Noncommutative fields in Reissner-Nordstr\"{o}m black hole background

Andjelo Samsarov; Biljana Nikoli\'c; Marija Dimitrijevi\'c \'Ciri\'c; Milorad Be\v{z}ani\'c; Nikola Konjik

REVIEW 3 major objections 4 minor 17 references

Noncommutative fields in Reissner-Nordstr\"{o}m black hole background

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The dual-metric description of noncommutative scalar and spinor fields in a Reissner-Nordström background fails for the electromagnetic field.

desk verdict Useful short paper with a real new computation—first-order NC Maxwell equations in RN—but the headline negative claim (no dual metric) rests on an assertion, not a proof. read the letter →

arxiv 2505.06181 v1 pith:FY25H2LO submitted 2025-05-09 hep-th

Milorad Bežanić , Marija Dimitrijević Ćirić , Nikola Konjik , Biljana Nikolić , Andjelo Samsarov This is my paper · ORCID

classification hep-th

keywords noncommutativespacetimeReissner-NordströmblackholeangulartwistSeiberg-Wittenmapelectromagneticperturbationsdualmetricquasinormalmodesgravity

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 works out, to first order in the deformation parameter $\Theta$, the equations for scalar, spinor, and electromagnetic fields in a Reissner–Nordström black hole background deformed by the angular twist. Its positive result is the explicit first-order equation (26) for electromagnetic perturbations. Its negative result is that this equation cannot be rewritten as a commutative Maxwell equation in an effective or dual metric, whereas the scalar and spinor equations can. The identified reason is the nonlinearity of the Seiberg-Witten map for the gauge field, which produces cubic terms in the field strength. If correct, the result fixes the starting point for noncommutative electromagnetic quasinormal-mode calculations and shows that the dual-geometry interpretation does not extend to gauge fields.

What carries the argument

The load-bearing object is the angular twist $\mathcal{F} = \exp\{-\tfrac{i}{2}\theta^{\alpha\beta} X_\alpha \otimes X_\beta\}$, with $X_1=\partial_t$ and $X_2=\partial_\varphi$, which deforms the product of fields while leaving the RN metric untouched because the two vector fields are Killing. Around that twist the Seiberg-Witten map expresses the noncommutative fields in commutative variables, giving the field expansions (10)-(13). The critical comparison is between the vector equation of motion (26) and the dual-metric form (27): for the scalar and spinor fields a dual metric (18) and dual vierbein (23) exist, while for the electromagnetic field the terms proportional to $\Theta eQ/r^3$ multiplying derivatives of $F$ spoil the comparison. This failure is attributed to the cubic $\hat{F} \wedge_\star \hat{F}$ structure produced by the nonlinear SW map of the gauge field.

What would settle it

Construct a candidate dual metric $g'_{\mu\nu}=g_{\mu\nu}+\Theta h_{\mu\nu}$ and check whether the $\Theta$-dependent derivative terms in equations (26) can be rewritten as connection corrections of the form $-\left(\Gamma'^{\nu}_{\mu\rho}-\Gamma^{\nu}_{\mu\rho}\right)F^{\mu\rho}$; if any $h_{\mu\nu}$ works, the paper's negative claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a boundary on the dual-metric description of noncommutative fields. After reviewing that charged NC scalar and spinor perturbations in the Reissner–Nordström background can be seen as commutative fields propagating in a modified RN metric $g'_{\mu\nu}$, the paper derives the first-order NC equations of motion for electromagnetic perturbations, equations (26). It then claims that these equations cannot be brought to the form $\partial_\mu F^{\mu\nu} + \Gamma'^{\nu}_{\mu\rho} F^{\mu\rho} = 0$ with Christoffel symbols of a dual metric. The obstruction is that the SW map for the gauge field is nonlinear, so the NC action (24) contains terms cubic in the field strengths; after variation these yield terms of the form $\Theta\, \partial(F F)$ that are not of the required Christoffel type. The scalar and spinor cases avoid this because their SW maps are linear in the corresponding fields.

