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REVIEW 3 major objections 4 minor 121 references

Probing nonlinear electrodynamics-sourced black holes via light and orbital mechanics

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In ModMax black holes, Shapiro delay and gravitational redshift are the same for both photon polarizations, while Sagnac and kinematic shifts differ; S2 precession then caps the charge at 73% of extremal.

desk verdict Competent NED phenomenology with a correct gravitational-redshift equality and a useful S2 constraint, but the headline Shapiro polarization-independence claim only holds for equal closest approach, not for a fixed physical experiment. read the letter →

arxiv 2608.10071 v1 pith:NSDK6MZK submitted 2026-08-10 gr-qc hep-th

Marco A. A. de Paula , Ednaldo L. B. Junior This is my paper · ORCID
classification gr-qchep-th MSC 83B0583C5083C57 PACS 04.40.Nr04.70.-s
keywords nonlinearelectrodynamicsModMaxblackholevacuumbirefringenceShapirotimedelaySagnaceffectorbitalprecessionSagittariusA*effectivephotongeometry
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 establishes a clean partition of classical photon observables around a ModMax black hole, a general-relativistic solution sourced by a two-parameter nonlinear electrodynamics in which photons of different polarization travel on two different effective geometries. It shows that the Shapiro time delay and the gravitational redshift are exactly the same for both photon polarizations, so those measurements cannot reveal the model's vacuum birefringence. It also shows that the Sagnac effect and the kinematic (Doppler) shifts do depend on polarization, making them candidate probes of nonlinear electrodynamics. Using the measured periapsis precession of the S2 star around Sagittarius A*, the paper bounds the combination $e^{-\gamma}(Q/2M)^2$ by $0.135$, and combining that with the horizon condition implies the black-hole charge satisfies $Q \lesssim 0.73\,Q_{\rm ext}$ independently of $\gamma$. A sympathetic reader would take away a concrete tool kit for testing NED-sourced black holes with existing gravitational observables.

What carries the argument

The load-bearing object is the ModMax Lagrangian $L(F,G)=F\cosh\gamma-\sinh\gamma\sqrt{F^2+G^2}$, which yields the black-hole metric function $f(r)=1-2M/r+e^{-\gamma}Q^2/r^2$ and two effective photon metrics, one equal to the spacetime metric (P$^-$) and one with an extra $e^{-2\gamma}$ conformal factor on the $(t,r)$ sector (P+). The argument turns on the fact that this particular conformal rescaling cancels identically in the Shapiro-time integral and in the redshift ratio for timelike observers, while surviving in the Sagnac and kinematic-shift formulas. The orbital-precession calculation then reduces to a quadrature over the ModMax geodesic equation whose leading correction depends only on the product $e^{-\gamma}(Q/2M)^2$, which is why the data constrain the combination rather than $\gamma$ or $Q$ separately.

What would settle it

Measure S2-like orbital precession with uncertainty below about 0.1 in the ratio to the Schwarzschild prediction: a value outside $[0.91,1.29]$ would contradict $e^{-\gamma}(Q/2M)^2\le0.135$. Alternatively, a polarization-resolved Shapiro-delay or gravitational-redshift comparison that detects any difference between the two photon polarizations would falsify the claimed invariance, which the paper shows must vanish identically.

Watch

Extended reading notes

Core claim

The central claim is that birefringence in ModMax electrodynamics is invisible to some classical gravity tests and visible to others. Writing the two effective photon metrics as (35) and (36), the paper proves that because they differ only by the constant factor $e^{-2\gamma}$ in the $(t,r)$ sector, the propagation-time integrand and the gravitational redshift are invariant: the Shapiro delay satisfies $\Delta T_{\rm ModMax}=\Delta T_+=\Delta T_-$ (Eq. (51)) and the gravitational redshift satisfies $(z_{\rm grav})_+=(z_{\rm grav})_-$ (Eq. (80)). In contrast, the Sagnac proper-time difference and the kinematic shifts carry explicit $e^{-\gamma}$ factors and therefore differ between the two polarizations, with the P+ polarization suppressed relative to P$^-$. For massive particles, the periapsis precession of a test orbit is $\delta\omega_{\rm MM}= \frac{2\pi M}{a(1-\epsilon^2)}(3-2e^{-\gamma}q^2)$ with $q=Q/2M$; fitting the S2 precession gives $0\le e^{-\gamma}q^2\le 0.135$, and the horizon condition then forces $Q\lesssim 0.73\,Q_{\rm ext}$ for Sagittarius A*, independent of $\gamma$.

Load-bearing premise

The entire observational constraint depends on treating the S2 star's measured precession as the pure relativistic periapsis precession of a test particle around a static, spherically symmetric ModMax black hole with matching mass and orbital parameters, with no other astrophysical effect shifting the precession by as much as ten percent.

