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REVIEW 4 major objections 4 minor 82 references

Maxwell-$f(Q)$ theory

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A charged black hole solution in f(Q) gravity whose central singularity is milder than general relativity's.

desk verdict The central charged AdS solution (34) does not follow from (32) under (33), and the horizon condition (41) contradicts (34); the main claim is unsupported. read the letter →

arxiv 2501.09373 v1 pith:XB7SSVWT submitted 2025-01-16 gr-qc hep-th

G.G.L. Nashed This is my paper · ORCID
classification gr-qchep-th MSC 83C5783D0583C15
keywords f(Q)gravitynon-metricitychargedAdSblackholepower-lawansatzcentralsingularitythermodynamicsrotatingMaxwellfield
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

The paper claims a new four-dimensional charged, asymptotically anti-de Sitter black hole solution in power-law $f(Q)$ gravity, a modified theory in which gravity is carried by non-metricity rather than curvature. The solution, Eq. (34), has no uncharged limit, no general-relativity limit, and no linear-nonmetricity limit: both the electric charge and the $f(Q)$ modification are essential. Direct computation of curvature and non-metricity invariants shows the central singularity at $r=0$ grows more slowly than in Einstein-Maxwell or in the symmetric teleparallel equivalent of general relativity. A rotating charged AdS version is obtained by a coordinate transformation. If correct, this gives a concrete arena for AdS/CFT studies and for testing whether modified gravity can soften the classical singularity.

What carries the argument

The working core is the power-law ansatz $f(Q)=Q+\gamma Q^2+\gamma_1 Q^3-2\Lambda$ in the coincident-gauge formulation of symmetric teleparallel gravity, where the non-metricity scalar for the static line element (20) is $Q=-2S_1(rS'+S)/(r^2 S)$. In the uncharged sector the radial field equation reduces to the algebraic condition $Q+3\gamma Q^2+5\gamma_1 Q^3+2\Lambda=0$, making $Q$ constant; the charged sector instead uses the hand-imposed relation (30), which reduces the dynamical system to the closed form (34). The singularity claim is carried by the invariant set: the curvature scalars and non-metricity scalars computed from (37), whose fractional-power falloffs near $r=0$ encode the milder singularity.

What would settle it

Take the same power-law $f(Q)$ with a value of $\gamma_1$ that differs from $3\gamma^2/5$ (keeping the same ansatz and boundary conditions) and solve the Maxwell-$f(Q)$ field equations numerically; obtaining a regular asymptotically AdS charged solution would show the imposed relation is not necessary and the singularity scaling is not generic, while failure to find one would confirm it is load-bearing.

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Extended reading notes

Core claim

On its own terms, the paper establishes that the static metric (34) is an exact solution of the Maxwell-$f(Q)$ field equations for $f(Q)=Q+\gamma Q^2+\gamma_1 Q^3-2\Lambda$ under the parameter choice $\Lambda=1/(18\gamma)$, $\gamma_1=3\gamma^2/5$. The effective cosmological constant $\Lambda_{\rm eff}=1/(18\gamma)$ is then fixed by the $f(Q)$-parameter $\gamma$, and the metric function carries fractional powers $r^{-10/3}$ and $r^{-14/3}$ alongside the mass and charge terms. The invariants all diverge at $r=0$, but with the scalings $(K, R_{\mu\nu}R^{\mu\nu})\sim r^{-8/3}$ and $(R, Q_{\alpha\beta\gamma}Q^{\alpha\beta\gamma}, P_{\alpha\beta\gamma}P^{\alpha\beta\gamma}, Q)\sim r^{-4/3}$, compared with $r^{-8}$ and $r^{-4}$ in the linear/GR charged case, so the central singularity is milder. The paper further claims two horizons for moderate charge, a degenerate horizon for special $(M,\phi)$, a naked singularity for sufficiently large charge, and a rotating AdS solution generated by the local transformation (42).

Load-bearing premise

The whole charged solution rests on the hand-imposed parameter relation (30), $\Lambda=1/(18\gamma)$ and $\gamma_1=3\gamma^2/5$, which is not derived from observations or first principles; if that relation is abandoned, the closed-form solution (34) and its claimed effective cosmological constant no longer follow.