Load-bearing premise

The argument depends on the unproven claim that the extra derivative terms in the electromagnetic equations cannot be absorbed by adjusting the connection of a modified geometry; no exhaustive check over all possible modified geometries is given.

Editorial extensions

If this is right

The first-order NC electromagnetic perturbations in the Reissner-Nordström background are governed by equations (26), so quasinormal-mode computations for this sector must start from those equations rather than from a dual metric.
The effective/dual metric interpretation is not universal: it covers the scalar and spinor sectors under the angular twist but fails for the vector sector.
Because the obstruction comes from the nonlinear SW map of the gauge field, matter fields with linear SW maps remain compatible with a dual geometry, while gauge fields do not.
The gravity equations (31) and (33)-(36) give the starting point for studying NC gravitational perturbations; the first nontrivial corrections there are quadratic in $\Theta$, not linear.

Reading between the lines

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

Editorial inference: the failure of the effective-metric description for the vector field is likely to extend to any field whose Seiberg-Witten map is nonlinear, so the dual-metric duality may be a special feature of scalar and spinor matter.
Editorial inference: the negative result may be twist-dependent; another twist or gauge choice could restore an effective-geometry description for electromagnetic perturbations, so the conclusion should be read as a property of the angular-twist model.
Editorial inference: computing electromagnetic quasinormal modes from (26) and comparing the resulting frequency shifts with scalar and spinor shifts would give a concrete observational test of whether the dual-metric failure leaves a physical signature in black-hole ringdown.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The genuinely new thing here is Eq. (26): the first-order noncommutative electromagnetic perturbation equations in the Reissner-Nordström background. That is a concrete, usable result for anyone wanting to compute NC quasinormal modes for vector fields, and it is not in the cited literature. The corresponding negative claim—that these equations cannot be recast as a commutative field in a dual effective metric—is also new and physically interesting if true. The scalar and spinor parts are reviews of earlier work, cleanly summarized, and the gravity section is explicitly a preview of future work.

What the paper does well: the setup is standard, the action (24) is structurally consistent with the Seiberg-Witten expansion, and the final equations (26) have the right zeroth-order Maxwell limit plus terms linear in the NC parameter. The logic that the cubic terms from the SW map for the gauge field spoil the dual description is plausible and worth stating. The paper is honest about what it is: a proceedings contribution with one new calculation.

The soft spots are real but, I think, addressable. Most importantly, the claim after Eq. (27) that the θ∂F terms 'cannot be absorbed' in any effective Christoffel symbols is asserted by inspection, not proven. The space of candidate dual metrics is not exhausted by the scalar/spinor example (18); a general first-order ansatz g' = g + θ h with off-diagonal components could in principle generate the same derivative structure. The stress-test note is right: what is needed is a solvability check of the linear system equating Eq. (26) to the θ-expansion of Eq. (27) for an arbitrary symmetric h. Without that, the central negative result is unsupported, even though Eq. (26) itself may well be correct. There are also minor presentation issues: the dual metric (18) has an off-diagonal r-φ component while the line element (19) shows a θ-φ term, and the e/Q notation is sloppy in places. The algebra from (24) to (26) is omitted; for a proceedings paper that is acceptable, but a referee would want at least the key intermediate steps or an ancillary file.

Who gets value: people working on noncommutative quasinormal modes, especially if they want to extend the scalar and spinor results to the electromagnetic case. It deserves a serious referee: the calculation is concrete, the method is established, and the negative claim is important enough to warrant a careful check. I would send it to review, with the request that the no-dual-metric claim be either proven or downgraded to a conjecture.

