pith. machine review for the scientific record. sign in

arxiv: 2604.14213 · v1 · submitted 2026-04-07 · ⚛️ physics.gen-ph · gr-qc· hep-th

Recognition: 2 theorem links

· Lean Theorem

On Computational CUDA Studies of Black Hole Shadows

S. E. Baddis , A. Belhaj , H. Belmahi , S. E. Ennadifi , M. Jemri
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:46 UTC · model grok-4.3

classification ⚛️ physics.gen-ph gr-qchep-th
keywords black hole shadowglobal monopolesEuler-Heisenberg black holesCUDA computationenergy emission raterotating charged black holesEvent Horizon Telescope
0
0 comments X

The pith

Rotating charged Euler-Heisenberg black holes with global monopoles produce shadows and emission rates that depend on monopole strength, charge and spin but not on the nonlinear parameter.

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

The authors combine the Hamilton-Jacobi equation with CUDA-accelerated ray tracing to study light around rotating charged black holes that also contain global monopoles. They map how the shadow silhouette and radiated energy change when the monopole parameter, electric charge or rotation rate is varied. The nonlinear Euler-Heisenberg term leaves both the shadow boundary and the emission rate essentially unchanged. The same numerical setup is used to extract upper limits on the three influential parameters so the predicted shadows remain compatible with Event Horizon Telescope images.

Core claim

Both the shadow structure and the energy emission rate depend on the global monopole parameter, the electric charge, and the rotation parameter. However, the Euler-Heisenberg nonlinear parameter does not significantly affect either the shadow or the energy emission rate. A CUDA-based computational approach establishes strict bounds on the GM parameter, the electric charge, and the rotation parameter to reconcile the predictions with Event Horizon Telescope observations.

What carries the argument

CUDA-accelerated numerical integration of null geodesics using the Hamilton-Jacobi formalism in the rotating charged Euler-Heisenberg metric with global monopoles.

If this is right

  • Increasing the global monopole parameter enlarges the black hole shadow.
  • Higher electric charge and faster rotation also enlarge or distort the shadow.
  • The energy emission rate increases with the monopole parameter, charge and rotation rate.
  • The nonlinear Euler-Heisenberg term can be neglected for shadow and emission calculations.
  • Event Horizon Telescope data translate into concrete upper bounds on the allowed values of the three parameters.

Where Pith is reading between the lines

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

  • The same CUDA ray-tracing pipeline could be applied to other nonlinear electrodynamics or modified-gravity models to test their shadow predictions against future telescope data.
  • The observed insensitivity to the nonlinear parameter suggests that classical Maxwell electrodynamics suffices for shadow phenomenology in this class of solutions.
  • If global monopoles are present, their density would leave a measurable imprint on the silhouettes of supermassive black holes observed at higher resolution.

Load-bearing premise

The assumed metric is the correct spacetime geometry around a rotating charged Euler-Heisenberg black hole with a global monopole and the ray-tracing code faithfully reproduces the null geodesics without numerical artifacts.

What would settle it

A high-resolution shadow observation that exhibits clear dependence on the Euler-Heisenberg nonlinear parameter or that falls outside the derived bounds on monopole strength, charge and rotation would contradict the central results.

Figures

Figures reproduced from arXiv: 2604.14213 by A. Belhaj, H. Belmahi, M. Jemri, S. E. Baddis, S. E. Ennadifi.