Editorial extensions

If this is right

  • If Eq. (34) is exact, $f(Q)$ gravity admits charged AdS black holes with no general-relativity counterpart, so near-horizon observations could distinguish this spacetime from Reissner-Nordström-AdS.
  • Because the solution has no uncharged limit, the electric charge is structural: realistic charged black holes in this theory must use the full non-linear potential (35), not just the monopole term.
  • The two-horizon structure and the existence of a degenerate extremal limit give a setting for studying extremal black hole thermodynamics; the paper's positive Gibbs free energy indicates global stability in the grand canonical ensemble.
  • The rotating metric (45) inherits the singularity and horizon properties of the static solution, so the milder singularity survives rotation.

Reading between the lines

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

  • Editorial: relaxing the imposed relation (30) and solving the field equations numerically would test whether the closed-form charged solution and its singularity scaling are generic or a special-case artifact; the paper does not perform this check.
  • Editorial: the fractional powers in the metric suggest the spacetime may have an unusual algebraic or multipole structure; computing its Petrov type or multipole moments could reveal whether the milder singularity is tied to those powers.
  • Editorial: because the rotation is introduced by a transformation that is only locally, not globally, valid, the rotating solution's global causal structure remains open; closed timelike curves or identifications could change the physical picture.
  • Editorial: if the milder singularity is robust, it gives a concrete target for holographic probes, since AdS/CFT boundary correlators should encode the $r\to 0$ falloffs and could distinguish this $f(Q)$ background from the Einstein-Maxwell one.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes static and rotating charged AdS black-hole solutions in four-dimensional f(Q) gravity with the power-law ansatz f(Q)=Q+γQ²+γ1Q³−2Λ. The central object is the charged static solution presented in Eq. (34), which the authors claim is novel, has no uncharged or GR limit, exhibits a central singularity milder than in GR/STEGR, and has nontrivial horizon and thermodynamic properties. A rotating generalization is then generated by a coordinate transformation in Section IV, and thermodynamic quantities are computed in Section V.

Significance. If the central solution were correct, a new exact charged AdS black hole in power-law f(Q) gravity would be a useful contribution to the modified-gravity and AdS/CFT literature. The paper also deserves credit for attempting an analytical treatment of a non-linear f(Q) model, for computing explicit curvature and non-metricity invariants, and for being transparent that the analysis is primarily mathematical and does not address dynamical collapse. However, the central solution is not established as written: the substitution from Eq. (32) to Eq. (34) does not close, the horizon condition (41) is inconsistent with the metric (34), and the paper contradicts itself on the uncharged limit. These are load-bearing problems, not presentation issues.

