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REVIEW 4 major objections 5 minor 48 references

Novel charged black hole solutions in conformal Killing gravity

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Pith's one-line read This paper derives three exact static, spherically symmetric charged black hole solutions in Conformal Killing Gravity coupled to nonlinear electrodynamics and shows their shadow radii can match the EHT Sgr A* constraints.

desk verdict New exact CKG-NED black hole metrics, but the shadow analysis uses a different Lagrangian than the one advertised, which undermines the parameter bounds. read the letter →

arxiv 2502.00589 v1 pith:VBQU6JGW submitted 2025-02-01 gr-qc astro-ph.HEhep-th

Ednaldo L. B. Junior , José Tarciso S. S. Junior , Francisco S. N. Lobo , Manuel E. Rodrigues , Luís F. Dias da Silva , Henrique A. Vieira This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th
keywords conformalKillinggravitynonlinearelectrodynamicschargedblackholesholeshadowSagittariusA*eventhorizonKretschmannscalarcosmologicalconstant
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 tries to establish that conformal Killing gravity (CKG), a modified theory in which the cosmological constant emerges as an integration constant, admits new exact black hole solutions when coupled to nonlinear electrodynamics. The authors present three static, spherically symmetric metrics, each generated by a different NED Lagrangian, and verify that each has a curvature singularity at $r=0$, reduces to Schwarzschild or Reissner-Nordström in suitable limits, and tends to Maxwell theory as $F\to 0$. The observational point is the shadow: after tuning parameters, each model's shadow radius falls within the Event Horizon Telescope's bounds for Sagittarius A*, at 1-$\sigma$ for model 1 and at 2-$\sigma$ for models 2 and 3. If the paper is right, these geometries are viable candidates for the Milky Way's central black hole and give concrete targets for further shadow, horizon, and stability tests.

What carries the argument

The reconstruction formula (14) is the central object: obtained by solving the CKG-NED field equations for the ansatz $\mathrm{d}s^2 = A(r)\,\mathrm{d}t^2 - A(r)^{-1}\,\mathrm{d}r^2 - r^2\,\mathrm{d}\Omega^2$, it expresses the NED Lagrangian $L$ and its derivative $L_F$ directly in terms of the metric function $A(r)$ and the constants $f_0$, $f_1$, and magnetic charge $q$. This formula lets the authors start from three proposed Lagrangians, integrate to find $A(r)$, and then check self-consistency. The shadow calculation then runs through the effective metric (35)-(36), where photon geodesics follow $g^{\mathrm{eff}}_{\mu\nu} = L_F\,g_{\mu\nu} - L_{FF}\,F_{\mu\alpha}F^{\alpha}_{\ \nu}$; this effective metric changes the photon-sphere condition and therefore the shadow radius that is compared with the EHT bounds.

What would settle it

Substitute each proposed $A(r)$ from Eqs. (16), (25), and (29) back into the field equations (12)-(13) and check that the residuals vanish for all $r$; any nonzero residual would refute the solution claim. Independently, recompute the shadow radii using standard metric null geodesics instead of the effective metric (35) and see whether the claimed 1-sigma and 2-sigma compatibility with Sgr A* survives.

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Extended reading notes

Core claim

The central claim is that the three metric functions in Eqs. (16), (25), and (29) are exact solutions of the CKG field equations sourced by nonlinear electrodynamics, with Lagrangian densities (15), (24), and (28) respectively. Each solution is static and spherically symmetric with $C(r)=r^2$, has a curvature singularity at $r=0$ as shown by the Kretschmann scalar, and generically admits multiple horizons (Cauchy, event, and cosmological) whose merging is controlled by critical values of mass, charge, cosmological constant, or model parameters. The authors also derive the explicit NED Lagrangian in CKG for each solution; each one differs from its general-relativity counterpart by terms involving the constant $f_1$, and each reduces to Maxwell theory as $F\to 0$. The final claim is observational: using the effective NED metric for photon propagation and the Sgr A* mass-distance priors, the shadow radius of model 1 is within the 1-$\sigma$ EHT bound for suitable parameters, while models 2 and 3 are within the 2-$\sigma$ bound.

Load-bearing premise

The argument assumes that the reconstruction formula (14) fully solves the CKG-NED field equations for the metric ansatz, so that the three proposed $A(r)$ are genuine solutions, and that the EHT shadow-size bounds can be applied to these non-Kerr spacetimes with photons following the effective metric (35).

