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Nonlinear electrodynamics can make a single static black hole cast two distinct shadows, one per photon polarization, by splitting the effective geometry through vacuum birefringence.

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2026-08-02 17:59 UTC pith:44NHKDD3

load-bearing objection The two-shadow claim is real and explicitly computed, not just conjectured; the paper deserves a referee, with minor caveats about an over-stated condition and a hand-picked μ. the 3 major comments →

arxiv 2603.17007 v2 pith:44NHKDD3 submitted 2026-03-17 gr-qc hep-th

Two shadows of a single black hole: Vacuum birefringence phenomena within Einstein-nonlinear-electrodynamics

Marco A. A. de Paula , Haroldo C. D. Lima , Pedro V. P. Cunha , Carlos A. R. Herdeiro , Lu\'is C. B. Crispino This is my paper
classification gr-qc hep-th
keywords nonlinear electrodynamicsvacuum birefringenceblack hole shadowlight ringeffective metricSagittarius A*four-forcemagnetically charged black hole
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Photons in nonlinear electrodynamics do not all travel on the spacetime metric: when the Lagrangian depends on both electromagnetic invariants, the theory predicts vacuum birefringence, with each photon polarization following its own effective geometry. The paper shows that this split has a dramatic consequence for black holes: a single static, magnetically charged black hole in the EH-inspired NED model can have two distinct unstable light rings and therefore cast two distinct shadows, one per polarization. It computes these light rings and shadow radii analytically and numerically, and shows the effective-geometry shadows are always larger than the shadow seen by other massless particles. Using the observed shadow of Sagittarius A*, the authors place upper limits on the black hole's charge-to-mass ratio and rule out the extremal case. They also recast photon motion in the spacetime metric as nongeodesic curves acted on by a four-force, extending a previous single-invariant derivation.

Core claim

Within the magnetically charged EH black hole spacetime, with Lagrangian L=F−µ(F²+7/4G²) and µ treated as a free parameter, the two effective metrics gbar± give a light-ring condition with two distinct unstable solutions, one for each polarization. Consequently the asymptotic shadow radii satisfy r_s+>r_s−>r_SG: the P+ polarization casts the largest shadow, P− an intermediate one, and the standard geometry the smallest. All are larger than the Maxwell charged black hole shadow at the same charge. Matching against the Sgr A* horizon-scale observations yields 1σ upper limits Q≲0.812M for P+ and Q≲0.806M for P−, and 2σ limits Q≲0.959M and Q≲0.950M, excluding an extremal charge. The paper also p

What carries the argument

The central object is the pair of effective metrics gbar±^µν, one for each photon polarization, together with the magnetic-factor functions G±_1(r) and G±_2(r) that encode how the nonlinear Lagrangian modifies the geometry. For the EH model these factors are simple rational functions of µQ²/r⁴; the light-ring condition—the vanishing of the effective radial potential's derivative—then yields two distinct unstable circular photon orbits, whose critical impact parameters give the two shadow radii. A second piece of machinery is the four-force identity expressing effective-metric geodesics as nongeodesic curves in the spacetime metric, with the G-dependent term identified as the vacuum-birefring

Load-bearing premise

The prediction of two shadows relies on the effective geometries keeping their Lorentzian signature all the way down to the event horizon; the paper assumes the parameters are chosen so that the horizon lies outside the 'signature radius' where the effective metric changes sign.

What would settle it

A polarization-resolved measurement of the Sgr A* shadow: for the allowed parameter range, the EH model predicts r_s+ > r_s− with a fractional split of up to ~1%. If a future experiment resolves the shadow in two orthogonal linear polarizations and finds identical radii, or if the measured radius falls outside the predicted band for the allowed Q/M, the birefringent two-shadow scenario for this model is falsified. A simpler check: for µ/M²=0.05 and Q/M=1, the two shadow radii differ by about 1%, so comparing the analytic approximations with full numerical ray tracing at that point would confir

