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REVIEW 4 major objections 4 minor 41 references

Light propagation and gravitational lensing effects in charged Kalb-Ramond spacetime in nonlinear electrodynamics

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper derives analytic light-deflection and lensing formulas for charged Kalb-Ramond black holes with nonlinear electrodynamics, controlled by parameters l and γ.

desk verdict A careful Bozza lensing package for the KR-ModMax metric, but the Δφ−π subtraction is unjustified in a conical spacetime and injects an l-dependent artifact that dominates the observables. read the letter →

arxiv 2602.22905 v2 pith:TS566OKC submitted 2026-02-26 gr-qc hep-th

C. F. S. Pereira , Marcos V. de S. Silva , A. R. Soares , A. A. Araújo Filho , R. L. L. Vitória , H. Belich This is my paper · ORCID
classification gr-qchep-th
keywords gravitationallensinglightdeflectionKalb-RamondblackholesModMaxelectrodynamicsstrong-fieldlimitweak-fieldLorentzsymmetryviolationphotonsphere
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 exactly how light bends around an electrically charged black hole whose spacetime is modified by a Kalb-Ramond field (a Lorentz-symmetry-violating antisymmetric tensor background) and by ModMax nonlinear electrodynamics, controlled by the parameters l and γ. It derives closed-form weak-field deflection, a strong-field deflection split into a logarithmic divergent piece and a numerical regular piece, and the standard lensing observables built from them. If correct, the formulas show that even infinitely distant light picks up a constant deflection from the spacetime's conical asymptotics, and that the relativistic-image separation and flux ratio respond characteristically to l and γ. The paper also gives Einstein-ring radii and angular positions for weak-field lensing, with numerical tables for a galactic-center-sized black hole.

What carries the argument

The load-bearing object is the photon-sphere structure of the metric f(r)=1/(1−l)−2M/r+ξQ²e^{−γ}/((1−l)²r²), encoded by the combination D=ξQ²e^{−γ}/(M²(1−l)³). Photon geodesics are studied through the effective potential V_eff=L²f(r)/r²; the turning-point equation gives the impact parameter β, and the outer photon-sphere radius r_m2 controls the strong-field divergence. The technical engine is the strong-field expansion that rewrites the deflection integral in the variable z=1−r0/r, isolates the logarithmic divergence at r0→r_m2, and expresses the deflection as Δφ_D+Δφ_R, with Δφ_R evaluated numerically; the coefficients ã, b̃, and β_c then feed the lensing observables.

What would settle it

Numerically integrate the exact deflection integral (Eq. 28) for the metric in Eq. (2) without the strong-field expansion, for the same parameter choices as the paper's tables, and compare with δφ=Δφ_D+Δφ_R−π; if the difference does not vanish near the photon sphere, or if δφ does not diverge logarithmically as β→β_c, the central claim fails. A simpler check is whether Δφ_R stays finite and the coefficient of the log diverges as predicted when r0→r_m2.

Watch

Extended reading notes

Core claim

The central claim is that the charged Kalb-Ramond–ModMax black hole, with metric function f(r)=1/(1−l)−2M/r+ξQ²e^{−γ}/((1−l)²r²), produces a deflection angle whose weak-field limit is δφ≈π(−1+√(1−l))+4M(1−l)²/β+..., and whose strong-field limit is δφ=Δφ_R+Δφ_D−π, with Δφ_D a logarithmic divergence set by the photon-sphere parameters and Δφ_R computed numerically. From these the paper constructs the angular separation s and flux ratio r̃ of the relativistic images, plus the Einstein-ring angle and radius in the weak field. In the Schwarzschild limit l→0, γ→0, Q→0, all formulas reduce to the known 4M/β deflection and the standard Einstein ring.

Load-bearing premise

The load-bearing premise is that the standard strong-field logarithmic-divergence expansion, developed for asymptotically flat black holes, remains valid when the metric approaches the nonzero constant 1/(1−l) at infinity; the paper assumes this and does not prove that the divergence form or the definitions of β_c and the regular part survive in that non-flat limit.

