied to coordinate-dependent operator expressions. This paper tries to show that the commuting symmetry operators behind that separation are geometric objects: they descend from Killing tensors on a spacetime obtained by adding two extra dimensions via the Eisenhart-Duval lift. On the lifted spacetime, the scalar equation that encodes the Maxwell field becomes an ordinary massless Klein-Gordon equation, and its hidden symmetries become Killing vectors and tensors. The paper establishes that the resulting operators on the original spacetime coincide with the previously known coordinate operators up to first-order terms, and that the same procedure supplies a symmetry operator for the Teukolsky equation whose eigenvalue is the separation constant.

What carries the argument

The Eisenhart-Duval lift is the engine: given an equation of the form $H\Phi=[g^{ab}(\nabla_a-iqA_a)(\nabla_b-iqA_b)-2V]\Phi=E\Phi$, one forms the metric $\tilde{g}=g_{ab}dx^a dx^b+2qA_a dx^a du+2du\,dv-2V\,du^2$ and writes solutions as $\tilde{\Phi}=e^{iEu/2}e^{iv}\Phi(x)$. The wave operator upstairs is then the massless Klein-Gordon operator $\tilde{\Box}$, and any Killing vector or Killing tensor on the lifted metric satisfying the anomaly-free and Schouten-Nijenhuis conditions projects to a symmetry operator on the base. Benenti's inverse-Stäckel form is the tool that produces the Killing tensors: it turns the single-variable structure of the inverse lifted metric into $D$ mutually commuting tensors. The main calculation is checking the commutativity conditions of Section 2.5 for those tensors.

What would settle it

For a concrete generic choice, say $D=4$ with $X_1=x_1^2-2mx_1+a^2$ and $X_2=a^2-x_2^2$, evaluate the right-hand side of condition (2.37) for the Killing tensors in (4.26) using computer algebra. The paper's claim is false if any component of $\tilde\nabla_A\tilde{m}^{AB}_{(i,j)}$ is nonzero for $i\neq j$, or if the operators (4.35) fail to commute when applied to a separated mode solution.

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Extended reading notes

Core claim

On the Kerr-NUT-(A)dS family, the reduced Maxwell equation (the LFKK equation) can be embedded into a massless Klein-Gordon equation on a lifted, two-dimension-higher spacetime. The lifted metric inherits a separability structure from the base, so it admits a full set of Killing vectors $\tilde{L}_{(i)}$ and Killing tensors $\tilde{K}_{(i)}^{AB}$. The corresponding first- and second-order operators commute with the lifted d'Alembertian and with one another, and projecting them back to the base gives symmetry operators $L_{(i)}$ and $K_{(i)}$ of the LFKK equation. The main claim is that these operators are the covariant version of the coordinate-form operators of [7]: they agree up to first-order operators generated by Killing vector fields. For the four-dimensional Teukolsky equation, the same construction gives an operator $K$ with $[H,K]=0$ and $K\Phi=\kappa\Phi$, where $\kappa$ is the separation constant.

Load-bearing premise

The load-bearing premise is that the cancellations reported in Appendix C are complete: each component such as $\tilde{m}^{(i,j)}_{\mu\hat\mu}$, $\tilde{m}^{(i,j)}_{a-}$, and $\tilde{m}^{(i,j)}_{+-}$, and each divergence such as $\delta U_+^{(i,j)}$, really vanishes. If even one of these identities fails for a generic choice of metric functions $X_\mu$, the operators $K_{(j)}$ would not commute and the separation constants would not be independent. The proof of these identities is delegated to 'straightforward calculation' rather than exhibited.

Editorial extensions

If this is right

The LFKK separation of variables for Maxwell fields on Kerr-NUT-(A)dS is explained by hidden symmetries of a lifted spacetime, not by a coordinate accident.
The Teukolsky equation for Maxwell perturbations in four dimensions acquires a genuine second-order symmetry operator, and its eigenvalue is the separation constant $\kappa$, so the operator maps solutions to solutions.
The construction applies to any equation fitting the form (2.6), so spin-$s$ perturbation equations, including gravitational perturbations ($s=2$) in four dimensions, are natural candidates for the same treatment.
The Lorenz gauge condition becomes an algebraic constraint on the separation constants, showing that admissible modes satisfy a polynomial condition in the polarization parameter $\beta$.