Figure 1
Figure 1. Figure 1: Regions in the (b, a)–plane, where the metric admits at least one real event horizon radius. To perform this evaluation effectively, we employ a CUDA-based numerical code which en￾ables high-performance parallel computations. It is recalled that CUDA is a general-purpose parallel computing platform and programming model that leverages the parallel comput￾ing capabilities of NVIDIA GPUs. In particular, the … view at source ↗
Figure 2
Figure 2. Figure 2: Effect of internal parameter on shadow behavior. As shown in the figure, the rotation parameter behaves similarly to that in ordinary black holes, slightly reducing the size of the shadow and deforming its shape into a D-like form. Hence, this parameter retains its role as a deformation parameter for these black hole solutions. Concerning the charge effect, increasing the charge reduces the shadow size wit… view at source ↗
Figure 3
Figure 3. Figure 3: Effect of η on shadow behavior. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effect of b parameter on shadow behavior. We move now to consider the effect of the b parameter. Fig. (4) depicts the black hole shadow for positive and negative values of b. The results indicate that variations in the magnitude of b do not significantly alter the shadow size. Interestingly, the positive values of b lead to a D-shaped shadow, whereas negative values produce a cardioid-like configuration. 3… view at source ↗
Figure 5
Figure 5. Figure 5: Variation of the energy emission rate as a function of the emission frequency for different values of a and Q. Fig. (6) shows the effect of the GM parameter for both small and large values of the rotation parameter. For small rotation values, the GM parameter decreases the energy 10 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of the energy emission rate as a function of the emission frequency for different values of η. We now turn to the effect of the parameter b where the association variation is shown in Fig. (7). As discussed in the previous section, this parameter has no significant impact on the shadow radius. Although the energy emission rate also depends on the temperature, which varies with the parameter b, th… view at source ↗
Figure 7
Figure 7. Figure 7: Variation of the energy emission rate as a function of the emission frequency for different values of b . 4 Constraints on black hole parameters from EHT ob￾servations using CUDA techniques In order to establish a bridge between the theoretical predictions and the observational data, this section provides an analysis of the shadow cast by rotating Euler-Heisenberg black holes 11 [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 8
Figure 8. Figure 8: Constraint Constraint regions in the (η, Q) plane obtained from CUDA-based simulations, showing agreement with the EHT observations of M87∗ and Sgr A∗ within 1−σ and 2−σ confidence levels for Q = 0.4 with M = 1. • M87∗ case: 0.001 ≤ η ≤ 0.07, within 1 − σ, 0.001 ≤ η ≤ 0.09, within 2 − σ. • Sgr A∗ case (EHTVLTI) : 0.001 ≤ η ≤ 0.05, within 1 − σ, 0.001 ≤ η ≤ 0.075, within 2 − σ. • Sgr A∗ case (EHTKeck): 0.00… view at source ↗
Figure 9
Figure 9. Figure 9: Constraint Constraint regions in the (η, Q) plane obtained from CUDA-based simulations, showing agreement with the EHT observations of M87∗ and Sgr A∗ within 1−σ and 2−σ confidence levels for a = 0.5 with M = 1. larger values of Q improve the agreement between theoretical shadow predictions and the EHT data. Taking Q = 0.4, the black hole metric still allows a wide range of valid horizons. However, only sp… view at source ↗
read the original abstract

Combining high-performance CUDA numerical codes with the Hamilton--Jacobi formalism, we investigate the shadows properties of rotating charged Euler--Heisenberg black holes in the presence of global monopoles. Then, we discuss the associated energy emission rate by varying the involved black hole parameters. As a result, we show that both the shadow structure and the energy emission rate depend on the global monopole parameter, the electric charge, and the rotation parameter. However, we observe that the Euler--Heisenberg nonlinear parameter does not significantly affect either the shadow or the energy emission rate. In order to reconcile the present theoretical predictions with the shadow observations reported by the Event Horizon Telescope collaboration, we employ a CUDA-based computational approach to establish strict bounds on the GM parameter, the electric charge, and the rotation parameter.

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 combines CUDA numerical codes with the Hamilton-Jacobi formalism to study shadows and energy emission rates of rotating charged Euler-Heisenberg black holes with global monopoles. It claims that both the shadow structure and energy emission rate depend on the global monopole parameter, electric charge, and rotation parameter, but are insensitive to the Euler-Heisenberg nonlinear parameter. Bounds on the first three parameters are derived to match Event Horizon Telescope observations.

Significance. If the numerical results are reliable, the reported insensitivity to the nonlinear parameter would be a useful simplification for modeling such black holes, and the parameter bounds could constrain exotic solutions against real data. The CUDA approach facilitates exploration of complex metrics beyond analytic reach.

major comments (2)
  1. Numerical Methods section: The CUDA implementation of Hamilton-Jacobi ray-tracing is not validated against known analytic limits, such as the Schwarzschild photon-sphere radius r=3M or the Kerr shadow boundary. This validation is required to establish that the reported null dependence on the nonlinear parameter and the quoted EHT bounds are not due to truncation errors or incorrect metric insertion in the effective potential.
  2. Section on observational constraints: The bounds on the global monopole, charge, and rotation parameters are obtained by tuning the model until the computed shadow matches the EHT measurement. This direct fitting procedure, rather than an independent prediction from the metric, weakens the claim that the CUDA runs 'establish strict bounds' and introduces circularity between the theoretical setup and the target observable.
minor comments (2)
  1. Abstract: The phrase 'strict bounds' is used without specifying the quantitative matching criterion (e.g., shadow diameter within a stated sigma of the EHT value) or reporting uncertainties from the numerical runs.
  2. Figures: Shadow plots for varying parameters would be clearer if they overlaid the EHT observational contour or error region for direct visual assessment of the claimed agreement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the manuscript. We address each major point below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: Numerical Methods section: The CUDA implementation of Hamilton-Jacobi ray-tracing is not validated against known analytic limits, such as the Schwarzschild photon-sphere radius r=3M or the Kerr shadow boundary. This validation is required to establish that the reported null dependence on the nonlinear parameter and the quoted EHT bounds are not due to truncation errors or incorrect metric insertion in the effective potential.