Editorial extensions

If this is right

  • Each solution is an exact, analytically given charged black hole in CKG, extending Schwarzschild and Reissner-Nordström-AdS, so the theory now has concrete strong-field testbeds.
  • The CKG-NED Lagrangian for each model contains a term in $f_1$ that is absent in general relativity and approaches Maxwell theory as $F\to 0$, making the nonlinear electrodynamic content partly testable.
  • The multi-horizon structure and extreme transitions give explicit parameter regimes where Cauchy and event horizons merge, which is relevant for causal structure and cosmic censorship studies.
  • The parameter ranges identified in the conclusion, for example $q<0.6$, $f_k>2$, and $f_1<10^{-65}$ for model 1 and $q<0.155$ and $q<0.256$ for models 2 and 3, place these geometries within the EHT Sgr A* shadow bounds.
  • Because the cosmological constant enters these solutions as an integration constant, the horizon structure illustrates how $\Lambda$ can affect observables without being a fixed coupling of the theory.

Reading between the lines

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

  • Editorial inference: the shadow radius responds oppositely to charge in the three models (decreasing with $q$ in model 1, increasing in models 2 and 3), so future higher-resolution shadow measurements could distinguish among the three CKG-NED geometries and from Reissner-Nordström.
  • Editorial inference: if the reconstruction formula (14) is as general as it appears, the three solutions are only a sample of a much larger family of CKG-NED black holes, including possibly regular ones once scalar fields are added, as the authors propose for future work.
  • Editorial inference: the compatibility claim rests on using the effective NED metric for photons; using the standard metric instead would likely change the shadow radius and could change which parameter ranges survive the EHT comparison, so the effective-metric choice is itself a testable assumption.
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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 / 5 minor

Summary. The paper studies static, spherically symmetric charged black hole solutions in Conformal Killing Gravity (CKG) coupled to nonlinear electrodynamics (NED). It derives a general reconstruction formula for the NED Lagrangian from a chosen metric function A(r), then proposes three explicit metrics (Models 1–3), computes their Kretschmann scalars and horizon structure, and compares the predicted shadow radii with the EHT Sgr A* constraints. The central claim is that the three metrics are exact solutions of CKG for the three proposed Lagrangians (15), (24), and (28), and that the corresponding shadows are consistent with the EHT observations of Sgr A*.

Significance. If the central claim were established, the paper would provide new exact charged black hole solutions in a recent modified-gravity framework and a method for constraining CKG parameters with EHT data. The paper has useful ingredients: an explicit reconstruction formula, three analytic metric functions with explicit Kretschmann scalars, and a numerical shadow analysis that includes the effective NED metric. However, the identification between the proposed Lagrangians and the reconstructed Lagrangians used in the shadow analysis is flawed, so the main claim is not currently supported. The framework is salvageable, but the model definitions and the observational conclusions must be reworked.

