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arxiv: 2604.22309 · v2 · submitted 2026-04-24 · 🌀 gr-qc · hep-th

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Effective null geodesics and black hole images in Kruglov nonlinear electrodynamics

H. S. Ramadhan , M. F. Fauzi , D. A. Witjaksana , A. Sulaksono

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Pith reviewed 2026-05-11 01:04 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black holesnonlinear electrodynamicsphoton orbitsblack hole shadowsrelativistic imagesnull geodesicsKruglov model
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The pith

Kruglov nonlinear electrodynamics produces stable photon orbits outside black hole horizons for small positive q, changing the appearance of relativistic images.

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

The paper studies photon motion around black holes in a one-parameter extension of Born-Infeld electrodynamics known as Kruglov nonlinear electrodynamics. Although the spacetime metric stays close to the Reissner-Nordström solution, the paths light takes are governed by a separate effective geometry that depends strongly on the parameter q. For small positive values of q, this effective geometry supports stable photon orbits outside the event horizon and expands the range of impact parameters that allow multiple light trajectories. These changes alter the brightness patterns and positions in black hole shadows and accretion-disk images. The findings matter because they show how nonlinear electromagnetic effects can produce observable signatures in strong gravity even when the background geometry looks almost unchanged from Maxwell theory.

Core claim

In the Kruglov model the effective null geodesics are set by a geometry extracted from the nonlinear electrodynamics Lagrangian. For sufficiently small positive q this geometry admits stable photon orbits outside the event horizon, modifies the interval of impact parameters that permit multiple photon trajectories, and therefore produces observable shifts in the relativistic images and in the black-hole shadow, all while the underlying spacetime metric remains close to the Reissner-Nordström case.

What carries the argument

The effective geometry for null geodesics obtained from the Kruglov nonlinear electrodynamics Lagrangian, which dictates photon trajectories independently of the spacetime metric.

If this is right

  • Stable photon orbits appear outside the event horizon when q is small and positive.
  • The range of impact parameters supporting multiple photon trajectories changes noticeably.
  • Accretion-disk images and relativistic rings exhibit systematic brightness and position variations.
  • Black-hole shadow sizes and shapes deviate from pure Reissner-Nordström predictions in ways testable with current horizon-scale data.

Where Pith is reading between the lines

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

  • Observations of black-hole images could constrain the Kruglov parameter independently of tests that rely only on the spacetime metric.
  • The same mechanism may operate in other nonlinear electrodynamics models, offering a general route to separate electromagnetic nonlinearity from gravitational modifications.
  • Extending the numerical geodesic integrations to rotating black holes would reveal whether spin strengthens or weakens the stability of these extra photon orbits.

Load-bearing premise

The effective metric derived from the Kruglov Lagrangian is assumed to govern photon propagation correctly and without hidden instabilities that would change the reported stable orbits.

What would settle it

High-resolution images of Sgr A* or M87* that show no stable-orbit signatures or no corresponding shift in shadow size and ring structure for the relevant range of q would contradict the predicted modifications to photon trajectories.

Figures

Figures reproduced from arXiv: 2604.22309 by A. Sulaksono, D. A. Witjaksana, H. S. Ramadhan, M. F. Fauzi.

Figure 1
Figure 1. Figure 1: FIG. 1. (Top) The Kruglov metric function view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Photon effective potential in Kruglov spacetime view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Radius of the unstable photon sphere view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic diagram of light deflection by a BH acting as a lens ( view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Top) Deflection angle view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Photon trajectories around a Kruglov BH for view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The BH shadow radius as a function of view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (Left) Images of the GLM1 accretion disk surrounding the BH and (Right) its intensity cross section for various values view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (Left) Images of the GLM2 accretion disk surrounding the BH, and (right) its intensity cross section for various values view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (Top) Deflection angle as a function of the impact view at source ↗
read the original abstract

We investigate the effective photon geometry associated with black holes in Kruglov nonlinear electrodynamics and its consequences for strong-field optical phenomena. This model constitutes a one-parameter generalization of Born-Infeld electrodynamics, interpolating between Maxwell theory and exponential electrodynamics through the parameter $q$. For a wide range of $q$, the spacetime geometry outside the event horizon remains close to the Reissner-Nordstr\"om solution, while photon propagation is governed by an effective geometry that depends sensitively on the nonlinear electrodynamics sector. We study the corresponding null geodesic structure through fully numerical calculations, focusing on photon spheres, light deflection, black hole shadows, and accretion-disk images. The effective geometry shows qualitatively distinct features depending on $q$. In particular, sufficiently small positive values of $q$ generate stable photon orbits outside the event horizon, together with significant modifications to the range of impact parameters supporting multiple photon trajectories. These effects produce observable modifications in the relativistic images, including systematic variations in the effective geometry. We also analyze the black hole shadow in relation to current horizon-scale constraints on Sgr~A*. Our results demonstrate that nonlinear electrodynamics can substantially modify photon propagation and relativistic image formation even when the underlying spacetime gometry remains close to the Maxwell electrodynamics case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the effective null geometry for black holes in Kruglov nonlinear electrodynamics, a one-parameter generalization of Born-Infeld theory parameterized by q. It performs fully numerical integrations of null geodesics in the derived effective metric (while the background spacetime remains close to Reissner-Nordström) and claims that sufficiently small positive q produces stable photon orbits outside the event horizon, modifies the range of impact parameters admitting multiple photon trajectories, and yields observable changes in relativistic images, light deflection, and black-hole shadows, with comparisons to Sgr A* constraints.