Editorial extensions

If this is right

  • In the weak field, the constant term π(−1+√(1−l)) means the deflection does not vanish at large impact parameter; the spacetime behaves like a global-monopole-type conical spacetime, so l imprints itself even far from the black hole.
  • For fixed mass and charge, increasing positive l reduces weak-field deflection, negative l amplifies it, and the Schwarzschild result is recovered at l=0.
  • In the strong field, as l→1 the angular separation s between the outermost image and the packed inner images goes to zero while the flux ratio grows, so the relativistic images merge and brighten.
  • The nonlinearity parameter γ changes the strong-field observables in opposite directions for canonical (ξ=1) and phantom (ξ=−1) fields; in the canonical case the separation reaches a maximum and then develops an imaginary contribution, while in the phantom case it varies monotonically.
  • For a bulge star at fixed source and lens distances, the Einstein-ring radius and angle shrink by many orders of magnitude as l grows toward 1, giving a sharp weak-field discriminator for Lorentz violation.

Reading between the lines

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

  • The l-dependent constant deflection term suggests a clean test: measure lensing at very large impact parameters where the M/β term is negligible; any nonzero offset would be a direct signature of the conical asymptotics, independent of the electrodynamics parameters.
  • Because γ enters only through the combination D=ξQ²e^{−γ}/..., the model predicts a degeneracy between electric charge and nonlinearity; fitting θ∞, s, and r̃ together could in principle break it, but real data would need all three observables.
  • The strong-field expansion's applicability to this non-asymptotically-flat metric is the main open question; an independent derivation of the divergence structure for metrics with f(∞)≠1 would either validate or revise the numerical observables in the paper's tables.
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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 paper studies light deflection and gravitational lensing in a static, spherically symmetric black hole solution of Kalb-Ramond gravity coupled to ModMax nonlinear electrodynamics, with metric function (2). It presents a weak-field expansion of the deflection angle (Eq. (25)), a strong-field expansion using the Bozza/Tsukamoto method (Eqs. (38)-(43)), and then derives lensing observables: relativistic image separation and flux ratio in the strong-field limit (Eqs. (50)-(51), Tables I-V) and the Einstein-ring radius in the weak-field limit (Eq. (54), Table VI). The paper checks its Schwarzschild limits (Eqs. (26)-(27)) and considers both canonical (ξ=1) and phantom (ξ=-1) sectors. The central claim is that these formulas correctly predict the light deflection and observable signatures in the charged Kalb-Ramond-ModMax geometry.

Significance. If correct, this would be a useful addition to the lensing literature for Lorentz-violating black hole spacetimes, extending the standard geodesic and Bozza/Tsukamoto pipeline to a recently proposed charged solution with nonlinear electrodynamics. The paper's strengths are the explicit analytic weak-field expansion, the systematic application of the strong-field limit machinery, and the verification of Schwarzschild limits. The treatment of both canonical and phantom fields is also a positive feature. However, the significance is currently undermined by an unresolved issue with the definition of the deflection angle in a spacetime that is not asymptotically flat (the metric function tends to 1/(1-l), not 1), and by internal inconsistencies in the observables tables and formulas. Because these issues affect the main quantitative results, the paper needs substantial revision before the claimed predictions can be accepted.