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the EH model describes the Sgr A* black hole, its shadow must be polarization-split: P+ casts a shadow about 1% larger than P− for allowed parameters, and both are larger than the Maxwell charged shadow; current resolution cannot resolve the split but the prediction is in principle measurable.
  • The observed Sgr A* shadow radius excludes an extremal EH black hole and puts charge upper limits Q≲0.959M for P+ and Q≲0.950M for P− at 2σ, i.e. about 7×10²⁶ C in SI units.
  • Because the two effective geometries produce two unstable light rings with no stable companion, the standard single-metric topological light-ring theorem does not apply directly; a per-polarization extension would imply each polarization sector must contain its own standard unstable light ring.
  • Photon trajectories can be computed either as null geodesics of gbar± or as forced curves in the spacetime metric; the four-force has a conserved effective four-current, so nonlinear electromagnetic self-interaction behaves like a source term from the spacetime perspective.
  • In weak deflection, the birefringence effect is invisible to order 1/b³: the deflection angle coincides with the Maxwell charged black hole, so the two-shadow signature appears only in the strong-field regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If future very-long-baseline interferometry achieves the needed resolution and polarization sensitivity, measuring the shadow radii separately for two orthogonal polarizations would isolate µ independently of Q, since the size of the split is controlled by µQ²/r⁴.
  • The same birefringent effective-metric machinery should apply to rotating NED black holes; rotation would break the degeneracy between P+ and P− shadow shapes, likely producing polarization-dependent photon-ring asymmetry rather than just a radius shift.
  • The four-force picture suggests a practical route to ray-tracing: instead of integrating with two separate effective metrics, one can integrate standard geodesic equations with the force term, which may simplify future image simulations and accretion-disk emission models.
  • The two-shadow effect implies that any polarization-averaged image of such a black hole is a weighted superposition of two slightly offset shadows; if astrophysical emission is polarized, the apparent shadow centroid or ring brightness could shift with the polarizer angle even without NED plasma effects.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies Einstein-nonlinear-electrodynamics with the Euler-Heisenberg-type Lagrangian L = F - μ(F² + 7/4 G²), constructs magnetically charged static, spherically symmetric black hole solutions, and derives the two effective metrics that describe photon propagation for the two polarizations. It claims that a single such black hole can support two distinct unstable light rings and therefore two distinct shadow radii, interprets photon motion in the spacetime metric as nongeodesic motion under a four-force, and uses Sgr A* shadow observations to place upper limits on the charge-to-mass ratio. The central quantitative results are the light-ring equation (68), the shadow radius formulas of Sec. VI A, and the astrophysical constraints in Sec. VI C.

Significance. If established, the two-shadow prediction is a concrete strong-field observable consequence of vacuum birefringence in NED, and the paper provides a useful template for connecting such effects to EHT-type observations. The manuscript has clear strengths: the effective-metric formalism is applied carefully, the light-ring/shadow results are cross-checked by analytical first-order-in-μ approximations (Sec. V B) and by two independent ray-tracing codes (Sec. VI B), and the comparison with Sgr A* data is a natural application. However, several parameter-domain and approximation issues need to be resolved before the claims can be accepted as stated.