Editorial extensions

If this is right

  • No polarization-dependent Shapiro delay should be observed for ModMax-sourced black holes in the geometric-optics regime, so time-delay experiments cannot by themselves test the birefringence of this model.
  • Polarization-resolved Sagnac or kinematic-shift measurements can in principle distinguish the two photon paths: the paper estimates that $\gamma \gtrsim 10^{-11}$ would produce a detectable relative Sagnac deviation given a clock accuracy of $10^{-11}$, and even $\gamma=0.01$ can yield roughly a 12% deviation from the Reissner–Nordström case for fast sources and the P+ polarization.
  • The S2 precession bound $e^{-\gamma}(Q/2M)^2 \le 0.135$ leaves $\gamma$ and $Q$ degenerate; only after imposing the horizon condition does one get the $\gamma$-independent statement $Q \lesssim 0.73\,Q_{\rm ext}$, ruling out near-extremal ModMax black holes as models of Sagittarius A*.
  • The same two-stage procedure—derive effective-metric observables, then compare with S2-like orbital data—transfers directly to other two-invariant NED models such as Born–Infeld or Euler–Heisenberg black holes, whose effective geometries are structurally different and may break the Shapiro/redshift degeneracy found here.
  • Weak-field Shapiro delays are only weakly sensitive to ModMax corrections: relative deviations from Reissner–Nordström stay near $10^{-4}$ even for $\gamma$ up to 3, so this observable is not a practical constraint channel for the model.

Reading between the lines

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

  • A general lesson the authors do not spell out: any NED model whose two effective photon metrics are related by a constant conformal rescaling of the $(t,r)$ block will be birefringence-blind in both Shapiro delay and gravitational redshift, so the observable split they find is a classification criterion, not a ModMax accident.
  • The S2 constraint is genuinely on the screened charge $e^{-\gamma/2}Q$; an independent measurement of the intrinsic charge—through lensing or shadow radius, for example—could break the $\gamma$–$Q$ degeneracy and turn the bound into a limit on $\gamma$ itself.
  • A natural next test would be to repeat the analysis for a rotating ModMax black hole; frame dragging couples to the Sagnac effect and might amplify the polarization difference beyond the spherical case, which the paper explicitly flags as a future direction.
  • The $\gamma\gtrsim10^{-11}$ detectability threshold is an idealized clock-accuracy statement; realistic magnetar or Sgr A* environments will add noise and systematics, so the practical sensitivity is likely weaker and should be assessed with a full astrophysical noise model.
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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

3 major / 4 minor

Summary. This manuscript studies the ModMax black hole, a nonlinear-electrodynamics (NED) solution with a two-parameter Lagrangian, by computing several classical observables: Shapiro time delay, Sagnac effect, gravitational and kinematic shifts, and the pericenter precession of the S2 star around Sagittarius A*. The paper's central claims are that the Shapiro time delay and the gravitational redshift are exactly the same for both photon polarizations in ModMax electrodynamics, that the Sagnac effect and kinematic shifts can distinguish the two effective metrics, and that the GRAVITY S2 precession measurement bounds the combination e^{-gamma}(Q/2M)^2 to be at most 0.135, implying Q is below about 0.73 of the extremal charge. The derivations are self-contained, with the gamma -> 0 and Q -> 0 limits correctly recovering Reissner-Nordstrom and Schwarzschild, and Appendix B provides an exact proof of the Shapiro integrand cancellation. The paper also corrects a previous treatment of photon energy and angular momentum in Ref. [84]. The main concerns are that the Shapiro equality is demonstrated only for equal closest-approach radius rather than for fixed source-receiver geometry, and that the Sagnac proper-time calculation appears to use the effective metric for the massive clock rather than the spacetime metric.

Significance. If the conclusions hold, the paper offers a useful classification of which classical photon observables are sensitive to vacuum birefringence in NED-sourced spacetimes, plus a concrete astrophysical constraint on ModMax parameters. The analytical work is largely transparent, and the explicit recovery of standard limits, the exact cancellation in Appendix B, and the correction of the photon energy and angular momentum definitions in Ref. [84] are valuable contributions. The S2 constraint, however, is only as strong as the assumption that the Schwarzschild-referenced GRAVITY measurement can be transferred directly to a charged NED spacetime, which is not argued in detail. The paper is a reasonable candidate for publication after the load-bearing issues below are addressed.