major comments (4)
  1. [Sec. III.A, Eq. (25)] The deflection angle is defined as δφ = Δφ − π, but this spacetime is not asymptotically flat: f(r) → 1/(1−l) ≠ 1. Setting M=Q=0 in Eq. (24) gives Δφ = π√(1−l), not π. Therefore Eq. (25) contains a constant term π(−1+√(1−l)) that survives when the black hole is removed. This is not a gravitational deflection by the compact object; in the physical angular coordinate Φ = φ/√(1−l), the background geodesic has ΔΦ = π, and the constant disappears. The authors then use this δφ in the flat-space lens equation (40) and in the Einstein-ring derivation (Eqs. (52)-(54)), producing an l-dependent offset in Table VI that is a coordinate artifact. The authors need to define the deflection angle relative to the correct background (either by subtracting π√(1−l) or by working with the physical angle Φ) and rederive the lens equation and the β–θ relation accordingly. This is a load-bearing issue for all w
  2. [Sec. III.B, Eqs. (38)-(43)] The Bozza/Tsukamoto strong-field expansion is formulated for asymptotically flat spacetimes where δφ → 0 as β → ∞. In the present geometry, δφ tends to the nonzero constant π(−1+√(1−l)). The paper does not prove that the logarithmic divergence form in Eqs. (38)-(39), or the definitions of β_c, ā, and b̄ in Eqs. (42)-(43), remain valid in a conical asymptotic background. In particular, the regular part Δφ_R in Eq. (39) is computed numerically at the photon sphere, but the subtraction Δφ_R − π in Eq. (43) again inserts the flat-space reference. Consequently, the strong-field observables in Tables I–V depend on an unvalidated and likely incorrect constant offset. The authors should either extend the Bozza formalism to aspherical asymptotics or justify that the constant can be absorbed; the observables need to be recomputed with the corrected reference.
  3. [Sec. IV.A, Table I] There is a numerical/unit inconsistency in Table I. The text states that θ∞ = 26.5473 μas for the Schwarzschild black hole, but the l=0 row lists θ∞ = 0.00013 μas; other rows are also of order 10^-4 μas. The values of s (0.03322 μas at l=0) and r (6.82188 mag) match the known Schwarzschild strong-field observables, so the θ∞ column appears to be off by a large factor. Since θ∞ is used through Eq. (49) to set β_c and thus all other observables, the table as printed does not permit verification of the results. The unit error or computation error must be corrected, and the values in Tables I–V should be rechecked.
  4. [Sec. IV.B, Eq. (54)] The Einstein-ring formula (54) does not correctly reduce to the Schwarzschild result. Setting l=0 in Eq. (54) gives θ_E = (1/2)√(16M/DOL) = 2√(M/DOL) if the printed formula is used, whereas the correct Schwarzschild Einstein ring from Eq. (55) is θ_E = √(4M DLS/(DOS DOL)). The square-root term in Eq. (54) is missing a factor DLS/DOS multiplying 16M(1−l)^2/DOL. This algebra error propagates into Table VI, whose l=0 value (θ_E = 2.12 arcsec) matches Eq. (55) with the stated DOL=4 kpc, DOS=8 kpc, but not Eq. (54) as written. The formula must be corrected and the table recalculated.
minor comments (4)
  1. [Sec. III.A, Eq. (25)] The text states the expansion is kept 'up to the second order in M and charge Q', but Eq. (25) retains terms such as M Q^2/β^3 and M^2 Q^2/β^4, which are higher order if Q^2 counts as second order. The ordering convention should be stated explicitly.
  2. [Figs. 1-7] The plots are labeled in μarcsecs, but Eq. (25) contains a constant π(−1+√(1−l)) of order 0.1 rad ≈ 10^10 μas for l=0.1, which would dwarf the plotted values. The plotted curves appear to omit this constant term, contrary to Eq. (25). Either the plots should include the constant or the definition of δφ in the plots should be clarified.
  3. [Sec. IV.A, Tables II-V] The θ∞ columns in Tables II-V also contain values of order 10^-4 μas, which are inconsistent with the stated Sgr A* value of 26.5 μas. If the reported numbers are actually in different units, this should be stated; otherwise the tables need to be regenerated.
  4. [Sec. II, Eq. (2)] The metric is taken from Ref. [23] without re-derivation. For a self-contained lensing paper, a brief verification that this is a solution of the Kalb-Ramond-ModMax field equations would be helpful, although it is not strictly required for the lensing analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: weak- and strong-field deflection and lensing observables are analytic consequences of the external metric; self-citations are only limit checks.

full rationale

The derivation chain is open and non-circular. The line element (1)-(2) is taken from Ref. [23]; all subsequent quantities—conserved quantities (13)-(14), effective potential (17), photon sphere (20), orbit integrals (23)-(24), weak-field expansion (25), strong-field coefficients (31)-(43), and observables (47)-(56)—are obtained by direct substitution, series expansion, or numerical quadrature from that metric. The parameters l, gamma, xi, M, Q are inputs; none is fitted to the deflection angle, image positions, or Einstein radii. The self-citations [38] and [41] are used only to verify the l→0 Schwarzschild limits and the l=0 Einstein radius; those limits are standard independent results, and the citations are not the source of the central formulas. The metric's validity is inherited from the external Ref. [23], which is a citation dependency rather than a self-citation chain. The reviewer's concern about subtracting π in a conical spacetime with f(∞)=1/(1−l) is a physical correctness issue: the zeroth-order term π(−1+√(1−l)) follows directly from the chosen definition δϕ=Δϕ−π and may be a coordinate artifact, but it is not a fitted parameter dressed as a prediction, nor an assumption of the conclusion. Accordingly, no circular step is present.