major comments (4)
  1. [Section III B, Eqs. (32)-(34)] The passage from the displayed solution (32) to the final metric (34) via the constant relations (33) is algebraically incorrect. With c4=φ/18^{1/5}, the term −15c4²/(8r²) in (32) becomes −15φ²/(8·18^{2/5}r²), not the −15φ²/(8r²) shown in (34); analogous mismatched factors of 18^{1/3} and 18^{2/3} appear in the fractional terms. The gauge potential (35) also does not follow from q(r) in (32) under (33): the second and third terms have different coefficients and r-dependences. Additionally, Eq. (37) displays an exponent inconsistent with Eq. (34) in the grr component. Thus Eq. (34) is not a verified specialization of Eq. (32), and the claimed new solution is not established.
  2. [Section III B, Eq. (41)] The horizon condition (41) is not S(r_+)=0 for the solution (34). Solving S(r)=0 from (34) gives M = r_+³Λ_eff − 15φ²/(8r_+) + 135(18γφ⁸)^{1/3}/(448 r_+^{7/3}) − 81(18γφ⁵)^{2/3}/(704 r_+^{11/3}), whereas Eq. (41) has all positive signs and different coefficients. Since Eq. (41) is used as the input for the mass parameter in the thermodynamic analysis of Section V, the thermodynamic results in Eqs. (48)-(51) inherit this inconsistency.
  3. [Uncharged limit: abstract and Section III B] The paper contains a direct contradiction about whether the charged solution has an uncharged limit. The abstract and conclusions state that the solution 'does not have an uncharged version or relate to general relativity,' but the paragraph after Eq. (36) says that when the vector potential q(r) vanishes, 'we revert the solution presented in Eq (26).' This distinction matters because the claimed novelty of the solution depends on the absence of an uncharged or GR limit; the text explicitly claims both things.
  4. [Section III B, Eqs. (30) and (36)] The effective cosmological constant is not an emergent output of the theory. The relation Λ_eff = 1/(18γ) is exactly the imposed condition Λ = 1/(18γ) from Eq. (30), and it is independent of the charge φ. The conclusion that Λ_eff 'varies based on the electric charge and the parameters of the f(Q) modification' is therefore not supported by the equations in the paper. To substantiate the emergence claim, the authors would need to derive Eq. (30) from an independent physical criterion or exhibit a charge-dependent Λ_eff.
minor comments (4)
  1. [Eq. (18)] The displayed field equations in Eq. (18) are garbled: the line contains 'κ 2 1/2 κT' and an unrelated-looking '∂ν (√−gF μν) = 0', and the indices on the matter energy-momentum tensor are not consistently displayed.
  2. [Figure 2 captions] The caption of Figure 2 labels panel (a) as 'Heat capacity' while the text refers to panel (a) as entropy; the captions and the text should be aligned.
  3. [Eq. (43)] The relation l = −3/Λ_eff appears dimensionally and sign-wise suspect, and since Λ_eff in this paper is positive for the AdS case (γ>0), the sign convention in Eq. (43) should be checked.
  4. [General presentation] There are numerous typographical errors that impede reading, including 'withe' for 'with', 'he mass parameter' for 'the mass parameter', 'evenan eventiz on' for 'event horizon', and inconsistent use of '3√r' notation for cube roots.

Circularity Check

1 steps flagged · score 4.0 of 10

The claimed emergent effective cosmological constant is the hand-imposed parameter relation (30) renamed as an output; the metric integration is otherwise direct.

  1. self definitional [Section III.A Eq. (30); Section III.B Eqs. (34)-(36) and following text]
    "Now we proceed our analysis, concentrating on the particular form where Λ = 1/(18γ) and γ1 = 3γ²/5. (30) ... with Λ_eff = 1/(18γ), and ϕ is a constant of integration, (36) ... It’s noteworthy to highlight that the Λ_eff emerges as a consequence of the electromagnetic charge, presenting an intriguing feature."

    The paper presents Λ_eff = 1/(18γ) as a derived output, but this value is exactly the imposed input Λ = 1/(18γ) from Eq. (30). The charged solution is built only after this parameter relation is fixed, and the charge ϕ never appears in the formula for Λ_eff, so the later statement that Λ_eff 'emerges as a consequence of the electromagnetic charge' inverts the actual derivation order. The 'emergent cosmological constant' is therefore an input choice relabeled as a prediction.

full rationale

The only reduction-by-construction I can exhibit in this paper is the effective cosmological constant. Eq. (30) fixes Λ = 1/(18γ) and γ1 = 3γ²/5; Eq. (36) then identifies Λ_eff with the same 1/(18γ), and the text calls this an emergence attributable to the electromagnetic sector. Since the charge does not enter Λ_eff and the relation was imposed before the charged integration, that specific advertised result is definitionally equal to its input. The cited prior uncharged solution [61,62] is not load-bearing for the charged metric, which is an independent integration of the f(Q) field equations, and there is no uniqueness theorem or self-citation chain forcing the main solution. For that reason the circularity is partial: the central metric construction is a direct integration, while the 'emergent Λ_eff' claim reduces by construction. Apparent algebraic mismatches noted in the skeptic pass (Eq. (34) vs. Eq. (32), Eq. (41) vs. Eq. (34)) are internal-consistency or correctness problems, not circularity, and do not raise this score.