Referee Report

3 major / 4 minor

Summary. The manuscript studies first-order noncommutative corrections, induced by an angular twist, to matter and gravitational field equations on a Reissner-Nordström background. It reviews the authors' earlier results for charged scalar and spinor fields, including the dual description in which the corrected equations are interpreted as commutative fields propagating in an effective metric (Eqs. (17)-(19) and (22)-(23)). The new material is the derivation of the equation of motion for noncommutative electromagnetic perturbations, Eq. (26), and the claim that this equation cannot be rewritten as the Maxwell equation in any dual metric because terms proportional to a ∂F cannot be absorbed into Christoffel symbols. The paper then quotes, without derivation, first-order-in-θ² equations from two NC gravity models, SO(2,3)_★ gravity and braided NC gravity, as a starting point for future quasinormal-mode work. The paper is explicitly a short proceedings contribution.

Significance. If Eq. (26) is correct and the no-dual-metric claim can be made rigorous, the paper provides a concrete and falsifiable distinction between scalar/spinor and vector noncommutative perturbations: the effective-metric interpretation works for the former but fails for the latter, and Eq. (26) can serve as a starting point for noncommutative electromagnetic quasinormal-mode calculations. The presentation of the SO(2,3)_★ and braided-gravity equations, although taken from earlier work, usefully frames the open problem of gravitational perturbations. However, because the derivation of Eq. (26) is not shown and the central negative claim is supported only by inspection, the significance is at present conditional.

major comments (3)

[Section 3.3, Eqs. (25)-(26)] The step from the varied action to the explicit component equations is not shown. Equation (25) is the full varied equation, but the substitution F = F_RN + ε f, the cancellation of the zeroth order, and the collection of first-order terms into the four equations (26) are all omitted. Since Eq. (26) is the central new result and the basis for the paper's main conclusion, the authors should include the intermediate algebra or provide a published reference containing it. In addition, the index conventions in (26) need to be fixed: the equations mix upper- and lower-index field-strength components, and the last equation appears to lack terms of the same structure as the zeroth-order Maxwell equation, so a reader cannot check the signs and numerical factors without redoing the entire derivation.
[After Eq. (27)] The claim that 'it is obvious' that equations (26) cannot be rewritten in the form (27) is not established. The comparison is made only against the particular form (27) with the original field strength F^{μν}; a dual-metric description of a vector field would require, at minimum, a covariant equation ∇'_μ F'^{μν} = 0 with the field strength and indices raised with the dual metric g'_{μν} = g_{μν} + θ h_{μν}. The authors should formulate a general first-order ansatz for the symmetric tensor h_{μν}, expand the candidate dual equation to order θ, and show that the resulting linear system has no solution for the independent components of h_{μν}. The presence of terms of the form θ h ∂F and θ ∂h F means that the derivative structure alone does not rule out absorption. Without such a solvability check, the central negative conclusion — that the dual-metric description fails for the vector field — is unsupported, even if Eq. (26) itself is correct.
[Section 3.3, Eq. (24)] The action (24) is presented as the first-order SW-mapped NC Maxwell action, but the paper does not state which gauge of the Seiberg-Witten map or which boundary terms are used. Since Eq. (24) is cubic in the field strength, its variation contains terms of different derivative orders, and the subsequent equations (26) depend on those terms. A short appendix showing the SW-map expansion leading to (24), or a precise reference to the relevant formula in [9], would make the paper self-contained. As it stands, the reader cannot distinguish an algebraic error in (24) from a legitimate choice of SW-map convention.

minor comments (4)

[Throughout] The manuscript contains numerous typographical errors that should be corrected before publication, including 'becuase' after Eq. (1), 'deforamtion' after Eq. (6), 'paramter', 'nonommutative', 'transforamtions', 'dinamical', and 'unded' in Section 4.
[Section 2, Eq. (9) and footnote] The relation between the antisymmetric matrix θ^{αβ} and the scalar parameter a is unclear in the displayed matrix; please define θ^{tφ} = a, θ^{φt} = -a explicitly and use the notation consistently in Eqs. (17)-(26).
[Section 3.1, Eq. (14) and Section 3.3, Eq. (24)] The statement before Eq. (14) that the coupling constant e is absorbed into A_μ is not consistently applied in the vector action (24), where the factor 1/(4 e²) appears. Please specify whether A_μ and F_{μν} in Eq. (24) are rescaled by e and adjust the prefactor accordingly.
[Abstract and Section 4] The abstract's statement that 'typically for NC gravity models, the first nontrivial corrections are quadratic in the NC parameter' should be qualified as applying to the models considered here, since the paper does not survey all NC gravity models.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependency found: the NC electromagnetic equations of motion are derived in-paper, and the no-dual-metric conclusion, while underproved, is not a circular reduction to an input.

