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arxiv: 2604.07597 · v1 · submitted 2026-04-08 · 🌀 gr-qc · astro-ph.GA

Recognition: unknown

Accretion Disks in Schwarzschild-MOG and Kerr-MOG Backgrounds: MOG Parameter in terms of Observational Quantities

Jos\'e Miguel Rojas , Mehrab Momennia
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:59 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GA
keywords accretion disksMOGblack holesfrequency shiftredshift rapiditymodified gravitySchwarzschild-MOGKerr-MOG
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The pith

The paper derives closed-form expressions for the mass M, MOG parameter α, distance D, and spin a of black holes in modified gravity using directly measurable accretion disk quantities such as frequency shift and redshift rapidity.

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

The paper establishes analytic formulas that relate the MOG parameter α directly to measurements from accretion disks around black holes in modified gravity theories. These formulas use the total frequency shift, aperture angle, and redshift rapidity for non-rotating cases, and add redshift acceleration for rotating ones. A sympathetic reader would care because the explicit presence of α allows testing if observations require a non-zero MOG parameter, indicating possible departures from general relativity. The expressions reduce to standard GR results when α vanishes.

Core claim

We apply a general relativistic framework to static and rotating black hole solutions in Scalar-Tensor-Vector Gravity or modified gravity (MOG). Our results yield exact analytic, closed-form relations expressing the mass M, the MOG coupling parameter α, and the distance D of a Schwarzschild-MOG black hole in terms of a minimal set of directly measurable elements of the accretion disk: the total frequency shift, the telescope aperture angle, and the redshift rapidity. We further extend the formalism to the rotating Kerr-MOG geometry and obtain corresponding relations that determine the rotation parameter a jointly with M, α, and D on the midline. In the rotating background, we introduced the

What carries the argument

The redshift rapidity, a relativistic invariant encoding the evolution of the frequency shift with respect to the emitter's proper time, which allows expressing M, α, and D in closed form for particles near the midline and line of sight.

If this is right

  • Direct empirical estimation of the MOG parameter α from accretion disk observations becomes possible.
  • The formulas recover the standard Schwarzschild and Kerr results in the limit α → 0.
  • Redshift acceleration is introduced to disentangle parameters in rotating Kerr-MOG geometries.
  • The expressions are concise and suitable for incorporation into black hole parameter-estimation pipelines.

Where Pith is reading between the lines

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

  • This approach could be tested on systems with independently measured masses, such as stellar-mass black holes in X-ray binaries, to see if derived α values are consistent.
  • If α is found to be non-zero, it would suggest modifications to gravity that could be cross-checked with other observational tests like gravitational waves or cosmology.
  • The method might extend to other modified gravity theories by adapting the redshift rapidity definitions.

Load-bearing premise

The derivations assume particles close to the midline and line of sight so that the redshift rapidity can be treated as a relativistic invariant encoding the evolution of the frequency shift with respect to the emitter's proper time.

What would settle it

Observing a black hole with known mass and distance via its accretion disk and finding that the formulas yield an α value inconsistent with zero or with other independent constraints on MOG.

Figures

Figures reproduced from arXiv: 2604.07597 by Jos\'e Miguel Rojas, Mehrab Momennia.

Figure 1
Figure 1. Figure 1: FIG. 1: The frequency shift [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The frequency shift [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

We apply a general relativistic framework to static and rotating black hole solutions in Scalar-Tensor-Vector Gravity or modified gravity (MOG). Our results yield exact analytic, closed-form relations expressing the mass $M$, the MOG coupling parameter $\alpha$, and the distance $D$ of a Schwarzschild-MOG black hole in terms of a minimal set of directly measurable elements of the accretion disk: the total frequency shift, the telescope aperture angle, and the redshift rapidity. The resulting expressions are derived for particles close to the midline and line of sight, where the redshift rapidity is treated as a relativistic invariant encoding the evolution of the frequency shift with respect to the emitter's proper time in MOG spacetime. We further extend the formalism to the rotating Kerr-MOG geometry and obtain corresponding relations that determine the rotation parameter $a$ jointly with $M$, $\alpha$, and $D$ on the midline. In the rotating background, we introduced the redshift acceleration (general-relativistic version of jerk) to disentangle the spacetime parameters. Crucially, the explicit appearance of $\alpha$ in these formulas enables direct empirical estimation of this parameter, thereby providing a means to test for departures from standard general relativity. The previous results obtained in the standard Schwarzschild/Kerr backgrounds are recovered in the limit $\alpha \to 0$. The derived expressions are concise and suitable for incorporation into black hole parameter-estimation pipelines.

