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A scalar gravitational wave in MOG makes a black hole's shadow rhythmically change its area, while a delayed massive-vector wave knocks the shadow's center sideways — together breaking the static shadow degeneracy between MOG and general re

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2026-08-02 21:01 UTC pith:FQOFQVTU

load-bearing objection A plausible phenomenological template for time-dependent MOG shadows, but the headline breathing signal is unverified because the paper's own adiabatic assumption breaks down at the frequencies used in its figures. the 4 major comments →

arxiv 2602.21432 v3 pith:FQOFQVTU submitted 2026-02-24 gr-qc

Breathing Black Hole Shadows in Modified Gravity (MOG)

Nikko John Leo S. Lobos , Emmanuel T. Rodulfo This is my paper
classification gr-qc MSC 83C5783C3583D05 PACS 04.50.Kd04.70.Bw04.30.Nk98.62.Sb
keywords Modified Gravity (MOG)Scalar-Tensor-Vector Gravity (STVG)black hole shadowgravitational wave polarizationbreathing modemassive vector fieldquasinormal modesExtreme Mass Ratio Inspiral
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.

This paper tries to show that when a gravitational wave passes a Schwarzschild-MOG black hole, the shadow does not just deform in area-preserving ways as in general relativity; it breathes and then wobbles. The scalar field of MOG produces a time-dependent shadow radius R_sh(t) ≈ R̄_sh (1 + h_b(t)/2), giving a rhythmic area fluctuation that GR strictly forbids. The massive vector field arrives later because of dispersion and sources longitudinal metric perturbations, causing a delayed, asymmetric translation of the shadow center. If correct, these two time-dependent signatures provide a testable template for next-generation interferometry and a way to measure the vector-field mass.

Core claim

By perturbing the Hamilton-Jacobi equation for null geodesics with the scalar and massive-vector gravitational-wave content of Scalar-Tensor-Vector Gravity (MOG), the paper derives two distinct dynamic shadow signatures. The scalar breathing mode changes the critical impact parameter and hence the shadow radius as R_sh(t) ≈ R̄_sh (1 + h_b(t)/2), so the apparent area fluctuates as δA(t) ≈ π R̄_sh² h_b(t). The massive vector field, traveling slower than light by Δt ≈ D μ_v²/(2ω²), sources longitudinal metric perturbations that translate the shadow center by δX, δY, with amplitudes set by the photon-sphere radius and the delayed strain. The paper argues these time-dependent effects break the ob

What carries the argument

The central mechanism is the perturbed Hamilton-Jacobi equation for photon null geodesics in a Schwarzschild-MOG background. The scalar strain h_b(t) multiplies the effective potential as [1 − h_b(t)], leaving the photon-sphere radius r_p fixed but modulating the critical impact parameter; this produces the breathing mode. For the massive vector, a Proca-type dispersion relation v_g = sqrt(1 − μ_v²/ω²) yields the time delay, and the non-gauge longitudinal metric perturbations h_xz, h_yz couple line-of-sight and transverse momenta to generate the translational wobble. The photon-sphere size r_p appears as the characteristic interaction length in the thin-lens integrals for δX and δY.

Load-bearing premise

The derivation assumes the photon's total energy stays constant while the wave passes — the adiabatic approximation — which the paper itself notes breaks down for fundamental quasinormal modes with ω_b ~ 1/M, so the headline breathing formulas are leading-order estimates in exactly the regime used for the figures.

What would settle it

A direct numerical integration of the time-dependent null geodesic equations in the metric of Eq. (19) with h_b(t) = A_b e^{-(t−t₀)/τ} cos(ω_b t) would settle the breathing claim: if the resulting shadow area fluctuation departs from πR̄² h_b(t) or vanishes for ω_b ~ 1/M, the central signature is an artifact of the adiabatic approximation. Observationally, a well-resolved shadow during a strong EMRI burst that shows the breathing but no delayed centroid wobble at the predicted Δt = D μ_v²/(2ω²) would falsify the massive-vector echo.