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

No new entity or fitted constant is introduced; all model parameters are carried over from the input metric. The main ledger burden is the unproven transfer of asymptotically-flat strong-field lensing machinery to a conical spacetime, plus the acceptance of the KR-ModMax metric from prior work.

free parameters (4)
  • l (Lorentz-violation parameter)
    Input parameter of the metric Eq. (2), inherited from Ref. [23]; varied by hand in figures/tables, not fitted to data.
  • gamma (ModMax nonlinearity parameter)
    Input parameter controlling nonlinear electrodynamics; varied by hand from 0 to 1 in figures/tables; not fitted.
  • xi (canonical/phantom sign) = ±1
    Discrete sign in the metric that selects canonical or phantom field; chosen by hand, not fitted.
  • M/Q ratio = 1.2, 1.0, 0.8
    Illustrative mass-charge ratios used in Figs. 2-7; chosen by hand, not fitted.
assumptions (4)
  • domain assumption The metric in Eqs. (1)-(2) is a valid static spherically symmetric solution of Kalb-Ramond gravity coupled to ModMax electrodynamics.
    Paper takes f(r) directly from Ref. [23] and derives geodesics from it without validating the field equations.
  • ad hoc to paper Bozza/Tsukamoto strong-field expansion applies to a spacetime that is not asymptotically flat (f -> 1/(1-l)).
    The method in Ref. [34] is stated for asymptotically flat spacetimes; the present metric has a conical deficit/surplus, and the paper does not justify the extension. Used in Sec. III.B to obtain Eqs. (38)-(39).
  • standard math Null geodesics and conserved quantities from the Lagrangian Eq. (13) are standard; the deflection integral Eq. (23) is the correct geometric-optics expression.
    No derivation of the geodesic formalism is needed beyond the standard Euler-Lagrange treatment.
  • domain assumption The lens equation Eq. (40) with flat-space angular diameter distances remains valid in the asymptotic conical geometry.
    The spacetime is not flat at infinity, yet the observables use D_OL, D_OS, and D_LS as if in a flat lensing setup.

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Cite this review

Pith. "Pith review of Light propagation and gravitational lensing effects in charged Kalb-Ramond spacetime in nonlinear electrodynamics." pith.science (2026). https://pith.science/paper/TS566OKC

@misc{pith2026260222905,
  author       = {Pith},
  title        = {Pith review of: Light propagation and gravitational lensing effects in charged Kalb-Ramond spacetime in nonlinear electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TS566OKC}},
  note         = {Machine review of arXiv:2602.22905}
}
abstract

In this work, we theoretically investigate the deflection of light for strong- and weak-field regimes in the background of an electrically charged BH described in Kalb-Ramond gravity, which introduces the Lorentz symmetry violation parameter $l$, as well as the control of the degree of nonlinearity incorporated by electrodynamics through the parameter $\gamma$. We analytically constructed the expansion coefficients in both limits and used them as a basis to investigate gravitational lensing effects through observables, taking into account the variation of the parameters involved in the model, both for the canonical field and the phantom case.

Figures

Figures reproduced from arXiv: 2602.22905 by the authors.

Figure 1
Figure 1. FIG. 1: Light deflection in the weak-field regime. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: In the figures above, we fix the parameter [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: In the figures above, we fix the parameter [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4: In the figures above, we fix the parameter [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: In the figures above, we fix the parameter [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: In the figures above, we fix the parameter [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: In the figures above, we fix the parameter [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Angular deflection for a Schwarzschild-type BH in Kalb-Ramond gravity. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Angular deflection of light for a BH in Kalb-Ramond gravity, where in the panel on [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Angular deflection of light for a BH in Kalb-Ramond gravity, where in the panel on [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Light angular deflection diagram. [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Angular separation and 2 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Angular separation and 2 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Angular separation and 2 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

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Reviewed August 2, 2026 · model on record in the stance chip above.