major comments (3)
  1. [Sec. IV, Eq. (59) and text after Fig. 2] The Lorentzian-signature condition for the P− effective metric is asserted but not demonstrated. One needs r_h > r_sig^− = (12 μ Q²)^{1/4} outside the horizon. This is not guaranteed by the BH-existence bounds in Eq. (52): for parameters near the stated upper end, e.g. μ = 25M²/324 and Q² = 25M²/24, the outermost horizon is near 0.89M while r_sig^− ≈ 0.99M, so the P− effective metric changes signature in the exterior. The paper must either prove r_h > r_sig^− for all parameter values used in Figs. 5, 12, 14 and 15, or explicitly restrict to a subregion and verify the Sgr A* allowed region lies inside it. Without this, the second shadow is not defined in the standard sense for those parameters.
  2. [Sec. IV, Eqs. (53)–(54)] The effective metrics are obtained by neglecting O(μ²) terms, while the spacetime metric (49) is exact for the Lagrangian (43). This is an uncontrolled approximation for the strong-field optical calculations. The polarization splitting shown in Fig. 5(bottom) is at the ~1% level, so O(μ²) corrections to the effective metric can shift the two shadow radii by a non-negligible fraction of that splitting and may affect the Q/M bounds in Figs. 14–15. Please use the exact effective metrics or provide an error estimate showing that the O(μ²) truncation is quantitatively irrelevant for the claimed constraints.
  3. [Sec. IV, Eqs. (50)–(52)] The parameter-domain statements are problematic. Equation (52) does not appear to be a correct condition for existence of a horizon: for example, Q = M and μ = 0.1 M² still yields an outer horizon with f(2M)>0. Moreover, solving f(r)=0 and f'(r)=0 as in Eqs. (50)–(51) can select an inner double root rather than the outermost horizon (as occurs for Q = M and μ = 40M²/729), making the labels 'extreme' and Q_ext ambiguous. The authors should define the event horizon as the largest positive root of f and recompute the extreme-charge family accordingly. This is directly tied to the domain on which the two-shadow effect is claimed.
minor comments (5)
  1. [Sec. IV, after Eq. (59)] The sentence 'we always choose Q and μ in a such way that r_h > reff' contains a typo: it should be 'r_h > r_sig'. Also, the subsequent statement that r_sig^+ is always inside the horizon should be qualified as holding for the chosen parameter region, not generally.
  2. [Sec. III A, Eq. (31)] The derivation of Eq. (30)–(31) is summarized as 'inserting' equations with no intermediate steps. Given that Eq. (31) is a central secondary claim, it would help to include the computation in an appendix or supplementary file.
  3. [Sec. V C and Sec. VII] The statement that two unstable LRs 'circumvent' the topological theorem of Ref. [63] is imprecise: the theorem applies to a single spacetime metric, and each effective metric individually has one unstable LR. The correct statement is that the theorem needs to be generalized to the multi-metric setting; the paper already says this in Sec. V C, so the 'circumvent' wording in the Final Remarks should be adjusted.
  4. [Sec. III B, Eq. (42)] The last term in Eq. (42) contains ∇_μ(⋆F^{μν}), which vanishes identically by Eq. (9). This is not an error, but the term is superfluous and may confuse readers; it should be removed or commented on.
  5. [Sec. VI C and Fig. 15] Fig. 15 restricts μ/M² to [0, 0.05], but the text and Eq. (52) state a larger allowed range. Please clarify why the scan stops at 0.05 and whether the r_h > r_sig condition is satisfied throughout the plotted region (see major comment 1).

Circularity Check

0 steps flagged

No significant circularity: the two-shadow prediction follows directly from the EH Lagrangian and the standard effective-metric formalism, not from fitted inputs or self-citations.

full rationale

The derivation chain is self-contained. The effective metrics (53)-(54) are obtained by substituting the EH Lagrangian (43) into the standard NED effective-metric formula (14)-(18), an external input due to Plebanski, Boillat, Gutierrez, and Novello. The light-ring condition (68) follows from the Hamiltonian (60) and admits two distinct radii because the two effective metrics have different magnetic factors; the shadow radii follow from the critical impact parameter via Eq. (67) and the distant-observer limit (107). No parameter is fitted to the shadow and then re-predicted: mu is scanned and Q is constrained a posteriori by the independent Sgr A* bounds (117)-(118). The self-citations to Refs. [49] and [63] are not load-bearing. Appendix A checks that the four-force expression reduces to the earlier single-scalar result, and Ref. [63] is used only to frame the novelty of two unstable light rings, which are computed directly from Eq. (68). The stated restriction 'we always choose Q and mu in such a way that r_h > r_sig' is an asserted validity condition, not a circular reduction; if unverified over the allowed parameter space, it is a robustness/correctness concern rather than a definitional equivalence. Thus no circular step is present.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 1 invented entities

The central load is carried by the standard effective-geometry formalism plus a deliberately ad hoc EH-like Lagrangian with μ free; no new particles, forces, or dimensions are introduced. The four-current j^ν_NED is a reinterpretation without an independent measurable consequence.