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

If this is right

  • A scalar breathing mode directly changes the apparent area of a black hole shadow, a signature forbidden for pure tensor gravitational waves in standard GR.
  • The MOG static-shadow degeneracy with Schwarzschild shadows is broken dynamically, giving time-dependent observables rather than just a shifted radius.
  • The predicted time delay Δt ≈ D μ_v²/(2ω²) offers a direct, quantitative probe of the massive vector-field mass.
  • For EMRI sources, the local strain near the photon sphere scales as the mass ratio q ~ 10⁻⁵, placing the predicted fractional shadow area changes and wobbles within the target sensitivity of proposed space-based interferometry and ngEHT.
  • A full MOG gravitational-wave event would show a sequential timeline: simultaneous tensor stretching plus scalar breathing, a relaxation, then a delayed centroid wobble.

Where Pith is reading between the lines

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

  • The same adiabatic Hamiltonian perturbation technique could be extended to rotating MOG black holes, where the breathing and wobble may mix with frame-dragging and produce spin-dependent late-time oscillations.
  • The wobble amplitude, being proportional to r_p and the longitudinal strain, could be degenerate with gravitational-wave memory effects or with centroid shifts from microlensing; disentangling these may require simultaneous GW and EHT-style observations.
  • If the adiabatic approximation fails for the fundamental quasinormal-mode frequencies used in the figures, the predicted area oscillation may be reduced or replaced by a different pattern; a full geodesic integration would clarify whether the breathing mode survives as the paper's leading-order estimate suggests.

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

4 major / 6 minor

Summary. The manuscript considers a non-rotating Schwarzschild-MOG black hole and asks how its shadow responds to passing scalar and massive-vector gravitational waves. After deriving the static shadow radius, it perturbs the null Hamiltonian and obtains R_sh(t) ≈ R̄_sh(1 + h_b(t)/2) for a scalar breathing mode and a delayed translational wobble δX, δY from the massive vector field. The paper argues that these time-dependent signatures break the observational degeneracy between MOG and GR and offers a template for ngEHT and space-based interferometry.

Significance. If the dynamical predictions were established, the area-breathing signature would be a qualitatively new probe of scalar degrees of freedom in gravity, and the delayed wobble could, in principle, allow a measurement of the vector mass via Δt = D μ_v²/(2ω²). The paper is clearly written and honest about the adiabatic limitation. However, the central derivation is only a leading-order estimate in exactly the regime shown in the figures, and the vector-sector amplitudes are not derived from MOG parameters. The paper therefore currently establishes a phenomenological template rather than a proof.