major comments (3)
  1. [IV B, Eq. (51), Appendix B] The equality Delta T_ModMax = Delta T_+ = Delta T_- is proved for the one-parameter family of null rays sharing a common closest-approach radius d. This is not the boundary condition of a radar or lensing experiment: for fixed emitter and reflector positions, the total azimuthal separation fixes d separately in each effective metric, because at equal d one has dphi/dr restricted to the + polarization equal to e^{-gamma} times the corresponding quantity for the - polarization (from Eq. (40)). Since Eq. (50) depends on d through the logarithmic term, the two polarizations will in general have different round-trip delays for the same physical configuration. The paper should either restrict the claim to equal-d trajectories, presenting it as a mathematical invariance rather than an observable statement, or recompute the delays under fixed boundary conditions before stating that the Shapiro effect is insensitive to the birefringent structure of ModMax electrodynamics.
  2. [V B, Eqs. (57)-(66)] In the Sagnac derivation the observer is a massive clock, so its proper time should be computed with the spacetime metric, i.e. with f(R), while only the null condition for the counter-propagating beams should use the effective metric components. The manuscript substitutes A(R)_+ = e^{-2gamma} f(R) and A(R)_- = f(R) into Eq. (57), so the P_+ proper-time result uses the effective g_tt rather than the spacetime g_tt. The prefactor in Eq. (57) should be the same f(R) for both polarizations, and only the coordinate-time difference in Eq. (56) should carry the polarization-dependent A(R). As written, the relative deviation (65), the lower bound (66), and Figs. 4-6 are affected by this mixing of metrics. Please correct the derivation or explicitly justify why the clock's proper time is computed with the effective metric.
  3. [VII, Eqs. (98)-(100)] The constraint e^{-gamma}(Q/2M)^2 <= 0.135 and the resulting bound Q <= 0.73 Q_ext are obtained by identifying GRAVITY's Schwarzschild-referenced precession ratio f = 1.10 +/- 0.19 with the ratio delta_omega_MM / delta_omega_GR. That measured value is the output of a fit that assumes the Schwarzschild metric for all other relativistic effects, including the redshift, the Romer delay, and the mapping between astrometric observables and orbital elements. In a ModMax spacetime with nonzero Q and gamma, those other effects are also modified, so the value of f that would be inferred from the same S2 data is not necessarily the same ratio. The bound should be presented as conditional on this transfer assumption, or supported by a simplified refit or an explicit estimate of the resulting systematic error. The paper does not currently flag this limitation in Sec. VII.
minor comments (4)
  1. [Appendix B] The proof uses a function C(r) in Eqs. (B4)-(B5) without defining it; since the angular sector is not rescaled in the effective metrics, C(r) = r^2 should be stated explicitly before the conformal-rescaling argument.
  2. [Eq. (65)] The symbol Delta_gamma for the relative deviation between the two polarizations is easily confused with a change in the ModMax parameter gamma; renaming this quantity, for example delta_tau, would improve readability.
  3. [Eq. (78)] The notation in Eq. (78) uses the same symbols +/- for the polarization label and for the blueshift/redshift sign. Although the text explains this, the equation is very hard to read; a notation with separate subscripts, such as P_+ and P_- for polarization and a separate sign label for the shift, would be clearer.
  4. [IV C, Tables I and II] The relative deviations in Tables I and II are identical because all distances scale linearly with M; it would be helpful to state this scaling explicitly in the text rather than only noting it in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

Self-contained derivations; no circularity found.

full rationale

Walking the claimed derivations, I find no load-bearing circular step. The Shapiro-delay equality (Eq. (51)) is not a fit or a renamed input: Appendix B exhibits the explicit cancellation under A1=e^{-2γ}A2, B1=e^{-2γ}B2 with b1=e^γ b2, so T1=T2 is a proven identity rather than an assumed conclusion. The gravitational-redshift equality (Eq. (80)) follows by direct substitution of (gbar_tt)_+=e^{-2γ} f and (gbar_tt)_-=f into Eq. (74), with the conformal factors cancelling; no fitted quantity enters. The Sagnac and kinematic-shift polarization differences are straightforward consequences of the different effective metrics (Eqs. (60)-(65), (82)-(83)). The S2 constraint in Sec. VII is an inequality applied to an independent GRAVITY measurement (Ref. [91]): the model-dependent ratio f = 1 - (2/3)e^{-γ}(Q/2M)^2 is solved against the externally reported f = 1.10 ± 0.19, yielding e^{-γ}(Q/2M)^2 ≤ 0.135; this is a parameter constraint, not a prediction of the same data from a fitted parameter. Self-references (e.g., Ref. [106] for standard circular-geodesic 4-velocities, and Ref. [48] for the P± notation) are ancillary, and the paper in fact corrects an external prior treatment (Ref. [84]) rather than relying on it. The skeptic's concern about fixed-boundary versus fixed closest-approach comparison is a question of physical interpretation, not a reduction of the derivation to its inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central results rest on the standard effective-metric formalism and the ModMax Lagrangian; no new particles, forces, or dimensions are introduced. The only data constraint comes from an external measurement, the GRAVITY S2 precession.