    Authors: We agree that explicit validation against analytic limits is necessary to confirm the numerical reliability. In the revised manuscript we will add a new subsection to the Numerical Methods section that directly compares our CUDA ray-tracing output with the known Schwarzschild photon-sphere radius (r=3M) and with the Kerr shadow boundary for several spin values. These tests will verify correct insertion of the metric into the effective potential and will rule out truncation or implementation errors as the source of the reported insensitivity to the Euler-Heisenberg parameter. revision: yes

  2. Referee: Section on observational constraints: The bounds on the global monopole, charge, and rotation parameters are obtained by tuning the model until the computed shadow matches the EHT measurement. This direct fitting procedure, rather than an independent prediction from the metric, weakens the claim that the CUDA runs 'establish strict bounds' and introduces circularity between the theoretical setup and the target observable.

    Authors: We acknowledge the referee’s concern about the language used to describe the constraints. Our procedure evaluates the shadow radius over a grid of the three parameters and retains only those values whose predicted shadow lies inside the EHT uncertainty interval; this is a standard consistency check rather than a fit of the metric itself. Nevertheless, to remove any ambiguity we will revise the text to replace “establish strict bounds” with “derive observational constraints” and will add a short paragraph clarifying that the comparison is performed after the metric and ray-tracing are fully specified. This change addresses the perception of circularity while preserving the scientific result. revision: partial

Circularity Check

0 steps flagged

No significant circularity; standard numerical parameter exploration and observational constraints

full rationale

The paper applies the Hamilton-Jacobi formalism to null geodesics in a given rotating charged Euler-Heisenberg metric with global monopoles, then uses CUDA ray-tracing to numerically compute shadow boundaries and energy emission rates while varying the global monopole, charge, rotation, and nonlinear parameters. Dependence on parameters is reported from this exploration, and bounds are set by requiring the computed shadow to be consistent with EHT data. This constitutes standard model exploration and parameter constraint from observation, not a derivation that reduces to its own inputs by construction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling are present in the abstract or described chain. The numerical implementation may require external validation for correctness, but that does not create circularity in the logical structure.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The paper rests on the validity of the Euler-Heisenberg-plus-monopole metric, the applicability of the Hamilton-Jacobi equation to null geodesics, and the assumption that EHT shadow size is the only observable needed to constrain the three parameters. No new entities are postulated.

free parameters (3)
  • global monopole parameter
    Varied numerically and bounded by matching to EHT shadow size
  • electric charge
    Varied numerically and bounded by matching to EHT shadow size
  • rotation parameter
    Varied numerically and bounded by matching to EHT shadow size
axioms (2)
  • domain assumption The spacetime metric of a rotating charged Euler-Heisenberg black hole with global monopoles is given and correct
    Used as the background for all ray tracing
  • standard math Hamilton-Jacobi formalism yields the correct null geodesics for shadow calculation
    Invoked to compute the shadow boundary

pith-pipeline@v0.9.0 · 5447 in / 1591 out tokens · 65563 ms · 2026-05-10T17:46:27.136099+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 24 canonical work pages · 1 internal anchor

  1. [1]

    S. W. Hawking, D. N. Page,Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87 (4) 577 (1983)

    1983

  2. [2]

    Belhaj, A

    A. Belhaj, A. El Balali, W. El Hadri, E. Torrente-Lujan,On Universal Constants of AdS Black Holes from Hawking-Page Phase Transition, Phys. Lett. B 811 135871 (2020), arXiv:2010.07837. 15

    work page arXiv 2020

  3. [3]

    Belhaj, H

    A. Belhaj, H. Belmahi, M. Benali, W. El Hadri, H. El Moumni, E. Torrente-Lujan, Shadows of 5D Black Holes from string theory, Phys. Lett. B 812 136025 (2021), arXiv:2008.13478

    work page arXiv 2021

  4. [4]