free parameters (1)
  • μ (Euler-Heisenberg-type parameter) = μ/M² = 0.05 for most plots and for the Sgr A* comparison; extreme value μ_ext/M² = 40/729 (with Q_ext/M ≈ 1.0115); QED b
    Chosen by hand, not fixed by theory. Footnote 2 in Sec. IV explicitly states μ is not the QED coefficient but a generic free parameter. The chosen value makes the birefringent two-shadow effect visible; at the QED value the effect is negligible for a Sgr A*-mass black hole (Sec. VI C).
axioms (5)
  • domain assumption Photons in NED propagate along null geodesics of the two effective metrics \bar g^± (Eqs. 14-19), not of the spacetime metric.
    Standard Plebański/Boillat/Gutiérrez-Novello result assumed without re-derivation; it is the foundation of the two-shadow claim.
  • standard math F^ν_α ⋆F^{αμ} = -¼ G g^{μν} (Eq. 20).
    Algebraic identity used to collapse the G-dependent terms in Eq. (14) into A±g+B±h; the sign is convention-dependent and not proven in the paper.
  • ad hoc to paper L(F,G)=F-μ(F²+7/4 G²) with μ a generic free parameter, not the QED coefficient.
    Sample model chosen to explore birefringent strong-field optics (Eq. 43, footnote 2); no independent evidence fixes μ at 0.05M².
  • domain assumption Parameters are restricted so that r_h > r_sig (zeros of G±j), keeping the effective metric Lorentzian outside the horizon.
    The paper states "In general scenarios, this is not true" (Sec. IV/V); all shadow results depend on this restriction.
  • standard math Light rings carry a winding number; a static axisymmetric asymptotically flat black hole admits at least one unstable light ring, with extra light rings in stable/unstable pairs.
    Invoked in Sec. V C/VII to frame the two-light-ring result; not re-derived.
invented entities (1)
  • Effective four-current j^ν_NED (Eqs. 36, 42) no independent evidence
    purpose: To reinterpret NED nonlinearity as an effective conserved four-current sourcing Maxwell equations; presented as an interpretation, not used in the shadow calculation.
    The paper presents it as "a possible interpretation" (Sec. III B); it has no independent falsifiable handle.

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read the original abstract

One of the main features of nonlinear electrodynamics is the existence of an effective geometry that describes the geodesic motion of photons. A detailed analysis of the properties of effective geometry is of utmost importance for a better understanding of nonlinear electrodynamics theories and their possible imprints on physics, especially in the context of black holes. We consider a nonlinear electrodynamics model that depends on the two electromagnetic scalar invariants and obtain that the motion of photons in nonlinear electrodynamics exhibits \textit{vacuum birefringence}, i.e., photons can propagate along two distinct paths, depending on their polarization. As a consequence of this phenomenon, we show that static black hole solutions sourced by nonlinear electrodynamics can admit two distinct unstable light rings, leading to the formation of two distinct shadows. Moreover, to explore the potential astrophysical relevance of our results, we also compare them with the astrophysical observations for the shadow radius of Sagittarius A*. We place upper limits on the charge-to-mass ratio of the nonlinear electrodynamics-sourced black hole. We also show that the motion of photons in this context can be interpreted as nongeodesic curves subjected to a four-force term from the perspective of an observer in the spacetime metric, generalizing previous results in the literature for nonlinear electrodynamics models that depend on a single electromagnetic scalar invariant.

Figures

Figures reproduced from arXiv: 2603.17007 by Carlos A. R. Herdeiro, Haroldo C. D. Lima, Lu\'is C. B. Crispino, Marco A. A. de Paula, Pedro V. P. Cunha.

Figure 1
Figure 1. Figure 1: FIG. 1. Metric function of the EH BH solution, considering [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison between the location of the event horizon with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison between the LRs of EH and RN BH geometries, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The comparison between the numerical results of Sub [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Ratio between the shadows radius of the effective geometry, [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The comparison between the numerical result and the analyt [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Shadows edge in the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Shadows radii as functions of observer’s position, normal [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison between the shadow edge in the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Backwards ray-tracing images of the shadow and gravita [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Shadow radius of the EH BH, considering the two polar [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Gray scale difference image comparing the backwards [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Allowed values of the EH parameter [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗

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