major comments (4)
  1. [III.A, Eqs. (15) and (19)] The Lagrangian obtained by inserting the Model 1 metric (16) into the reconstruction formula (14) is not the proposed Lagrangian (15); it contains the additional term -q^3 f1/(\sqrt{2}\sqrt{F}) plus f0+\Lambda/2. Therefore the metric (16) is not demonstrated to solve the CKG-NED field equations for L(F)=a0 F^{fk}+F unless f1=0 and f0=-\Lambda/2. The same mismatch occurs for Model 2, where Eq. (26) differs from Eq. (24) by -2q^3 f1/(\sqrt{2}\sqrt{F})+\Lambda/2+f0, and for Model 3, where Eq. (30) differs from Eq. (28) by q(\lambda-2q^2 f1)/(2\sqrt{2}\sqrt{F})+\Lambda/2+f0. This invalidates the abstract's claim that the three proposed Lagrangians generate the three metrics.
  2. [IV.B, Eqs. (40)-(41)] The effective metric used for the shadow calculation is built from the derivative L_F of the reconstructed Lagrangians, not from the proposed Lagrangians (15), (24), and (28). For example, the f1 r^6 terms in Eqs. (40)-(41) arise from the \sim 1/\sqrt{F} term in Eq. (19). Consequently, the EHT constraints in Figs. 12-15 and the parameter bounds in the Conclusion, such as f1<10^{-65}, apply to the reconstructed Lagrangians (19), (26), and (30), not to the models defined by (15), (24), and (28). The paper's central observational claim is therefore mismatched with the models it purports to test.
  3. [Abstract and Sec. III, Eqs. (19), (26), (30)] The statement that the computed nonlinear Lagrangian densities agree with Maxwell theory in the limit F\to 0 is false for f1\neq 0, because the reconstructed Lagrangians (19), (26), and (30) contain terms proportional to f1/\sqrt{F} that diverge as F\to 0. Even apart from the constant and cosmological-constant terms, the models do not have the claimed Maxwell limit. This needs to be corrected either by setting f1=0 in the model definitions or by explicitly acknowledging and analyzing the singular F\to 0 behavior of the reconstructed Lagrangians.
  4. [IV.A, Eq. (38)] The EHT shadow-size limits 4.55 \lesssim r_s/M \lesssim 5.22 at 1 sigma are taken from Ref. [44] for a Schwarzschild comparison and are applied here to non-Kerr metrics with photons following the NED effective metric (35). The transferability of these limits to the effective-metric photon spheres is an assumption that is not justified in the manuscript. Since the consistency with Sgr A* is a central conclusion, the paper should either justify this transfer or compare the calculated shadow with the observed ring size through an explicit calibration procedure.
minor comments (5)
  1. [Throughout] There are several typographical errors: "imosing" should be "imposing" in Sec. III.A, "rigth" should be "right" in the caption of Fig. 6, "Srg A*" should be "Sgr A*" in the Conclusion, and "aac" should be "a0c" in Sec. III.A.
  2. [Eq. (30)] Equation (30) contains a parameter \lambda that is not defined anywhere; it should either be identified with \Lambda or removed.
  3. [Eqs. (12)-(13)] The derivation of the field equations (12) and (13) from Eq. (1) is not shown; the authors should provide the intermediate steps or cite a reference where this reduction is performed.
  4. [Eq. (37)] The symbol \omega_g in Eq. (37) is used without definition; it should be defined as the gravitational angular radius M/D used in the mass-distance prior.
  5. [Fig. 4] Figure 4, which is intended to illustrate the Maxwell-type behavior of Eq. (19), uses f1=0.1 and a finite range of F; the actual behavior for small F is divergent because of the f1/\sqrt{F} term, so the caption and the surrounding text should be reconciled with the analytic expression.

Circularity Check

2 steps flagged · score 6.0 of 10

The core CKG derivation is independent, but the EHT shadow consistency is a postdicted fit obtained with reconstructed Lagrangians that differ from the advertised models, so the observational claim partially reduces to a construction from the chosen metric.

  1. fitted input called prediction [Section IV.B-D and Conclusion]
    "To summarize, we have adjusted the parameters and shown that the three models can match the EHT observations for Sgr A⋆."

    The EHT shadow limits in Eq. (38) are the target against which rs/M is compared in Figs. 12-15. The parameters f1, q, a0, fk are varied until the computed shadow radius enters the 1σ or 2σ bands, and the conclusion then reports the scanned intervals as constraints and as evidence of consistency. Thus the agreement with EHT is a postdiction from a parameter search, not an independent prediction of the CKG-NED construction. The CKG field-equation derivation itself did not use EHT data, so this circularity is confined to the observational-consistency claim.

  2. self definitional [Eqs. (14), (19), (40)]
    "As a consistency check, we use the metric function (16) in Eq. (14) to obtain a Lagrangian density that also generates this solution. By doing so we get L(F) = a0F^fk − q^3f1/(√2√F) + f0 + F + Λ/2."

    The Lagrangian used to build the effective metric (40)-(41) for the shadow is obtained by substituting the chosen A(r) into the reconstruction formula (14), so L(F) is defined in terms of A(r) and the free integration constant f1. This reconstructed Lagrangian differs from the advertised model (15) whenever f1 ≠ 0, yet the shadow analysis uses f1 = 2×10^-65 or 2×10^-66. The EHT compatibility is therefore a property of the chosen metric plus the fitted f1 by construction, not a test of the independent Lagrangian (15). The same mismatch occurs for Model 2 between (24) and (26), and for Model 3 between (28) and (30).

full rationale

The derivation of the metric functions from the CKG field equations and the reconstruction formula (14) is self-contained and does not use the EHT data as an input; that part is not circular. However, the paper's shadow analysis is only loosely connected to the Lagrangians advertised in Eqs. (15), (24), and (28). The actual Lagrangians used in the effective metric, Eqs. (19), (26), and (30), are obtained by substituting the chosen metric into the reconstruction formula, and they contain a 1/√F term controlled by the integration constant f1. The shadow calculations then tune f1 and other parameters until the EHT bounds are met, and the conclusion explicitly states that the parameters were adjusted. Consequently, the claimed EHT consistency is a fit, and the associated parameter bounds apply to the reconstructed Lagrangians rather than to the originally proposed NED models. This is a partial circularity of the observational claim, not of the central CKG solution construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the untested CKG field equations, a restricted metric ansatz, the completeness of the two derived equations, the NED effective-metric photon prescription, and an extrapolation of EHT shadow bounds to non-Kerr spacetimes. The free parameters are mostly tuned to the shadow data.