full rationale

The paper's new result is the first-order NC electromagnetic perturbation equation (26), obtained by expanding the NC action (24) with the Seiberg-Witten map and varying; this derivation is self-contained in the paper and does not fit any parameter to a target result. The subsequent claim that (26) cannot be rewritten in the form (27) is asserted by inspection after Eq. (27), without an exhaustiveness proof over all possible dual metrics that would also include the index-raising effects of a deformed metric on F^{μν}; this is a completeness/correctness gap, not a circular step, because the narrow comparison to (27) is not derived from the conclusion. The scalar and spinor dual-metric results and the gravity equations are review material drawn from the authors' own prior work [6,8,10,13,15], but those citations are not load-bearing for the new EM calculation, which compares (26) to (27) directly. Self-citation without load-bearing does not constitute circularity, so no circular step is identified.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central derivation depends on the angular-twist model, the standard Seiberg-Witten map expansions, the semiclassical assumption that the RN geometry is undeformed, and the RN background field F_{r0}=eQ/r^2. The gravity equations additionally depend on model coefficients and truncations. No new physical entities are introduced; the dual metric is an effective mathematical device, not a new fundamental object.

free parameters (4)

a = unspecified (small)
The deformation parameter in the angular twist (1); all matter equations are first order in a and gravity corrections are second order. It is an input of the model, not fitted to data.
c1, c2, c3 = unspecified
Coefficients in the SO(2,3)_★ gravity action (28). The equations (30) and (31) depend on c2 and c3, and the paper does not fix these weights from a deeper principle.
l = unspecified
Length scale in the SO(2,3)_★ model; it appears as 1/l^2 in Eq. (30) and is not determined by the paper.
Lambda = unspecified
Cosmological term parameter in the braided gravity action (32); it is a model input and is not fixed or measured here.

assumptions (5)

domain assumption The twist vector fields ∂_t and ∂_φ commute and are Killing vectors of the RN metric, so the deformation does not change the background geometry.
Section 2 states that X1 and X2 are commuting Killing vectors for the metric (3), 'thus the twist (1) does not act on the RN metric.' The semiclassical treatment in Section 3 depends on this.
standard math The Seiberg-Witten map provides the first-order expansions (10)-(13) for the angular twist, and no new degrees of freedom or charge quantization issues arise.
The paper cites [9,11] for the all-order Seiberg-Witten map and uses (10)-(13) as input; it does not rederive them.
domain assumption The background electromagnetic field is the RN field with only F_{r0}=eQ/r^2, and perturbations are split as F_{μν}=F^RN_{μν}+ε f_{μν} with ε small.
Section 3.3 after Eq. (25) defines the split; the zeroth order is assumed to vanish because the RN field satisfies the Einstein-Maxwell equations.
domain assumption For the SO(2,3)_★ gravity model, only the low-energy sector with at most two derivatives, vanishing zeroth-order torsion, and background scalar fixed to lγ5 is considered.
Section 4 before Eq. (30) states the truncation and T^{(0)}=0; the gravity equations (30)-(31) hold only under these restrictions.
domain assumption In braided NC gravity, the braided gauge transformations close the Lie algebra, so the Seiberg-Witten map is not needed and the equations of motion are exact before expansion.
Section 4, paragraph on braided gravity, citing [15]; this justifies using the exact action (32) and then expanding in powers of a.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Noncommutative fields in Reissner-Nordstr\"{o}m black hole background." pith.science (2026). https://pith.science/paper/FY25H2LO