Reading between the lines

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

Inference: the same lift should produce commuting symmetry operators for massive vector (Proca) fields on Kerr-NUT-(A)dS, with the mass entering through $V$ in (2.6); a nonzero mass would likely break the exact coincidence with the operators of [7] by adding a $V$-dependent term.
Inference: the ambiguity in comparing $K_{(j)}$ with the earlier coordinate operators, namely differences proportional to first-order operators $L_{(i)}$, suggests the separation constants themselves are defined only up to shifts that depend on the Killing-vector charges.
Inference: a direct testable extension is to check whether the eigenvalue identity $K\Phi=\kappa\Phi$ persists for the $s=2$ Teukolsky equation; if it does, the method would supply a covariant symmetry operator for gravitational perturbations without needing a higher-dimensional LFKK analogue.

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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

Useful paper. The main result is a covariant construction, via the Eisenhart-Duval lift, of the commuting symmetry operators behind variable separation in the LFKK formulation of Maxwell on Kerr-NUT-(A)dS, plus a new symmetry operator for the four-dimensional Teukolsky equation. The authors say plainly that the LFKK operators coincide with Krtouš-Frolov-Kubizňák up to first-order pieces; what is genuinely new is the lift derivation and the covariant form, and for Teukolsky the operator K with [H,K]=0 and eigenvalue equal to the separation constant. The chain is mostly self-contained, with detailed appendices, and I think the reader's ACCEPT verdict is right. This is a real conceptual advance in understanding why these equations separate.

The soft spot is where the stress-test note points. Mutual commutativity of the K_(j) is delegated to Appendix C, and the decisive cancellations—vanishing of m^(i,j)_{μμ̂}, m^(i,j)_{a-}, m^(i,j)_{+-} and δU^(i,j)_+=0—are asserted as 'straightforward calculation' rather than exhibited. That is a fair thing to ask for, but I don't read it as a demonstrated flaw. The paper's own cross-check against [7] is independent and supportive: the operators match up to lower-order terms, which would not happen if the commutativity algebra were badly wrong. The stress-test worry that δm^(i,j)=0 is just handed to known base-space commutativity is also fair, but again it is a request for more detail, not evidence of failure. I would ask the authors to expand C.2 or provide a check in n=2 where a reader can verify by hand.

Minor: there is a sign/typo in the sentence about V before Eq. (4.21); the explicit formula for V is what matters and it checks out. The citation pattern is honest, β is a parameter of the ansatz rather than a fitted number, and there is no circularity.

Bottom line: people working on hidden symmetries and black-hole perturbation theory will want this. It deserves a serious referee; I would send it out and expect accept after minor revision, chiefly the expanded Appendix C.

Referee Report

2 major / 4 minor

Summary. The paper develops a covariant method, based on the Eisenhart-Duval lift, for constructing commuting symmetry operators for scalar equations of the form (2.6). It applies the method to the Teukolsky equation on the four-dimensional Kerr-NUT-(A)dS spacetime, obtaining a second-order symmetry operator K whose eigenvalue is the separation constant of the separated equations (Eq. (3.33)), and to the LFKK equation on the D-dimensional Kerr-NUT-(A)dS spacetime, obtaining mutually commuting operators K_(j) in the covariant form (4.35). The authors then compare these operators with the coordinate expressions of Frolov-Krtouš-Kubizňák [7] and show agreement up to first-order operators generated by Killing vectors, at least in even dimensions. The Lorenz gauge condition is also discussed in Section 4.5.

Significance. If the proof is completed, the paper gives a genuinely geometric explanation for the commutativity of the symmetry operators underlying separation of variables in the LFKK equation, and it provides a covariant form of the operators previously given only in coordinates in [7]. The method is potentially applicable to other field equations. The paper is carefully structured and includes extensive explicit formulas; the cross-check with [7] and the eigenvalue identity (3.33) for the Teukolsky operator are valuable independent confirmations of the main construction. The main weakness is that the commutativity proof in Appendix C is not fully exhibited, which is important because mutual commutativity is the load-bearing property behind the separation constants.

major comments (2)