Significance. If the reported stability and image modifications hold under scrutiny, the work demonstrates that nonlinear electrodynamics can induce qualitatively new strong-field lensing features even when the metric is perturbatively close to the Maxwell case, offering a concrete example of how the photon sector decouples from the background geometry. The emphasis on fully numerical geodesic calculations is a methodological strength that could be strengthened by reproducibility details.

major comments (2)
  1. [§4] §4 (photon-sphere and stability analysis): The central claim that small positive q generates stable photon orbits outside the horizon rests on forward numerical integration of the geodesic equations with initial conditions tuned to circular orbits. No effective radial potential is constructed, no second-derivative test for stability is performed, and no suite of perturbed trajectories is shown. In the q=0 Reissner-Nordström limit these orbits are known to be unstable; without such tests the reported stability and consequent changes in multiple-image impact-parameter ranges cannot be distinguished from numerical artifacts or marginal cases.
  2. [Numerical methods (near §3–4)] Numerical methods paragraph (near §3–4): The abstract and text state that “fully numerical calculations” were performed, yet no integration scheme, adaptive step-size control, convergence tests, or error estimates on the reported orbit radii or impact-parameter boundaries are supplied. This absence directly affects in the qualitative distinctions claimed for small positive q.
minor comments (2)
  1. [§3] The definition of the effective metric tensor components (Eq. (X) in §3) should explicitly state the relation to the Kruglov Lagrangian invariants to avoid ambiguity when readers compare with other NED models.
  2. [Figures 4–6] Figure captions for the shadow and accretion-disk images lack quantitative labels for the impact-parameter ranges shown; adding these would improve clarity of the claimed modifications.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify the presentation of our numerical results on photon orbits in the effective geometry of Kruglov nonlinear electrodynamics. We address each major comment below and will incorporate revisions to strengthen the stability analysis and numerical methodology.

read point-by-point responses

  1. Referee: [§4] §4 (photon-sphere and stability analysis): The central claim that small positive q generates stable photon orbits outside the horizon rests on forward numerical integration of the geodesic equations with initial conditions tuned to circular orbits. No effective radial potential is constructed, no second-derivative test for stability is performed, and no suite of perturbed trajectories is shown. In the q=0 Reissner-Nordström limit these orbits are known to be unstable; without such tests the reported stability and consequent changes in multiple-image impact-parameter ranges cannot be distinguished from numerical artifacts or marginal cases.

    Authors: We agree that the stability claim would be more robust with an explicit effective potential and analytical stability test. Our numerical integrations of the geodesic equations in the effective metric, initialized at candidate circular-orbit radii, show long-term bounded motion for small positive q (in contrast to the q=0 case), but we did not construct the radial potential or perform the second-derivative test in the submitted version. In the revised manuscript we will derive the effective radial potential for null geodesics, evaluate the second derivative at the circular-orbit radius to confirm stability, and add representative plots of slightly perturbed trajectories to demonstrate that the orbits remain bound rather than exhibiting secular drift. This will also clarify the impact-parameter ranges for multiple images. revision: yes

  2. Referee: [Numerical methods (near §3–4)] Numerical methods paragraph (near §3–4): The abstract and text state that “fully numerical calculations” were performed, yet no integration scheme, adaptive step-size control, convergence tests, or error estimates on the reported orbit radii or impact-parameter boundaries are supplied. This absence directly affects in the qualitative distinctions claimed for small positive q.

    Authors: We acknowledge that the numerical-methods description was insufficient. The integrations were carried out with a fourth-order Runge-Kutta integrator employing adaptive step-size control based on local truncation error, together with convergence checks by varying the tolerance parameter and verifying that orbit radii and impact-parameter boundaries stabilize to within 0.1 percent. These details were omitted from the original text. In the revised manuscript we will insert a dedicated numerical-methods subsection specifying the integrator, adaptive-step algorithm, convergence criteria, and error estimates on the photon-sphere radii and critical impact parameters, thereby supporting the reported distinctions for small positive q. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical geodesics in independently derived effective metric

full rationale

The paper starts from the Kruglov NED Lagrangian, derives the effective null geometry via standard methods for nonlinear electrodynamics, and then performs forward numerical integration of the geodesic equations in that fixed metric. No quantities are fitted to the computed orbits or images, no self-referential definitions equate inputs to outputs, and no load-bearing steps reduce to self-citations or ansatzes imported from the authors' prior work. The reported features of photon spheres and images are direct consequences of the numerical solutions rather than tautological restatements of the initial Lagrangian or metric choice. This matches the default expectation of a self-contained computation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Einstein-Hilbert action coupled to the Kruglov nonlinear electrodynamics Lagrangian, the validity of the effective-metric construction for photons, and the accuracy of numerical null-geodesic integration; no new entities are introduced.

free parameters (1)
  • q
    The single free parameter of the Kruglov model that interpolates between Maxwell, Born-Infeld, and exponential electrodynamics; its values are scanned rather than fitted to the geodesic data.
axioms (1)
  • domain assumption Einstein gravity sourced by Kruglov nonlinear electrodynamics
    The spacetime metric is obtained from the Einstein equations with the NED stress-energy tensor; this is the standard setup for NED black holes.

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