@misc{pith2026250506181,
  author       = {Pith},
  title        = {Pith review of: Noncommutative fields in Reissner-Nordstr\"om black hole background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FY25H2LO}},
  note         = {Machine review of arXiv:2505.06181}
}

abstract

In this short paper we discuss dynamics of noncommutative (NC) matter fields in the Reissner-Nordstr\"{o}m (RN) black hole background. After reviewing the propagation of charged NC scalar and spinor fields, we derive the equation governing the propagation of NC electromagnetic (EM) perturbation in the RN background. The propagation of NC scalar and spinor perturbation have a dual description in terms of the propagation of commutative fields in the effective/dual metric. Finally, we turn to the gravitational perturbations. We present equations of motion for the NC gravitational field obtained in two different models: $SO(2,3)_\star$ NC gravity and braided NC gravity. Typically for NC gravity models, the first nontrivial corrections are quadratic in the NC parameter. The obtained NC gravity equations are the starting point to discuss the propagation of NC gravitational perturbations and the validity of the dual description in terms of the effective metric.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

[9]

Aschieri and L

P. Aschieri and L. Castellani, Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map, JHEP 1207 184 (2012). 11 Noncommutative ﬁelds in Reissner–Nordström black hole bac kground Nikola Konjik

work page 2012
[1]

K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Ein steinschen Theorie, Sitzungsberichte der Königlich Preussischen Akademie de r Wissenschaften 7: 189–196, (1916)

work page 1916
[2]

LIGO Scientiﬁc, Virgo Collaboration, B. P. Abbott et al. , Observation of Gravitational Waves from a Binary Black Hole Merger , Phys. Rev. Lett. 116 no. 6, (2016) 061102. LIGO Scientiﬁc, Virgo Collaboration, B. P. Abbott et al., GW 170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0. 2, Phys. Rev. Lett. 118 no. 22, (2017) 22...

work page 2016
[3]

Akiyama et al., First M87 Event Horizon Telescope Results

Event Horizon Telescope Collaboration, K. Akiyama et al., First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole , Astrophys. J. 875 no. 1, (2019) L1

work page 2019
[4]

C. P. Herzog, Lectures on Holographic Superﬂuidity and Superconductivi ty, J. Phys. A42 (2009) 343001

work page 2009
[5]

Molla, H

N. Molla, H. Chaudhary, S. Capozziello, F. Atamurotov, G . Mustafa, U. Debnath, Observable Signatures of RN Black Holes with Dark Matter Halos via Stron g Gravitational Lensing and Constraints from EHT Observations, Phys. Dark Univ. 47 (2025) 101804

work page 2025
[6]

M. D. Ćirić, N. Konjik and A. Samsarov, Noncommutative scalar quasinormal modes of the Reissner–Nordström black hole, Class. Quant. Grav. 35 (2018) no.17, 175005

work page 2018
[7]

Aschieri and L

P. Aschieri and L. Castellani, Noncommutative D = 4 gravity coupled to fermions, JHEP, 0906, 086 (2009)

work page 2009

Show all 17 references

[8]

Dimitrĳević-Ćirić, N

M. Dimitrĳević-Ćirić, N. Konjik and A. Samsarov, Propagation of spinors on a noncommuta- tive spacetime: equivalence of the formal and the eﬀective approach, Eur. Phys. J. C 83 (2023) 5, 387

2023
[10]

M. D. Ćirić, B. Nikolić and V . Radovanović, NC /u1D446/u1D442(2, 3)★ gravity: noncommutativity as a source of curvature and torsion , Phys. Rev. D 96 (2017) 6, 064029

2017
[11]

Seiberg and E

N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032. B. Jurčo, L. Möller, S. Schraml, P. Schupp and J. Wess, Construction of non-Abelian gauge theories on noncommutative spaces, Eur. Phys. J. C21, 383 (2001)