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 derives exact analytic closed-form expressions for the mass M, MOG parameter α, and distance D of a Schwarzschild-MOG black hole (and jointly the spin a for Kerr-MOG) in terms of a minimal set of accretion-disk observables: total frequency shift, telescope aperture angle, and redshift rapidity (with redshift acceleration added for the rotating case). The relations are obtained under the assumption that particles lie close to the midline and line of sight, allowing redshift rapidity to be treated as a relativistic invariant encoding the proper-time evolution of the frequency shift. The explicit dependence on α permits its direct empirical estimation to test departures from GR; the standard Schwarzschild/Kerr results are recovered for α → 0. The expressions are presented as concise and suitable for parameter-estimation pipelines.

Significance. If the derivations hold and the geometric assumptions are validated, the work supplies a direct analytic route to extract the MOG coupling α from observations, offering a concrete, falsifiable test of modified gravity without post-hoc fitting. The closed-form character and explicit recovery of the GR limit are genuine strengths that could facilitate incorporation into existing pipelines. The significance for empirical tests is nevertheless limited by the lack of any quantitative check on how violations of the midline/line-of-sight restriction propagate into the inferred α.

major comments (2)
  1. [Abstract] Abstract: the claim that the formulas enable 'direct empirical estimation of this parameter' without further modeling rests on the midline/line-of-sight restriction (explicitly stated in the abstract). No quantitative assessment or error propagation is supplied showing how much the extracted α changes when particles are distributed through a disk of finite thickness or when the line of sight is imperfectly aligned. Because α enters the metric and therefore the redshift expressions directly, even modest violations can bias the inferred value, undermining the central claim of a model-independent test.
  2. [Kerr-MOG section] Kerr-MOG derivation (the section introducing redshift acceleration): the manuscript states that redshift acceleration is introduced 'to disentangle the spacetime parameters' and obtain joint expressions for a, M, α, D. No explicit check is given that this quantity is independent of the others under the MOG line element or that the resulting algebraic system remains solvable and closed once the midline assumption is imposed; without that verification the joint-estimation claim is not yet load-bearing.
minor comments (2)
  1. [Abstract] The abstract refers to 'redshift rapidity' and 'redshift acceleration' without an inline definition or reference to the defining equation; adding a brief parenthetical definition (e.g., d(ln ν)/dτ) would improve immediate readability.
  2. The statement that 'the previous results obtained in the standard Schwarzschild/Kerr backgrounds are recovered in the limit α → 0' is correct in principle but would benefit from an explicit side-by-side comparison of the MOG and GR expressions in an appendix or table.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise important points regarding the scope of our assumptions and the need for additional verifications. We respond point by point below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the formulas enable 'direct empirical estimation of this parameter' without further modeling rests on the midline/line-of-sight restriction (explicitly stated in the abstract). No quantitative assessment or error propagation is supplied showing how much the extracted α changes when particles are distributed through a disk of finite thickness or when the line of sight is imperfectly aligned. Because α enters the metric and therefore the redshift expressions directly, even modest violations can bias the inferred value, undermining the central claim of a model-independent test.

    Authors: We acknowledge that our expressions for direct estimation of α are derived under the explicit midline and line-of-sight assumptions stated in the abstract. These assumptions allow the redshift rapidity to be treated as an invariant, yielding closed-form relations. We agree that assessing the sensitivity to violations of these assumptions is important for practical application. In the revised version, we will add a discussion paragraph estimating the leading-order effects of small deviations from the midline (e.g., using a perturbative approach for small height-to-radius ratios in the disk). A full quantitative error propagation analysis would require numerical modeling of ray tracing in MOG spacetimes, which lies outside the analytic focus of the present work but will be flagged as future research. revision: partial

  2. Referee: [Kerr-MOG section] Kerr-MOG derivation (the section introducing redshift acceleration): the manuscript states that redshift acceleration is introduced 'to disentangle the spacetime parameters' and obtain joint expressions for a, M, α, D. No explicit check is given that this quantity is independent of the others under the MOG line element or that the resulting algebraic system remains solvable and closed once the midline assumption is imposed; without that verification the joint-estimation claim is not yet load-bearing.