    Belhaj, H

    A. Belhaj, H. Belmahi, M. Benali, Y. Hassouni, M. B. Sedra,Optical behaviors of black holes in Starobinsky-Bel-Robinson gravity, Gen.Rel.Grav. 55 110 (2023)

    2023

  5. [5]

    Belmahi,Constrained Deflection Angle and Shadows of Rotating Black Holes in Einstein-Maxwell-scalar Theory,arXiv:2411.11622

    H. Belmahi,Constrained Deflection Angle and Shadows of Rotating Black Holes in Einstein-Maxwell-scalar Theory,arXiv:2411.11622

    work page arXiv

  6. [6]

    Event Horizon Telescope Collaboration.First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, the Astrophysical Journal Letters, 875 1 L1 (2019)

    2019

  7. [7]

    Event Horizon Telescope Collaboration.First Sagittarius A Event Horizon Telescope results. I. The shadow of the supermassive black hole in the center of the Milky Way the Astrophysical Journal Letters, 930 2 L12 (2022)

    2022

  8. [8]

    Akiyama et al., First M87 Event Horizon Telescope Results

    K. Akiyama and al.,First M87 Event Horizon Telescope Results. IV. Imaging the Cen- tral Supermassive Black Hole, Astrophys. J. 875 L4 (2019),arXiv:1906.11241

    work page arXiv 2019

  9. [9]

    Akiyama and al.,First M87 Event Horizon Telescope Results

    K. Akiyama and al.,First M87 Event Horizon Telescope Results. V. Imaging the Central Supermassive Black Hole, Astrophys. J. 875 L5 (2019)

    2019

  10. [10]

    Akiyama and al.,First M87 Event Horizon Telescope Results

    K. Akiyama and al.,First M87 Event Horizon Telescope Results. VI. Imaging the Cen- tral Supermassive Black Hole, Astrophys. J. 875 L6 (2019)

    2019

  11. [11]

    R. A. Konoplya, A. Zhidenko,Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83 793 (2011),arXiv:1102.4014 [gr-qc]

    work page internal anchor Pith review arXiv 2011

  12. [12]

    Z. Li, C. Bambi,Measuring the Kerr spin parameter of regular black holes from their shadow, JCAP 01 041 (2014),arXiv:1309.1606 [gr-qc]

    work page arXiv 2014

  13. [13]

    T. W. B. Kibble,Topology of cosmic domains and strings, J. Phys. A 9, 1387 (1976)

    1976

  14. [14]

    Vilenkin,Cosmic strings and domain walls, Phys

    A. Vilenkin,Cosmic strings and domain walls, Phys. Rep. 121, 263 (1985)

    1985

  15. [15]

    Belhaj, M

    A. Belhaj, M. Jemri,On Thermodynamics of Charged Black Holes via Extended Space-time Derivatives,International Journal of Modern Physics A, 04 41 (2026), arXiv:2511.18407 [hep-th]

    work page arXiv 2026

  16. [16]

    Sedaghatnia, H

    P. Sedaghatnia, H. Hassanabadi, A. A. Ara´ ujo Filho,P. J. Porfirio, and W. S. Chung, Thermodynamical properties of a deformed Schwarzschild black hole via Dunkl general- ization,Int.J.Mod.Phys.A 40 07 2550019 (2025),arXiv:2302.11460

    work page arXiv 2025

  17. [17]

    Askour, A

    N. Askour, A. Belhaj, L. Chakhchi, H. El Moumni, K. Masmar,On M87 ∗ and SgrA∗ Observational Constraints of Dunkl Black Holes,JHEAp 46 100349 (2025), arXiv:2412.09196 [gr-qc]. 16

    work page arXiv 2025

  18. [18]

    S. E. Baddis, A. Belhaj, H. Belmahi, M. Jemri,Constraining Black Hole Shadows in Dunkl Spacetime using CUDA Numerical Computations,Journal of High Energy Astrophysics 51 100541 (2025),arXiv:2510.16460 [gr-qc]

    work page arXiv 2025

  19. [19]

    Elafrou, G

    A. Elafrou, G. Thomas Collignon,Introduction to CUDA Performance Optimization, Nvidia

  20. [20]

    Nvidia,CUDA C++ Programming Guide

  21. [21]

    S. E. Baddis, A. Belhaj, and H. Belmahi,CUDA Assisted Swampland and Black Hole Thermodynamics,arXiv:2508.12378 [hep-th]

    work page arXiv

  22. [22]

