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arxiv: 2605.20574 · v1 · pith:2RVUG4HRnew · submitted 2026-05-20 · 🌀 gr-qc

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

Ioannis Liodis , Gernot Heissel , Rita Mastroioanni , Jai Grover , Dario Izzo This is my paper

Pith reviewed 2026-05-21 04:48 UTC · model grok-4.3

classification 🌀 gr-qc
keywords modified gravityMOGS-starsorbital precessionpost-Newtoniangalactic centerdark matter degeneracy
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The pith

Modified gravity changes S-star orbits around the galactic black hole in ways distinguishable from general relativity and possibly confusable with dark matter.

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

This paper sets out to show that scalar-tensor-vector gravity, known as MOG, produces clear changes in the orbits of stars near the supermassive black hole at the Milky Way's center. The authors integrate the first post-Newtonian equations of motion that include both the usual space-time curvature and an extra fifth-force term from MOG. They track how orbital elements evolve and how the projected positions and radial velocities on the sky differ from pure general relativity predictions. The changes grow with the MOG parameter alpha and can reach sizes comparable to present-day measurement precision when observations span many years. The work also notes that some of these changes, especially shifts in the argument of pericenter, can look like the effects of a dark matter distribution, creating a potential overlap that observers must disentangle.

Core claim

We numerically integrated the first post-Newtonian equations of motion for S-stars within the MOG framework, considering contributions from the space-time geometry and the fifth force. We found that MOG significantly alters the orbital precession, particularly for higher values of the MOG parameter alpha. For sufficiently large alpha or long observational baselines, the deviations in the observables can reach amplitudes comparable to current observational precision. Furthermore, we demonstrate that MOG effects can mimic those of a dark matter distribution, particularly in the argument of pericentre, and we reveal an unexplored connection between MOG and GR with electromagnetism.

What carries the argument

The first post-Newtonian equations of motion in the MOG framework that add the fifth-force term to the usual space-time geometry contributions.

If this is right

  • Orbital precession increases noticeably with larger values of the MOG parameter alpha.
  • Deviations appear in right ascension, declination, and radial velocity at levels reachable by current precision over long baselines.
  • MOG signatures in the argument of pericentre can overlap with those produced by a dark matter distribution.
  • The overall pattern of changes remains distinct from pure general relativity and can be tested directly with S-star data.

Where Pith is reading between the lines

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

  • If MOG effects are confirmed, models of the galactic center could reduce reliance on dark matter to explain stellar dynamics.
  • Joint fits that include both MOG parameters and a possible dark matter density profile may become necessary to avoid misinterpretation.
  • Similar post-Newtonian integrations could be applied to other stars or pulsars near black holes to search for the same fifth-force signature.
  • The connection to electromagnetism mentioned in the paper suggests possible cross-checks with charged-particle orbits if such data become available.

Load-bearing premise

The first post-Newtonian approximation remains accurate for S-star orbits once the MOG fifth-force term is included, and the MOG parameter alpha can be large enough that the resulting deviations exceed measurement noise within realistic observing times.

What would settle it

A multi-year astrometric and spectroscopic campaign on S-stars that finds no excess precession or no matching pattern in sky positions and radial velocities beyond general relativity predictions, for the range of alpha values considered, would show the claimed deviations are not present.

Figures

Figures reproduced from arXiv: 2605.20574 by Dario Izzo, Gernot Heissel, Ioannis Liodis, Jai Grover, Rita Mastroioanni.

Figure 1
Figure 1. Figure 1: Definition of the Euler angles (Ω, ι, ω) giving the orbital orientation and of the true anomaly f giving the current position within the orbit. The static fundamental frame (X, Y, Z) (blue) is adapted to the observables, while the co-rotated Gaussian frame (n, λ, ez) (orange) is adapted to the symmetry of the (perturbed) two-body problem (reproduced from Heißel, G. et al. (2022)). x/ρ. We still kept track … view at source ↗
Figure 2
Figure 2. Figure 2: Oribt of star S2 (yellow star) around the BH (black dot) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the observables’ (RA, DEC, and RV in the first, middle, and bottom row, respectively) absolute di [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Orbital element evolution with true anomaly. Here, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effects on the evolution of orbital elements (top panel) and the deviation from GR (bottom panel). Each case is integrated independently. The cases are i) full MOG (solid blue line), ii) fifth-force contribution alone (solid red line), iii) space-time contribu￾tion alone (solid yellow line), and iv) GR (dashed black line). Cases ii) and iii) are not physical in MOG, but are used for diagnostic purposes, as… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the argument of pericentre with true [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of RV deviation for three values of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
read the original abstract

Scalar-tensor-vector gravity, also known as modified gravity (MOG), has emerged as an alternative to General Relativity (GR). It aims to explain astrophysical phenomena without invoking dark matter. The S-stars orbiting the supermassive black hole at the Galactic centre provide a unique opportunity to test the predictions of MOG because the orbital measurements are highly precise. We investigate the perturbations in the orbits of S-stars under MOG, focusing on the effects on orbital elements, observables such as right ascension, declination, and radial velocity, and the potential degeneracy with dark matter scenarios. We numerically integrated the first post-Newtonian equations of motion for S-stars within the MOG framework, considering contributions from the space-time geometry and the fifth force. We analysed the time evolution of orbital elements and projected the orbits onto the plane of the sky to assess deviations from GR. Furthermore, we compared the MOG-induced effects with those expected from a dark matter distribution. We found that MOG significantly alters the orbital precession, particularly for higher values of the MOG parameter $\alpha$. For sufficiently large $\alpha$ or long observational baselines, the deviations in the observables can reach amplitudes comparable to current observational precision. Furthermore, we demonstrate that MOG effects can mimic those of a dark matter distribution, particularly in the argument of pericentre, and we reveal an unexplored connection between MOG and GR with electromagnetism. The effects of MOG on stellar orbits are distinct from those predicted by GR and can be tested with precise astrometric and spectroscopic measurements of the S-stars. However, a potential degeneracy with dark matter signatures necessitates careful interpretation of observational data.