major comments (4)
  1. [§III, Eqs. (20)–(22)] The breathing formula rests on Eq. (20), which equates the peak of the time-dependent effective potential with the unperturbed squared energy E². The authors explicitly concede in §III that for fundamental QNMs with ω_b ∼ 1/M, dE/dλ ≠ 0. Fig. 1 uses ω_b = 1.5 (in units of 1/M), placing the plotted signal in the invalid regime. Thus R_sh(t) ≈ R̄_sh(1 + h_b/2) and δA(t) ≈ π R̄_sh² h_b are unverified leading-order estimates; a full null-geodesic integration is needed to determine whether the area oscillation survives with the quoted amplitude and phase.
  2. [§III, Eq. (15)] The perturbation h_b(t) is treated as a function of observer time only, with no dependence on the retarded time along the photon trajectory. For ω_b = 1.5/M the wavelength is λ_b ∼ 4M, comparable to the photon-sphere radius r_p ≃ 3.5M for α = 0.2. The neglect of retardation and spatial gradients is therefore not controlled in Fig. 1 and can change the sign and amplitude of the shadow response.
  3. [§IV, Eqs. (33)–(38)] The vector-wobble amplitude is not derived. Eq. (33) is introduced as an order-of-magnitude stress-energy estimate, but it contains no dependence on the vector-field amplitude δϕ, the frequency ω_v, or the coupling beyond α; its D^{-4} scaling is asserted without derivation. Since A_x and A_y in Eqs. (37)–(38) are free parameters, the delayed echo is a template rather than a prediction. To make the claimed observational template quantitative, the relation between A_x, A_y and MOG parameters must be supplied.
  4. [§IV, Eqs. (34)–(36)] The wobble is computed by integrating the transverse velocity shifts ẋ, ẏ along the unperturbed background path and assuming the strain is constant over Δz ≈ 2r_p (thin-lens approximation). For a time-dependent massive-vector perturbation, the photon trajectory itself is modified; the shadow-centre displacement should be obtained from the full perturbed null geodesic and the resulting impact-parameter shift. The constant-phase assumption λ_v ≫ r_p is also not checked against the QNM frequency range used elsewhere in the paper.
minor comments (6)
  1. [Abstract] The phrase “quantum vacuum dispersion” is misleading; the dispersion relation in Eq. (28) is a classical effect.
  2. [Section II heading] There is a typo in the heading: “MO)” should be “MOG”.
  3. [§III, Eqs. (23) and (25)] The Heaviside-step notation in Eq. (23) and the piecewise form in Eq. (25) are redundant; choose one for consistency.
  4. [Figures] In both figures, the axis label “Time t0” likely means “Time t”; specify units and parameters (M = 1, α = 0.2, etc.) in the captions.
  5. [References] Reference [28] duplicates reference [23] (the same Synge paper).
  6. [Eq. (33)] Eq. (33) should be explicitly marked as an order-of-magnitude estimate (e.g., with a tilde) and should state why it is independent of the field amplitude and frequency.

Circularity Check

0 steps flagged

No significant circularity: the dynamic shadow signatures are linear responses to assumed metric perturbations, with MOG inputs drawn from external references.

full rationale

The paper's derivation chain is not circular. The static Schwarzschild-MOG shadow (Eqs. 9-12) is a direct re-derivation of known results; the breathing-mode analysis inputs a specified conformal strain h_b(t) (Eq. 14) and derives the shadow-radius response R_sh(t) ≈ R̄_sh(1+h_b/2) from the perturbed Hamilton-Jacobi equation (Eqs. 15-22). This is a linear-response calculation: the output is a function of the assumed input, not a restatement of it. The vector-sector wobble likewise integrates assumed longitudinal strains h_xz, h_yz (Eqs. 34-38) to obtain displacements proportional to those strains; the time delay Δt = D μ_v²/(2ω²) follows from the standard massive dispersion relation. No parameter is fitted to the predicted observable and then used as the prediction. The paper contains no self-citations by Lobos/Rodulfo; all MOG background references (Moffat et al.) are external. The acknowledged adiabatic approximation in §III is a validity limitation—the paper explicitly states dE/dλ ≠ 0 for ω_b ∼ 1/M—but this is a correctness/regime concern, not a circular definition. Therefore, no step reduces to its own input by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

The central claim depends on the MOG black hole metric and MOG wave content, both cited from prior work. The paper's new contribution is a computed observable under an explicitly acknowledged adiabatic approximation, with waveform parameters chosen by hand. No data are fit; no new parameter values are derived.