free parameters (6)
  • f1 = 2e-66 to 2e-65 (in units of M=1)
    CKG integration constant in the reconstructed Lagrangians (19), (26), and (30); its value is tuned to keep shadow radii inside the EHT Sgr A* bands.
  • magnetic charge q = q < 0.876 (2-sigma) and q < 0.6 (1-sigma) for Model 1; q < 0.155 (2-sigma) and q < 0.038 (1-sigma) for Model 2; q <…
    Charge is varied as a free parameter and bounded by the shadow fits.
  • a0 = 0.5 (central); 0 < a0 < 2 allowed at 1-sigma
    Model 1 NED coupling constant; tuned to match the shadow radius.
  • fk = 2 (central); fk > 2 allowed at 1-sigma
    Model 1 NED exponent; tuned to match the shadow radius.
  • Lambda = 10^-52 or 10^-41 (in units of M=1)
    Cosmological constant integration constant, fixed from prior Kottler shadow constraints in Ref. [44] rather than fitted here.
  • f0 = 0
    Dimensionful integration constant in the reconstructed Lagrangians; set to zero in the shadow analysis.
assumptions (5)
  • domain assumption CKG field equations H_alpha mu nu = 8 pi G T_alpha mu nu correctly describe gravity.
    Adopted from Harada [27]; the theory is recent and not independently validated by experiment.
  • domain assumption The static spherically symmetric metric ansatz with C(r)=r^2 and a magnetic-only NED source is sufficient.
    Section II.B excludes electric charge, rotation, and scalar fields, so the solutions are a restricted family.
  • domain assumption The two field equations (12)-(13) are the only independent components for the ansatz.
    The paper does not show that all other components of H and T are automatically satisfied.
  • domain assumption Photon trajectories are governed by the NED effective metric (35).
    Section IV.A; standard in NED but an additional dynamical assumption for CKG.
  • ad hoc to paper EHT shadow-size limits (38)-(39) from Ref. [44] apply to these non-Kerr metrics.
    The limits were calibrated for Sgr A* assuming Kerr or spherical GR shadows; applying them to CKG-NED metrics is an extrapolation.

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Cite this review

Pith. "Pith review of Novel charged black hole solutions in conformal Killing gravity." pith.science (2026). https://pith.science/paper/VBQU6JGW

@misc{pith2026250200589,
  author       = {Pith},
  title        = {Pith review of: Novel charged black hole solutions in conformal Killing gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBQU6JGW}},
  note         = {Machine review of arXiv:2502.00589}
}
abstract

In this paper, we investigate static spherically symmetric solutions in the context of Conformal Killing Gravity, a recently proposed modified theory of gravity that offers a new approach to the cosmological constant problem. Coupling this new theory with nonlinear electrodynamics, we derive the corresponding field equations and study their behavior under different parameter choices. We analyze three different models, each focusing on different key parameters. Our results reveal a rich causal structure with multiple horizons and transitions between extreme and non-extreme solutions depending on the parameter values. Moreover, we compute the nonlinear Lagrangian density for each model and find that it agrees with Maxwell theory in the limit $F \rightarrow 0$. We also confirm the existence of a central curvature singularity via the Kretschmann scalar. To connect our theoretical results with observational prospects, we study the black hole shadows associated with each model. The analysis shows that the calculated shadow size and shape of the three proposed models are consistent with the data for the supermassive object at the center of our galaxy and are therefore possible candidates for modeling this structure.

Figures

Figures reproduced from arXiv: 2502.00589 by the authors.

Figure 1
Figure 1. Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Graphical representation of the Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Graphical representation of the Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: Graphical representation of the function [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: Graphical representation of the Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 12
Figure 12. Figure 12: Shadow radius rs of the model of a black hole (red curve) metric function given (40) and in units of BH mass M, as a function of a/M. In the left plot we have used the values: fk = 2, q = 0.5, Λ = 10−41 and f1 = 2 × 10−65. The right plot depicts the fixed values: q = …
Figure 13
Figure 13. Figure 13: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]
Figure 14
Figure 14. Figure 14: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]
Figure 15
Figure 15. Figure 15: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]

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