    Berczik, R

    P. Berczik, R. Spurzem, L. Wang, S. Zhong, O. Veles, I. Zinchenko, S. Huang, M. Tsai, G. Kennedy, S. Li, L. Naso, and C. Li,Up to 700k GPU cores, Kepler, and the Ex ascale future for simulations of star clusters around black holes,arXiv:1312.1789 [astro-ph.IM]

    work page arXiv

  23. [23]

    A. G. M. Lewis, H. P. Pfeiffer,GPU-Accelerated Simulations of Isolated Black Holes, Class. Quant. Grav. 35 095017 (2018),arXiv:1804.09101 [gr-qc]

    work page arXiv 2018

  24. [24]

    Ginjupalli and G

    R. Ginjupalli and G. Khanna,High-Precision Numerical Simulations of Rotating Black Holes Accelerated by CUDA,arXiv:1006.0663 [physics.comp-ph]

    work page arXiv

  25. [25]

    Janis-Newman algorithm: generating rotating and NUT charged black holes

    H. Erbin,Janis- Newman algorithm: generating rotating and NUT charged black holes, Universe 3 19 (2017),arXiv:1701.00037 [gr-qc]

    work page Pith review arXiv 2017

  26. [26]

    Kamenshchik, P

    A. Kamenshchik, P. Petriakova,Newman–Janis algorithm’s application to regular black hole models, Phys. Rev. D107, 124020 (2023),arXiv:2305.04697 [gr-qc]

    work page arXiv 2023

  27. [27]

    Z. Cai, Z. Ban, Q.-Q. Liang, H. Feng, Z.-W. Long,Construction of Rotating Sen Black Holes via the Newman-Janis Algorithm and Applications to Accretion Disks and Shad- ows,JCAP 09 041 (2025),arXiv:2509.01226 [gr-qc]

    work page arXiv 2025

  28. [28]

    Magos and N

    D. Magos, N. Bret` on,Thermodynamics of the Euler-Heisenberg-AdS black hole, Phys. Rev. D 102, 084011 (2020),arXiv:2009.05904 [gr-qc]

    work page arXiv 2020

  29. [29]

    Hamil, B

    B. Hamil, B. C. L¨ utf¨ uo˘ glu, F. Ahmed, Z. Yousaf,Thermodynamic properties of Quantum-Corrected AdS Black Hole with Phantom Global Monopoles,Nucl.Phys.B 1014, 116861 (2025),arXiv:2404.16674 [gr-qc]

    work page arXiv 2025

  30. [30]

    Charged AdS Black Holes and Catastrophic Holography

    A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers,Charged AdS Black Holes and Catastrophic Holography, Phys. Rev. D 60, 064018 (1999),arXiv:hep-th/9902170

    work page Pith review arXiv 1999

  31. [31]

    S. W. Wei, Y. C. Zou, Y. X. Liu, R. B. Mann,Curvature radius and Kerr black hole shadow, JCAP 08 030 (2019),arXiv:1904.07710. 17

    work page arXiv 2019

  32. [32]

    D ˆA´ecanini, A

    Y. D ˆA´ecanini, A. Folacci, G. Esposito-Far‘ese,Universality of high-energy absorp- tion cross sections for black holes, Phys. Rev. D 83 044032 (2011),arXiv:1101.0781 [gr-qc]

    work page arXiv 2011

  33. [33]

    Wei, Y.-X

    S.-W. Wei, Y.-X. Liu,Relationship between high-energy absorption cross section and strong gravitational lensing for a static and spherically symmetric black hole, Phys. Rev. D 84 041501 (2011),arXiv:1103.3822 [hep-th]

    work page arXiv 2011

  34. [34]

    Perlick and O

    V. Perlick,Calculating black hole shadows: Review of analytical studies,Phys. Rep. 924 1 (2022),arXiv:2105.07101 [gr-qc]

    work page arXiv 2022

  35. [35]

    Kocherlakota et al

    P. Kocherlakota et al. [Event Horizon Telescope],Constraints on black-hole charges with the 2017 EHT observations of M87*, Phys. Rev. D 103, no.10 104047 (2021)

    2017

  36. [36]

    Chakhchi, H

    L. Chakhchi, H. El Moumni and K. Masmar,Signatures of the accelerating black holes with a cosmological constant from the Sgr A* and M87* shadow prospects, Phys. Dark Univ. 44 101501 (2024)

    2024

  37. [37]

    D. J. Gogoi and S. Ponglertsakul,Constraints on quasinormal modes from black hole shadows in regular non-minimal Einstein Yang Mills gravity, Eur. Phys. J. C 84, no.6 652 (2024). 18

    2024