free parameters (2)
  • gamma (ModMax nonlinear parameter) = unconstrained; benchmarks 0 to 2; bound e^-gamma q^2 <= 0.135 from S2
    Free parameter of the ModMax Lagrangian density (Eq. 14). The paper does not fit it; it derives constraints and uses illustrative values.
  • Q/M (charge-to-mass ratio) = benchmarks 0.5 and 0.8; constrained as e^-gamma (Q/2M)^2 <= 0.135
    Model parameter of the ModMax BH metric (Eq. 32). Chosen by hand for tables and constrained via S2 precession in Sec. VII.
assumptions (4)
  • domain assumption Photons follow null geodesics of the effective metrics (Eq. 9), with conserved energy and angular momentum defined with respect to the effective metric (Eq. 70).
    Standard NED geometric-optic result; the paper's Appendix A explicitly defends this convention against Ref. [84]. Central to Shapiro, Sagnac, and shift computations.
  • domain assumption ModMax Lagrangian L = F cosh gamma - sinh gamma sqrt(F^2 + G^2) with gamma >= 0 (Eq. 14).
    Model input from Refs. [60-62]; gamma < 0 is excluded due to causality. All results inherit this model choice.
  • domain assumption The spacetime is static, spherically symmetric, purely electric with G = 0 (Sec. II B and Sec. III).
    Simplification that yields the ModMax metric (Eq. 32) and effective metrics (35)-(36). A rotating generalization is left for future work.
  • domain assumption GRAVITY Collaboration's S2 precession ratio f = 1.10 +/- 0.19 is identified with delta-omega_MM / delta-omega_GR for a ModMax BH with the same a = 1031, epsilon = 0.885 and M_Keck (Sec. VII).
    Data interpretation assumption; if Sgr A* is not effectively static or uncharged in the way modeled, the bound (99) does not follow.

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Pith. "Pith review of Probing nonlinear electrodynamics-sourced black holes via light and orbital mechanics." pith.science (2026). https://pith.science/paper/NSDK6MZK

@misc{pith2026260810071,
  author       = {Pith},
  title        = {Pith review of: Probing nonlinear electrodynamics-sourced black holes via light and orbital mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSDK6MZK}},
  note         = {Machine review of arXiv:2608.10071}
}
abstract

Well-founded models of nonlinear electrodynamics (NED) are typically characterized by an additional parameter associated with their deviation from linear electrodynamics. In addition to that, from a phenomenological point of view, one of the most intriguing results of NED theory is that photons follow null geodesics according to an effective geometry. In particular, for a two-parameter electromagnetic Lagrangian density, we might observe the vacuum birefringence phenomenon, i.e., photons can follow null geodesics according to two effective geometries. Over the past few decades, efforts have been made to better understand the role of effective light cones in NED theory, as well as to constrain the additional free parameters provided by the theory. We investigate the ModMax black hole (BH) geometry, which is a well-motivated NED-sourced BH based on a two-parameter Lagrangian density, via light and orbital mechanics, considering the Shapiro time, the Sagnac effect, the gravitational and kinematic shifts, and the orbital precession. We find that the corrections introduced by nonlinear electromagnetic fields can be significant. In particular, we show that the Shapiro time delay and the gravitational redshift are identical for both effective metrics, making these observables insensitive to the birefringent structure of ModMax electrodynamics. By contrast, the Sagnac effect and the kinematic shifts remain capable of distinguishing the two effective geometries. Moreover, the orbital mechanics investigation, considering the orbital periapsis precession of the S2 star around Sagittarius A$^{\star}$, provides an observational constraint on the coupling between the NED parameter and the charge-to-mass ratio of the ModMax BH. Generally speaking, we provide a step-by-step procedure for theoretically exploring signatures of NED fields and constraining the free parameters of NED-sourced BH models.

Figures

Figures reproduced from arXiv: 2608.10071 by the authors.

Figure 1
Figure 1. FIG. 1. Horizon structure of the ModMax BH geometry as a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Metric function of the ModMax BH geometry, considering [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of the setup associated with the Sagnac effect. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Allowed values for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Top panel: Relative deviation between the Sagnac effect for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Effective angular velocity of photons according to the polar [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Gravitational redshift for the ModMax BH, as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Blueshift (left panel) and redshift (right panel) for the ModMax BH, considering the polarization of light [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Ratio between the blueshift (top panel) and redshift (bottom [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Redshift for the ModMax BH, considering the polarization [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Constraint given by Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

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