    Authors: We thank the referee for this precise comment. The redshift acceleration is introduced as the second proper-time derivative of the frequency shift to provide the additional equation needed to solve for the four unknowns (a, M, α, D). In the revised manuscript, we will explicitly demonstrate the independence by deriving the expression for redshift acceleration from the MOG metric and showing that it introduces a new functional dependence not reducible to the lower-order quantities. We will also confirm that the resulting system of four equations in four unknowns remains algebraically closed and solvable under the midline approximation, with the explicit solutions already provided in the text. revision: yes

standing simulated objections not resolved
  • Full numerical error propagation and bias quantification for non-ideal disk geometries and alignments, as this would necessitate extensive simulations beyond the analytic derivations of the current manuscript.

Circularity Check

0 steps flagged

Derivation chain self-contained; no circular reductions identified

full rationale

The paper derives closed-form expressions for M, α, D (and a) directly from the MOG line element by applying the definitions of total frequency shift, redshift rapidity, and aperture angle under the stated midline/line-of-sight geometric restriction. α enters explicitly as a free parameter of the input metric and is algebraically solved for in terms of observables; the resulting formulas are not fitted post hoc nor defined in terms of the target quantities. No self-citation load-bearing steps, self-definitional loops, or ansatz smuggling appear in the derivation. The α → 0 limit simply recovers the prior Schwarzschild/Kerr results without circularity. The central claim that the explicit α dependence enables empirical estimation follows from the algebraic inversion of the metric-derived redshift expressions and does not reduce to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the MOG metric solutions, the geometric restriction to midline/line-of-sight particles, and the identification of redshift rapidity as a proper-time invariant; no new free parameters are introduced beyond the MOG α that is being solved for.

axioms (2)
  • domain assumption The Schwarzschild-MOG and Kerr-MOG line elements are the correct background metrics for the accretion flow.
    Invoked throughout the derivation of the frequency-shift relations.
  • domain assumption Redshift rapidity is a relativistic invariant that directly encodes the evolution of frequency shift with emitter proper time.
    Stated explicitly for particles close to the midline and line of sight.

pith-pipeline@v0.9.0 · 5569 in / 1510 out tokens · 52664 ms · 2026-05-10T16:59:45.723820+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

43 extracted references · 36 canonical work pages

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    The right panels of Fig

    whenα= 0. The right panels of Fig. 1 display ˙z M OG as a function of the azimuthal angleφfor several values of the MOG parameterα. Unlike the case of the total frequency shift, the rapidity either increases or decreases withα, depending on the emitter radius. In the upper-right panel we setr e = 10r ISCO (α) andD= 10 5rISCO (α); these plots show how the ...

    2023

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    1 + p ˜P− ˜N 1−2 ˜P+ ˜N ˜a+ ˜∆3/2 cosδ mq ˜∆2 cos2 δm + (1−2 ˜P+ ˜N) sin2 δm # , (A1b) 1 +z (m) KM OG2 = 1q 1−3 ˜P+ 2 ˜N+ 2˜a p ˜P− ˜N

    The starting5×5system At the midline, the system to be solved is given by (93)-(97) re =Dtanδ m, (A1a) 1 +z (m) KM OG1 = 1q 1−3 ˜P+ 2 ˜N+ 2˜a p ˜P− ˜N " 1 + p ˜P− ˜N 1−2 ˜P+ ˜N ˜a+ ˜∆3/2 cosδ mq ˜∆2 cos2 δm + (1−2 ˜P+ ˜N) sin2 δm # , (A1b) 1 +z (m) KM OG2 = 1q 1−3 ˜P+ 2 ˜N+ 2˜a p ˜P− ˜N " 1 + p ˜P− ˜N 1−2 ˜P+ ˜N ˜a− ˜∆3/2 cosδ mq ˜∆2 cos2 δm + (1−2 ˜P+ ˜N...

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