[Appendix C, Eqs. (C.14)–(C.19)] The mutual commutativity of the LFKK symmetry operators is the central claim of the paper, but Appendix C verifies only the divergence condition (2.37) for the two-form \tilde m^(i,j); it does not explicitly verify the Schouten-Nijenhuis commutativity conditions (2.33) for the Killing tensors (4.25). Moreover, the decisive identities \tilde m^(i,j)_{\mu\hat\mu}=0, \tilde m^(i,j)_{a-}=0, \tilde m^(i,j)_{+-}=0, and \delta U^(i,j)_+=0 are stated as following from 'straightforward calculation' without the intermediate algebra. Since an error in any of these cancellations would invalidate [K_(i),K_(j)]=0 and hence the independence of the separation constants, please display the calculation in full, or state and prove a lemma that the Benenti-constructed tensors (4.25) automatically satisfy (2.33), and then supply the remaining component checks.
[Section 4.4, Eq. (4.40)] The comparison with the operators of [7] is made only in the even-dimensional case (\epsilon=0), where it is stated that \check S_\mu coincides with the operators \tilde C_\mu of [7]. The abstract and Section 4, however, claim agreement for the D-dimensional Kerr-NUT-(A)dS spacetime without parity restriction. The odd-dimensional case should either be compared explicitly with the corresponding expressions in [7] or the claim should be restricted to the cases actually verified.

minor comments (4)

[Section 2.4 and Figure 1] The procedure and Figure 1 say to check 'Eqs. (2.23), (2.33) and (2.38)'; Eq. (2.38) is the definition of \tilde m^(i,j), not a condition to be checked. The condition should be Eq. (2.37).
[Introduction] The phrase 'bland-new technique' should read 'brand-new technique'.
[Section 4.4] The sentence 'In the even-dimensional case (\epsilon=0), \check S_\mu coincides with the operators \tilde C_\mu of [7]' would be more useful if it cited the specific equation in [7] being compared, and if it stated explicitly how the odd-dimensional case is handled.
[Equation (4.24)] The notation '\tilde g^{AB} = \sum_{\mu=1}^n \zeta^{AB}_\mu Q_\mu (A=B\neq\mu)' is confusing because the condition 'A=B\neq\mu' mixes index labels with the summation index; please rephrase the index range and the exceptional case more clearly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, with cited previous results used as external benchmarks rather than as load-bearing inputs.

full rationale

The paper's central claim is that Eisenhart-Duval lift provides a covariant construction of the LFKK commuting symmetry operators, with the Teukolsky symmetry operator as a four-dimensional illustration. The construction does not fit any parameter to the target result: β is an ansatz parameter of the LFKK polarization tensor, and s is the physical spin. The symmetry operators are obtained by reading g, A, V out of the equation of motion, lifting the metric, applying Benenti's method to find Killing tensors, and then projecting the resulting operators back to the base spacetime. The comparison with operators of [7] is explicitly a cross-check, not an input: 'Up to differences proportional to L(i), these operators coincide with those of [7]' and 'we devised a method to express the symmetry operators of [7] in a covariant manner and this is the main result of this work.' The Teukolsky eigenvalue statement KΦ = κΦ is verified against the explicitly separated equations, not imposed. The main verification gap is in Appendix C, where the decisive commutativity conditions are reduced to statements such as 'After a straightforward calculation, we find that the components ... are vanishing.' This is an omitted or compressed calculation, not a circular reduction: the conditions being checked are exactly the independent conditions (2.33) and (2.37), and the paper does not define the operators in terms of their own commutativity. Self-citations to [24] and [30] are used for background facts about Killing equations and hidden symmetries, and the crucial commutativity of the base-space Killing tensors is attributed to the external reference [25]; none of these citations replaces the paper's own construction of the uplifted metric and Killing tensors. The result is therefore essentially self-contained against external benchmarks, and no equation or parameter is shown to reduce by construction to an input of the derivation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The ledger is small because the paper works within established theory: it uses standard mathematical tools (Benenti canonical forms, Eisenhart-Duval lift, Killing tensor theory) and inherits the domain result that the LFKK equation describes Maxwell fields on Kerr-NUT-(A)dS. The only parameter entering the construction is β, a constant of the ansatz, not a fitted number. No new physical entities are introduced.

free parameters (1)

β = not fitted; constant in the polarization ansatz
β enters the polarization tensor B, the ansatz (1.4)-(1.5), and the uplifted metric (4.22). It is not fitted to data; it is a constant of the construction. The Lorenz gauge condition (4.42) later gives an algebraic equation for β with D-2 nontrivial roots.