1999
[12]

Herceg, N

N. Herceg, N. Konjik, A. Naveena Kumara, A. Samsarov, in preparation
[13]

Dimitrĳević Ćirić, N

M. Dimitrĳević Ćirić, N. Konjik, T. Jurić, A. Samsarov a nd I. Smolić, Noncommutative Reissner–Nordström Black Hole from Noncommutative Charge d Scalar Field , Symmetry 17 (2025) no.1, 54

2025
[14]

M. D. Ćirić, D. Djordjević, D. Gočanin, B. Nikolić and V . Radovanović, NC /u1D446/u1D442(2, 3)★ gauge theory of gravity, Eur. Phys. J. ST 232 (2023) 23-24, 3747-3760. P. Aschieri and L. Castellani, Noncommutative gauge and gravity theories and geometric Seiberg–Witten map, Eu...

2023
[15]

Dimitrĳević Ćirić, G

M. Dimitrĳević Ćirić, G. Giotopoulos, V . Radovanović and R. J. Szabo, Braided /u1D43F∞-algebras, braided ﬁeld theory and noncommutative gravity , Lett. Math. Phys. 111 (2021) 148

2021
[16]

K. S. Stelle and P. C. West, Spontaneously Broken De Sitter Symmetry and the Gravitatio nal Holonomy Group, Phys. Rev. D 21, 1466 (1980). West, P. C. (1978). A geometric gravity Lagrangian. Physics Letters B, 76(5), 569-570

1980
[17]

Herceg, T

N. Herceg, T. Jurić, A. N. Kumara, A. Samsarov and I. Smol ić, Noncommutative quasinormal modes of Schwarzschild black hole, [arXiv:2409.01402 [gr-qc]]. N. Herceg, T. Jurić, A. Samsarov and I. Smolić, Metric perturbations in noncommutative gravity, JHEP 06 (2024), 130. 12

2024 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [9]

Aschieri and L

P. Aschieri and L. Castellani, Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map, JHEP 1207 184 (2012). 11 Noncommutative ﬁelds in Reissner–Nordström black hole bac kground Nikola Konjik

work page 2012

[2] [1]

K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Ein steinschen Theorie, Sitzungsberichte der Königlich Preussischen Akademie de r Wissenschaften 7: 189–196, (1916)

work page 1916

[3] [2]

LIGO Scientiﬁc, Virgo Collaboration, B. P. Abbott et al. , Observation of Gravitational Waves from a Binary Black Hole Merger , Phys. Rev. Lett. 116 no. 6, (2016) 061102. LIGO Scientiﬁc, Virgo Collaboration, B. P. Abbott et al., GW 170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0. 2, Phys. Rev. Lett. 118 no. 22, (2017) 22...

work page 2016

[4] [3]

Akiyama et al., First M87 Event Horizon Telescope Results

Event Horizon Telescope Collaboration, K. Akiyama et al., First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole , Astrophys. J. 875 no. 1, (2019) L1

work page 2019

[5] [4]

C. P. Herzog, Lectures on Holographic Superﬂuidity and Superconductivi ty, J. Phys. A42 (2009) 343001

work page 2009

[6] [5]

Molla, H

N. Molla, H. Chaudhary, S. Capozziello, F. Atamurotov, G . Mustafa, U. Debnath, Observable Signatures of RN Black Holes with Dark Matter Halos via Stron g Gravitational Lensing and Constraints from EHT Observations, Phys. Dark Univ. 47 (2025) 101804

work page 2025

[7] [6]

M. D. Ćirić, N. Konjik and A. Samsarov, Noncommutative scalar quasinormal modes of the Reissner–Nordström black hole, Class. Quant. Grav. 35 (2018) no.17, 175005

work page 2018

[8] [7]

Aschieri and L

P. Aschieri and L. Castellani, Noncommutative D = 4 gravity coupled to fermions, JHEP, 0906, 086 (2009)

work page 2009