free parameters (4)
  • MOG deformation parameter α = 0.2 in figures
    Controls static shadow size and all response amplitudes; taken from previous MOG papers, not fit here.
  • Vector field mass μ_v = not specified
    Central to the time delay Δt = D μ_v²/(2ω²); no numerical value or prior bound is given.
  • Scalar wave amplitude A_b, frequency ω_b, damping τ, phase Φ_0 = ω_b=1.5, τ=8.0 in Fig. 1; A_b, Φ_0 not stated
    The strain h_b(t) is posited as a damped sinusoid; values chosen for illustration.
  • Vector wobble amplitudes A_x, A_y, frequency ω_v, damping τ_v, phases Φ_x, Φ_y = not specified; used in Fig. 2
    The longitudinal strains are modeled as damped sinusoids without deriving amplitudes from field equations.
axioms (6)
  • domain assumption The static Schwarzschild-MOG metric (Eqs. 1–4) is the correct black hole solution of STVG.
    Invoked in §II as the starting point; taken from Moffat 2015, not re-derived.
  • ad hoc to paper A passing scalar wave perturbs the inverse metric as g^{xx}=g^{yy}=1/r² − h_b(t)/r² (Eq. 15).
    No derivation from the MOG scalar-field action; assumes no radial dependence of the perturbation.
  • ad hoc to paper Photon energy conservation during wave passage (adiabatic approximation).
    Eq. (20) sets the peak of the time-dependent potential equal to the unperturbed E²; the text admits this fails for QNM-scale frequencies.
  • ad hoc to paper Massive vector wave sources longitudinal metric perturbations h_xz, h_yz with amplitudes A_x, A_y.
    Eqs. (31)–(38): the existence and form of these strains are asserted; Eq. (33) is heuristic and dimensionally inconsistent.
  • domain assumption Standard-model photons do not couple to the MOG vector field.
    Stated in §IV before Eq. (31); ensures the wobble is metric-mediated rather than direct.
  • domain assumption Thin-lens approximation: deflection accumulates over Δz ≈ 2r_p with constant wave phase.
    Used to evaluate Eqs. (35)–(36); sets wobble amplitude proportional to A_x,y r_p.
invented entities (1)
  • No new entity invented; MOG's scalar and massive vector fields are inherited from prior literature. no independent evidence
    purpose: The paper invokes the MOG scalar and vector fields as the sources of the breathing and wobble signatures.
    The fields themselves are not new to this paper. The time-delay prediction is in principle falsifiable, but no numerical value of μ_v is given, so no independent prediction is made here.

pith-pipeline@v1.3.0-alltime-deepseek · 12505 in / 8611 out tokens · 62129 ms · 2026-08-02T21:01:07.918605+00:00 · methodology

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read the original abstract

In this paper, we investigate the dynamic phenomenological signatures of a Schwarzschild-MOG black hole shadow perturbed by passing gravitational waves. By perturbing the Hamilton-Jacobi equation for photon null geodesics, we demonstrate that the unique field content of MOG breaks the observational degeneracy with standard General Relativity. We mathematically prove two distinct, time-dependent signatures. First, the massless MOG scalar field induces a volumetric ``breathing mode'' polarization, causing the total apparent area of the shadow to rhythmically expand and contract. Second, the massive MOG vector field undergoes quantum vacuum dispersion, arriving at the observer with a predictable time delay. This delayed massive wave sources secondary longitudinal metric perturbations that manifest as a sudden, asymmetric translational wobble of the shadow on the celestial screen. These dynamic geometric shifts offer a robust observational template for next-generation interferometry to strictly test the existence of massive force carriers and scalar fields in gravity.

Figures

Figures reproduced from arXiv: 2602.21432 by Emmanuel T. Rodulfo, Nikko John Leo S. Lobos.

Figure 1
Figure 1. Figure 1: FIG. 1. Simultaneous evolution of the black hole shadow ra [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The translational displacement (wobble) of the black hole [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Post-Newtonian orbital mechanics around a black hole in modified gravity

    gr-qc 2026-05 unverdicted novelty 5.0

    MOG produces distinct orbital precession and sky-projected deviations for S-stars that grow with the parameter α and can resemble dark matter effects while remaining testable against GR.

  2. Dynamical Black Hole Thermodynamics in Modified Gravity

    gr-qc 2026-04 unverdicted novelty 5.0

    In modified gravity, dynamical Schwarzschild black holes under scalar waves exhibit non-thermal particle creation while preserving the generalized second law and forming stable zero-temperature remnants at the extremal bound.

  3. Dynamical Black Hole Thermodynamics in Modified Gravity

    gr-qc 2026-04 unverdicted novelty 5.0

    In Modified Gravity, a scalar breathing-mode perturbation yields non-thermal radiation and second-order entropy production, while a running MOG parameter α(M) quenches Hawking radiation into a stable remnant.

Reference graph

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