assumptions (5)

standard math Benenti's canonical form construction yields Killing tensors for metrics of the form (3.8).
Used in Secs. 3.2, 4.2, 4.3 to construct Killing tensors from the Stackel matrix; this is an established result in the theory of separable systems (Benenti [21,27]).
standard math The Eisenhart-Duval lift maps the equation of motion (2.6) to the massless Klein-Gordon equation (2.7) on the uplifted spacetime.
The lift construction (2.2)-(2.10) is a standard technique (Eisenhart [15], Duval et al. [17-19]).
domain assumption The LFKK equation (1.7) is the correct scalar reduction of the Maxwell equation on Kerr-NUT-(A)dS via the ansatz (1.4)-(1.5).
The paper takes the LFKK equation as given from Lunin [5] and Frolov-Krtous-Kubiznak [6,7]; the validity of the reduction is inherited from prior literature.
domain assumption The Kerr-NUT-(A)dS spacetime admits the principal tensor (4.9) and the commuting Killing tensors (4.12), with Schouten-Nijenhuis commutativity from Kolar-Krtous [25].
Used in Secs. 4.2-4.4 to identify hidden symmetries and to compare with [7].
standard math The anomaly-free condition (2.23) is sufficient for a Killing tensor to define a commuting symmetry operator of the scalar d'Alembertian (Carter [23]).
Invoked in Secs. 2.4, 3.4, and checked in Appendix C.

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Pith. "Pith review of On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime." pith.science (2026). https://pith.science/paper/RQCBIAPO

@misc{pith2026190810250,
  author       = {Pith},
  title        = {Pith review of: On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQCBIAPO}},
  note         = {Machine review of arXiv:1908.10250}
}

read the original abstract

We focus on the method recently proposed by Lunin and Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k to solve the Maxwell equation on the Kerr-NUT-(A)dS spacetime by separation of variables. In their method, it is crucial that the background spacetime has hidden symmetries because they generate commuting symmetry operators with which the separation of variables can be achieved. In this work we reproduce these commuting symmetry operators in a covariant fashion. We first review the procedure known as the Eisenhart-Duval lift to construct commuting symmetry operators for given equations of motion. Then we apply this procedure to the Lunin-Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k (LFKK) equation. It is shown that the commuting symmetry operators obtained for the LFKK equation coincide with the ones previously obtained by Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k, up to first-order symmetry operators corresponding to Killing vector fields. We also address the Teukolsky equation on the Kerr-NUT-(A)dS spacetime and its symmetry operator is constructed.

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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

On Generalized (Conformal) Killing Tensors
gr-qc 2026-07 conditional novelty 6.0 of 10

Mixed-symmetry generalized Killing tensors, plus a null-geodesic conformal-like version, are constructed and shown to exist in Kerr-NUT-AdS black holes.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

Krtouˇ s, V

P. Krtouˇ s, V. P. Frolov and D. Kubizˇ n´ ak,Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes, Nucl. Phys. B934 (2018) 7 [ 1803.02485]

arXiv 2018

[2] [1]

R. P. Kerr, Gravitational ﬁeld of a spinning mass as an example of algebr aically special metrics , Phys. Rev. Lett. 11 (1963) 237

work page 1963

[3] [2]

R. C. Myers and M. J. Perry, Black Holes in Higher Dimensional Space-Times , Annals Phys. 172 (1986) 304

1986

[4] [3]

S. A. Teukolsky, Rotating black holes: Separable wave equations for gravita tional and electromagnetic perturbations, Phys. Rev. Lett. 29 (1972) 1114

work page 1972

[5] [4]

S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equa tions for gravitational electromagnetic and neutrino ﬁeld perturbations , Astrophys. J. 185 (1973) 635

work page 1973

[6] [5]

Lunin, Maxwell’s equations in the Myers-Perry geometry , JHEP 12 (2017) 138 [ 1708.06766]

O. Lunin, Maxwell’s equations in the Myers-Perry geometry , JHEP 12 (2017) 138 [ 1708.06766]

arXiv 2017

[7] [6]

V. P. Frolov, P. Krtouˇ s and D. Kubizˇ n´ ak,Separation of variables in Maxwell equations in Pleba´ nski-Demia´ nski spacetime, Phys. Rev. D97 (2018) 101701 [ 1802.09491]

arXiv 2018

[8] [8]

V. P. Frolov, P. Krtouˇ s, D. Kubizˇ n´ ak and J. E. Santos,Massive Vector Fields in Rotating Black-Hole Spacetimes: Separability and Quasinormal Modes , Phys. Rev. Lett. 120 (2018) 231103 [ 1804.00030]

arXiv 2018