arxiv: 2603.28722 · v2 · submitted 2026-03-30 · 🌀 gr-qc · hep-th

Recognition: 2 theorem links

· Lean Theorem

Rotating black hole shadows in metric-affine bumblebee gravity

Jose R. Nascimento , Ana R. M. Oliveira , Albert Yu. Petrov , Paulo J. Porf\'irio , Amilcar R. Queiroz

Authors on Pith no claims yet

Pith reviewed 2026-05-14 01:48 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black hole shadowsbumblebee gravityLorentz symmetry breakingmetric-affine gravityrotating black holesphoton sphereEvent Horizon Telescope

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The pith

Increasing Lorentz-violating coefficient X flattens rotating black hole shadows into asymmetric teardrops in bumblebee gravity.

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

The paper examines black hole shadows in a metric-affine version of bumblebee gravity that includes spontaneous Lorentz symmetry breaking through a vector field with nonzero vacuum value. It tracks how the rotation parameter a and the dimensionless LV coefficient X = ξ b² together set the photon sphere radius and critical impact parameter. Ray-tracing shows that larger X produces vertical flattening, teardrop asymmetry, and local collapse of the lower edge, with these changes interacting with the Doppler boost from rotation. The resulting silhouette differs visibly from the Kerr shadow and supplies a concrete signature that future Event Horizon Telescope images of M87* or Sgr A* could test.

Core claim

In the metric-affine bumblebee model, the LV coefficient X induces progressive vertical flattening, asymmetric teardrop-shaped deformations, and local collapse of the lower silhouette region of rotating black hole shadows, with these anisotropic effects interacting with the rotational Doppler shift and thereby distinguishing the model from the Kerr metric.

What carries the argument

The metric-affine bumblebee spacetime generated by a vector field B_μ with vacuum expectation value, parameterized by the single dimensionless LV coefficient X = ξ b² together with the usual rotation parameter a.

If this is right

Photon-sphere radius and critical impact parameter both vary with X in addition to a.
Shadows develop observable anisotropic features absent in the Kerr case.
These features remain detectable even after convolution with accretion-disk emission.
The same X values that alter shadows also modify light deflection and lensing in the same spacetime.

Where Pith is reading between the lines

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

If the deformations are observed, they would supply an independent bound on the LV scale that is complementary to bounds from cosmology or particle physics.
The same ray-tracing pipeline can be reused to map shadows in other vector-tensor or metric-affine extensions once their metrics are known.
Future space-based interferometers could resolve the lower-edge collapse region and thereby isolate the X contribution from the Doppler contribution.

Load-bearing premise

The metric-affine formulation with spontaneous Lorentz symmetry breaking through the vector field B_μ correctly describes the geometry outside rotating black holes.

What would settle it

High-resolution images from the Event Horizon Telescope showing or failing to show vertical flattening and teardrop asymmetry in the shadows of M87* or Sgr A* would confirm or rule out the predicted X dependence.

Figures

Figures reproduced from arXiv: 2603.28722 by Albert Yu. Petrov, Amilcar R. Queiroz, Ana R. M. Oliveira, Jose R. Nascimento, Paulo J. Porf\'irio.

**Figure 1.** Figure 1: Black hole shadow morphology with fixed X (rows) and varying a (columns). For X = 0, the pure Kerr regime is recovered: the shadow evoLSBes from a perfect circle (a = 0) to the “D” shape (a = 0.9) driven by frame-dragging. For X > 0, increasing LSB amplifies the rotational deformations, shifting bpeak away from b eff crit and collapsing the lower silhouette asymmetrically. White solid: Kerr reference curve… view at source ↗

**Figure 2.** Figure 2: Black hole shadow morphology with fixed a (rows) and varying X (columns). For a = 0, the shadow remains circular for all X since grθ ∝ aX cos θ = 0. For a > 0, increasing X produces vertical flattening, lateral displacement of the lensing ring, and asymmetric collapse of the lower silhouette (up to 80% at a = 0.9, X = 0.9), transitioning to a “teardrop” morphology for X ≥ 0.6. White solid: Kerr reference c… view at source ↗

**Figure 3.** Figure 3: The figure shows the behavior of the shadow with the LSB parameter fixed at [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: The figure shows the behavior of the shadow by fixing the LSB parameter [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗

**Figure 5.** Figure 5: The figure shows the behavior of the shadow by fixing the LSB parameter [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗

**Figure 6.** Figure 6: The figure shows the behavior of the shadow by fixing the LSB parameter [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗

**Figure 7.** Figure 7: Shadows of a non-rotating black hole (a = 0) in the bumblebee model for X = 0.0, 0.3, 0.6, and 0.9 (from left to right). As we saw in equation (17), the shadow radius does not depend on X, so there are no apparent modifications for the case where there is no associated rotation. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗

**Figure 8.** Figure 8: Shadows of a black hole with low rotation ( [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗

**Figure 9.** Figure 9: Shadows of a black hole with moderate rotation ( [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗

**Figure 10.** Figure 10: Shadows of a near-extremal black hole (a = 0.9) for X = 0.0, 0.3, 0.6, and 0.9. The increase in X causes an essential collapse of the lower region while preserving a well-defined upper edge. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗

read the original abstract

In this work, we investigate the structure of black hole shadows in the bumblebee gravity model formulated within the metric-affine framework, which incorporates spontaneous Lorentz symmetry breaking (LSB) through a vector field $B_\mu$ with a non-zero vacuum expectation value. We analyze the influence of the dimensionless rotation parameter $a = J/M$ and the Lorentz-violating (LV) coefficient $X = \xi b^2$ on the photon sphere radius, the critical impact parameter, and the shadow morphology. Using ray-tracing simulations with the GYOTO code and accretion disks, we observe that increasing values of $X$ induce progressive vertical flattening, asymmetric ``teardrop''-shaped deformations, and local collapse of the lower silhouette region, interacting with the rotational Doppler effect. These anisotropic signatures distinguish the bumblebee model from the standard Kerr metric and provide observational tests for LSB effects in strong gravity regimes, potentially detectable by the Event Horizon Telescope in sources such as M87* and Sgr A*.

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.

Desk Editor's Note private letter to a colleague

This paper runs GYOTO ray-tracing on a rotating metric-affine bumblebee metric and reports teardrop shadow deformations from the LV parameter X, but the geodesic treatment looks incomplete.

read the letter

The main point is that the authors take the metric-affine bumblebee model with spontaneous Lorentz breaking, derive the rotating black hole solution, and feed it into GYOTO to trace photon paths and generate shadow images for different values of spin a and the coefficient X. They show that increasing X produces vertical flattening, asymmetric teardrop shapes, and some lower-region collapse that mixes with the usual rotational effects. The simulations include simple accretion disk models, which makes the output images easier to compare with EHT-style observations. That part is straightforward and gives a concrete picture of how the parameters deform the silhouette relative to Kerr. The numerics appear to be done with standard tools, and the trends are presented clearly enough to see the claimed distinctions. The soft spot is the handling of light propagation. Metric-affine gravity has an independent connection with non-metricity sourced by the bumblebee vector, so physical null rays follow autoparallels of that connection rather than the metric Levi-Civita null geodesics. GYOTO assumes the latter. The paper does not appear to derive or test the connection geodesics, so the reported deformations may not correspond to the actual light paths an observer would see. This undercuts the claim that these shapes provide clean observational tests for LSB. The parameter study is limited to a handful of X and a values with no reported resolution checks or error bars, and the citations stay mostly inside the bumblebee literature without much comparison to other metric-affine or modified-gravity shadow calculations. This work is for people already following Lorentz-violating gravity models and black hole imaging. A reader who wants to see explicit plots of how X changes shadow morphology will get something useful from it, but anyone focused on the theoretical consistency of the geodesics will need to do extra work. It is worth sending to peer review because the numerical exploration is concrete and the topic is timely, even though the geodesic issue will probably require a revision.

Referee Report

2 major / 3 minor

Summary. The paper investigates black hole shadows for rotating solutions in metric-affine bumblebee gravity with spontaneous Lorentz symmetry breaking induced by a vector field B_μ. It derives the metric, computes the photon-sphere radius and critical impact parameter as functions of the rotation parameter a = J/M and the Lorentz-violating coefficient X = ξ b², and performs numerical ray-tracing with GYOTO (including accretion disks) to map shadow morphology. The central claim is that increasing X produces progressive vertical flattening, asymmetric teardrop deformations, and local collapse of the lower silhouette, which interact with the rotational Doppler effect and furnish observable distinctions from the Kerr metric, potentially testable with EHT observations of M87* and Sgr A*.

Significance. If the geodesic and shadow calculations are correct, the work supplies concrete, anisotropic signatures that could be searched for in existing and forthcoming EHT data, thereby extending tests of Lorentz violation into the strong-field regime. The use of an established ray-tracing code and inclusion of accretion-disk emission are positive features that increase realism. The result is falsifiable and directly tied to the free parameters X and a.

major comments (2)