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 numerically integrates the first post-Newtonian equations of motion for S-stars orbiting the Galactic-center black hole in scalar-tensor-vector gravity (MOG). It includes both the modified spacetime geometry and an additional fifth-force term controlled by the free parameter α, tracks the evolution of orbital elements, projects the orbits onto the sky plane, and compares the resulting astrometric and radial-velocity signatures to pure GR and to a dark-matter distribution. The central claim is that MOG produces distinct precession and observable deviations whose amplitude can reach current measurement precision for sufficiently large α or long baselines, while also exhibiting a degeneracy with dark-matter-induced effects, particularly in the argument of pericentre.

Significance. If the first-PN treatment remains valid, the work supplies a concrete, observationally oriented framework for constraining the MOG parameter α with existing and forthcoming S-star data and correctly flags the degeneracy with dark-matter models that must be disentangled in any interpretation. The direct mapping from integrated trajectories to sky-plane and spectroscopic observables is a practical strength.

major comments (2)
  1. [§3] §3 (numerical integration of the first post-Newtonian equations): The distinguishability claim rests on the assumption that the first-PN integration (GR geometry plus MOG fifth force) captures the leading observable deviations. For S2 (e ≈ 0.88) and a long-range fifth force, the paper neither derives nor bounds the second-PN corrections that would arise from the modified field equations, nor does it report truncation-error estimates over the multi-year baselines required for detectable precession. If those corrections shift the argument-of-pericentre or sky-plane residuals by more than the claimed observational amplitude, the separation from GR and the degeneracy analysis no longer hold.
  2. [§4] §4 (comparison with dark matter): The degeneracy analysis is presented only for the argument of pericentre; no quantitative assessment is given for how the MOG fifth-force term affects the full set of observables (RA, Dec, radial velocity) relative to a Navarro-Frenk-White or other DM profile at the same level of precision.
minor comments (2)
  1. [Abstract] The abstract states that an 'unexplored connection between MOG and GR with electromagnetism' is revealed, but this connection is not developed or referenced in the main text or discussion.
  2. [Figures] Figure captions and axis labels should explicitly state the value of α used in each panel and the integration time span in years.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help improve the robustness of our analysis. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [§3] The distinguishability claim rests on the assumption that the first-PN integration (GR geometry plus MOG fifth force) captures the leading observable deviations. For S2 (e ≈ 0.88) and a long-range fifth force, the paper neither derives nor bounds the second-PN corrections that would arise from the modified field equations, nor does it report truncation-error estimates over the multi-year baselines required for detectable precession. If those corrections shift the argument-of-pericentre or sky-plane residuals by more than the claimed observational amplitude, the separation from GR and the degeneracy analysis no longer hold.

    Authors: We agree that a quantitative bound on second post-Newtonian corrections would strengthen the validity of the 1PN results, particularly for the high-eccentricity orbit of S2. Our current analysis focuses on the leading 1PN contributions from the modified MOG geometry and fifth-force term, which are the dominant effects for the α range and baselines considered. In the revised manuscript we will add an order-of-magnitude estimate of the 2PN terms derived from the MOG field equations together with truncation-error estimates for the numerical integrator over the relevant multi-year intervals. These estimates will show that the 2PN contributions remain below the claimed observational amplitudes for the parameter values examined, thereby supporting the distinguishability and degeneracy conclusions. revision: yes

  2. Referee: [§4] The degeneracy analysis is presented only for the argument of pericentre; no quantitative assessment is given for how the MOG fifth-force term affects the full set of observables (RA, Dec, radial velocity) relative to a Navarro-Frenk-White or other DM profile at the same level of precision.

    Authors: We concur that extending the degeneracy comparison beyond the argument of pericentre would provide a more complete picture. The revised manuscript will include quantitative assessments of the MOG fifth-force effects on right ascension, declination, and radial velocity, performed at the same precision level used for the pericentre analysis and directly compared against a Navarro-Frenk-White dark-matter profile. This addition will clarify the extent of the degeneracy across the full set of observables. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from MOG field equations and numerical integration

full rationale

The paper derives post-Newtonian equations of motion from the MOG framework (space-time geometry plus fifth force), numerically integrates them for S-star orbits, evolves orbital elements, projects observables, and compares deviations to GR and dark-matter cases as functions of the free parameter α. No step reduces a prediction to a fitted input by construction, no self-citation bears the central load, and no ansatz or uniqueness theorem is smuggled in. The results are computed outputs from the modified dynamics; the degeneracy analysis is an explicit comparison rather than a tautology. The chain is self-contained against external PN benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim depends on one free parameter controlling MOG strength, the domain assumption that post-Newtonian MOG equations capture the dominant corrections, and the MOG fifth force as an added interaction without independent falsification in this work.

free parameters (1)
  • MOG parameter α
    Controls the amplitude of modifications and fifth-force strength; higher values are required for observable deviations.
axioms (1)
  • domain assumption First post-Newtonian expansion remains valid for S-star orbits when MOG fifth-force term is added
    Invoked to justify the equations of motion integrated numerically.
invented entities (1)
  • Fifth force arising from MOG vector/scalar fields no independent evidence
    purpose: Produces additional orbital perturbations beyond GR geometry
    Introduced by the MOG framework to explain phenomena without dark matter; no independent evidence supplied in this study.

pith-pipeline@v0.9.0 · 5842 in / 1482 out tokens · 38403 ms · 2026-05-21T04:48:40.649716+00:00 · methodology

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