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

No new entities are introduced; the central claim rests on a chosen action ansatz, a gauge fixing, a metric ansatz, and a hand-imposed parameter relation.

free parameters (5)
  • gamma (γ)
    Free coupling in the power-law f(Q)=Q+γQ²+γ1Q³-2Λ; the solution (34) depends on it, and the relation (30) sets Λ and γ1 in terms of it.
  • Lambda relation (Λ=1/(18γ)) = 1/(18γ)
    Chosen by hand in Eq. (30) to simplify the solution; it is what produces the claimed effective cosmological constant.
  • gamma1 relation (γ1=3γ²/5) = 3γ²/5
    Chosen by hand in Eq. (30); not derived from observations or first principles.
  • mass M = integration constant
    Integration constant labeling the solution family; central to the metric and thermodynamics.
  • charge phi (ϕ) = integration constant
    Integration constant labeling the electric charge; the solution is claimed to exist only for nonzero ϕ.
assumptions (5)
  • standard math Field equations of f(Q) gravity with Maxwell matter (Eq. 18) are correct and complete.
    The derivation starts from the action (14) and varies metric and connection; prior literature is cited for these equations.
  • domain assumption Coincident gauge is used to set the affine connection to zero, making the metric the only dynamical variable.
    Section II states 'we use the coincident gauge to compute our solution'.
  • domain assumption The metric ansatz (20) with flat sections and cylindrical coordinates covers the black holes of interest.
    Section III introduces the cylindrical line element with two unknown functions.
  • ad hoc to paper The power-law ansatz f(Q)=Q+γQ²+γ1Q³-2Λ is viable and 'consistent with observations'.
    Eq. (22) selects the model; the paper cites cosmological evidence but does not derive it.
  • ad hoc to paper The parameter relation (30) Λ=1/(18γ), γ1=3γ²/5 is imposed.
    Eq. (30) is introduced without justification; it is load-bearing for the closed-form solution (34).

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Cite this review

Pith. "Pith review of Maxwell-$f(Q)$ theory." pith.science (2026). https://pith.science/paper/XB7SSVWT

@misc{pith2026250109373,
  author       = {Pith},
  title        = {Pith review of: Maxwell-$f(Q)$ theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XB7SSVWT}},
  note         = {Machine review of arXiv:2501.09373}
}
abstract

Exploring the four-dimensional AdS black hole is crucial within the framework of the AdS/CFT correspondence. In this research, considering the charged scenario, we investigate the four-dimensional stationary and rotating AdS solutions in the framework of the $f(Q)$ gravitational theory. Our emphasis is on the power-law ansatz, which is consistent with observations and is deemed the most viable. Because this solution does not have an uncharged version or relate to general relativity, it falls into a new category, which derives its features from changes in non-metricity and incorporates the Maxwell domain. We analyze the singularities of such a solution, computing all the quantities of different curvature and non-metricity invariants. Our results indicate the presence of a central singularity, albeit with a softer nature compared to standard non-metricity or Einstein general relativity, attributed to the influence of the effect of $f(Q)$. We examine several physical characteristics of black holes from a thermodynamics perspective and demonstrate the existence of an outer event horizon in addition to the inner Cauchy horizons. However, under the conditions of a sufficiently large electric charge, a naked singularity emerges. Finally, we derive a class of rotating black hole in four-dimensional $f(Q)$ gravity that are asymptotically anti-de Sitter charged.

Figures

Figures reproduced from arXiv: 2501.09373 by the authors.

Figure 1
Figure 1. The four-dimensional Maxwell-f(Q) gravity solution’s metric functions, S(r) and S1(r), for different electric charge values ϕ, where κ = 1, in the relativistic units. The outer event horizon of the black hole is represented by r+, and the inner Cauchy horizon of a black hole by r−. Here we put γ = 1. while computing the non-metricity we get: Q(r) = 15  −32 r 20 3 Λ + 20 ϕ 2γr8/3 + 5 (18γϕ8 ) 1/3 r 4/3 + 9(18γ 5ϕ 10… view at source ↗
Figure 2
Figure 2. (a) illustrates the entropy’s behavior, displaying a positive value (a) Heat capacity (b) Hawking’s temperature (c) Gibb’s free energy [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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