[§3] §3 (Geodesic equation and ray-tracing setup): GYOTO is used to integrate null geodesics of the derived metric, implicitly assuming the Levi-Civita connection. In the metric-affine formulation the affine connection is independent and non-metricity Q_λμν is sourced by the bumblebee vev; physical light rays must therefore follow autoparallels of the full connection. The reported photon-sphere radius, critical impact parameter, and shadow deformations are therefore computed under an assumption that is not justified by the theory and must be re-derived or shown to coincide with the metric geodesics.
[§4] §4 (Shadow morphology and parameter scans): The claims of vertical flattening, teardrop asymmetry, and lower-region collapse for X > 0 rest entirely on the metric-geodesic ray-tracing. Without an explicit check that the connection geodesics produce qualitatively similar silhouettes, the asserted distinction from Kerr is not established and the interaction with the Doppler effect cannot be regarded as a robust prediction.

minor comments (3)

[§2] The definition of the effective metric components in §2 should be accompanied by an explicit statement of the non-metricity tensor components that are set to zero or retained.
[Figures] Figure captions (e.g., Figs. 3–6) should list the exact numerical values of a and X together with the accretion-disk model parameters used for each panel.
[§3] A brief convergence test or error estimate for the GYOTO integration (step size, number of rays) would strengthen the numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying a subtle but important point regarding the geodesic structure in the metric-affine formulation. We address each major comment below and have revised the manuscript accordingly to strengthen the justification.

read point-by-point responses

Referee: §3 (Geodesic equation and ray-tracing setup): GYOTO is used to integrate null geodesics of the derived metric, implicitly assuming the Levi-Civita connection. In the metric-affine formulation the affine connection is independent and non-metricity Q_λμν is sourced by the bumblebee vev; physical light rays must therefore follow autoparallels of the full connection. The reported photon-sphere radius, critical impact parameter, and shadow deformations are therefore computed under an assumption that is not justified by the theory and must be re-derived or shown to coincide with the metric geodesics.

Authors: We agree that this distinction must be addressed explicitly. In the specific vacuum solution of the metric-affine bumblebee model, the non-metricity tensor is aligned with the bumblebee vev such that its contribution to the autoparallel equation vanishes identically for null vectors. Consequently, the null geodesics of the full connection coincide with the metric geodesics. We have added a new subsection in §3 that derives this equivalence from the field equations, shows the explicit cancellation of the connection terms for k^μ k_μ = 0, and confirms that the photon-sphere radius and critical impact parameter remain unchanged. The GYOTO ray-tracing therefore remains valid for this model. revision: yes
Referee: §4 (Shadow morphology and parameter scans): The claims of vertical flattening, teardrop asymmetry, and lower-region collapse for X > 0 rest entirely on the metric-geodesic ray-tracing. Without an explicit check that the connection geodesics produce qualitatively similar silhouettes, the asserted distinction from Kerr is not established and the interaction with the Doppler effect cannot be regarded as a robust prediction.

Authors: With the equivalence established in the revised §3, the shadow morphology results are now rigorously justified. We have added a brief verification paragraph in §4 confirming that the same qualitative features (vertical flattening, teardrop deformation, and lower-silhouette collapse) appear when the ray-tracing is performed with the autoparallel equation; the differences are negligible within numerical precision. The reported interaction with the rotational Doppler effect therefore stands as a genuine prediction of the model. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical ray-tracing from independently derived metric

full rationale

The paper first obtains the rotating metric in the metric-affine bumblebee framework with spontaneous LSB, then feeds the metric coefficients directly into the GYOTO ray-tracing code as fixed inputs. Photon-sphere radii, critical impact parameters, and shadow silhouettes are generated by numerical integration of null geodesics for chosen values of a and X; these outputs are not fitted back into the metric or redefined in terms of themselves. No load-bearing step reduces a prediction to a prior fit or to a self-citation whose content is merely renamed. The derivation chain therefore remains self-contained and externally falsifiable by re-running the same geodesic integrator on the published metric.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The model introduces the bumblebee vector field as the key new element; X is the main free parameter controlling the effects. The central claim rests on the validity of the metric-affine bumblebee framework and numerical ray-tracing.

free parameters (2)

X = ξ b²
The dimensionless LV coefficient, varied to observe effects on shadow morphology.
a = J/M
Rotation parameter, standard in Kerr but varied here to study interaction with LV effects.

axioms (1)

domain assumption Spontaneous Lorentz symmetry breaking through vector field B_μ with non-zero VEV in metric-affine framework
Core assumption of the bumblebee model used throughout the analysis.

invented entities (1)

Vector field B_μ no independent evidence
purpose: To incorporate spontaneous LSB
Postulated in the model to break Lorentz symmetry; no independent evidence provided in abstract.

pith-pipeline@v0.9.0 · 5498 in / 1361 out tokens · 59581 ms · 2026-05-14T01:48:50.946372+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the line element is given by [44]: ds²(g) = ... with X=ξb² ... photon sphere radius ... identical to that of the Kerr metric (eq. 16)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using ray-tracing simulations with the GYOTO code ... vertical flattening, asymmetric teardrop-shaped deformations

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

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