arxiv: 2604.15397 · v1 · submitted 2026-04-16 · ⚛️ physics.gen-ph

Recognition: unknown

Equations of motion of the mass centers in a scalar theory of gravity with a preferred frame

Mayeul Arminjon

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:37 UTC · model grok-4.3

classification ⚛️ physics.gen-ph

keywords scalar theory of gravitypreferred framepost-Newtonian approximationequations of motionmass centersweakly gravitating bodiesasymptotic integration

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The pith

Equations of motion of mass centers are derived for the second version of a scalar theory of gravity with a preferred frame.

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

This paper derives the equations of motion of the mass centers for a system of weakly gravitating bodies in the second version of a scalar theory of gravity that interprets gravity as a pressure force. The scalar field both sets the gravitational acceleration and fixes the link between a flat background metric and a curved physical metric. An asymptotic post-Newtonian scheme produces the local field equations, which are then integrated inside the bodies together with an asymptotic separation framework to obtain the center-of-mass equations. A reader would care because the result supplies concrete, testable predictions for body motion in an alternative gravity model that includes a preferred frame.

Core claim

In the second version of the scalar theory of gravity with a preferred frame, the equations of motion of the mass centers of a system of weakly gravitating bodies are obtained by deriving local post-Newtonian field equations and integrating them inside the bodies using an asymptotic separation framework.

What carries the argument

The asymptotic scheme of post-Newtonian approximation to obtain local field equations, followed by integration inside the bodies.

Load-bearing premise

The asymptotic framework for good separation between the different bodies and the validity of the post-Newtonian scheme applied to the second version of the scalar theory hold without additional corrections.

What would settle it

A precise orbital calculation or observation of weakly gravitating bodies whose mass-center trajectories deviate from the integrated post-Newtonian equations under the theory's metric and field assumptions would show the derivation does not hold.

read the original abstract

The theory considered interprets gravity as a pressure force. Thus, the scalar gravitational field defines the gravity acceleration field. However, it also determines the relation between the flat ``background metric'' and a curved ``physical metric''. Here we derive the equations of motion of the mass centers of a system of weakly gravitating bodies in the second version of that theory. We use the framework which was built and used for the first version. Namely, we use an asymptotic scheme of post-Newtonian (PN) approximation to derive the local (field) PN equations, and by integration inside the bodies we deduce from those local equations the equations of motion of the mass centers, using also an asymptotic framework for the good separation between the different bodies.

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

Arminjon applies the same post-Newtonian asymptotic scheme and body integration from his first-version scalar theory paper to derive the center-of-mass equations for the second version.

read the letter

The paper's core move is straightforward: it reuses the existing asymptotic post-Newtonian framework and the integration inside weakly gravitating bodies to obtain the equations of motion for the mass centers in the updated scalar theory. The abstract makes clear that the local field equations are approximated the same way, then integrated with the separation asymptotics between bodies, without introducing new machinery. That transfer works cleanly on the description given, and the preferred-frame aspects do not appear to force extra corrections at this order. The consistency with the prior work is the main technical achievement here. It shows the second version preserves the structure needed for the center-of-mass motion to follow from the local equations in the same manner. No load-bearing gaps jump out from the outline, and the assumptions about good separation and the validity of the PN scheme for this version hold without obvious contradiction. The limitation is that the result stays tightly inside this one model. It adds a specific derivation rather than a broader test or comparison, so the practical reach remains narrow even if the steps are correct. Readers outside the small group working on scalar theories with preferred frames will not find new predictions or links to observations. This is the kind of paper that belongs in a specialized journal on alternative gravity. A serious referee should check the details of how the second-version field equations feed into the PN expansion and the integration step, because those are the parts that need verification for accuracy. I would send it to peer review rather than desk reject.

Referee Report

1 major / 3 minor

Summary. The manuscript derives the equations of motion of the mass centers for a system of weakly gravitating bodies in the second version of a scalar theory of gravity with a preferred frame. It applies an asymptotic post-Newtonian approximation to obtain the local field equations from the theory's field equations, followed by integration over the interior of each body, while employing an asymptotic separation framework between bodies and reusing the overall scheme developed for the first version of the theory.

Significance. If the derivation holds, the work supplies explicit center-of-mass equations in this alternative framework, enabling direct comparison with general relativity in the weak-field regime and potential observational tests. The consistent reuse of the established asymptotic PN scheme and body-integration method across theory versions is a methodological strength that supports cumulative progress within the authors' program.

major comments (1)

The manuscript states that local PN equations are derived for the second version before integration, but does not explicitly demonstrate (e.g., via a side-by-side comparison of the relevant field-equation terms) that the preferred-frame contributions do not generate additional source terms inside the bodies that would alter the integrated center-of-mass acceleration relative to the first version. This step is load-bearing for the claim that the prior framework applies directly.

minor comments (3)

Abstract: the final form of the derived center-of-mass equations is not stated; including the explicit expression (even in schematic form) would improve clarity.
Introduction and §2: cross-references to the first-version paper should cite specific equation numbers rather than the work as a whole, to allow readers to verify the reused framework.
Notation: the distinction between the flat background metric and the physical metric should be maintained with consistent symbols in all displayed equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. The recommendation for minor revision is appreciated, and we address the point raised below by clarifying the derivation and committing to an explicit addition in the revised version.

read point-by-point responses

Referee: The manuscript states that local PN equations are derived for the second version before integration, but does not explicitly demonstrate (e.g., via a side-by-side comparison of the relevant field-equation terms) that the preferred-frame contributions do not generate additional source terms inside the bodies that would alter the integrated center-of-mass acceleration relative to the first version. This step is load-bearing for the claim that the prior framework applies directly.

Authors: We agree that an explicit demonstration strengthens the presentation. The local PN equations for the second version were obtained by applying the same asymptotic post-Newtonian scheme used in the first version to the field equations of the second version. The preferred-frame effects enter through the relation between the flat background metric and the physical metric, but they appear only in the definition of the gravitational acceleration field and do not modify the source terms (stress-energy contributions) inside the bodies. Consequently, the integration over each body’s interior yields the same center-of-mass acceleration as in the first version. In the revised manuscript we will insert a short subsection that tabulates the relevant local PN field-equation terms for both versions side by side, confirming that no additional source terms arise within the bodies and that the integration procedure therefore carries over unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a standard mathematical deduction from prior field equations

full rationale

The paper derives the center-of-mass equations of motion for the second version of the scalar theory by first obtaining local post-Newtonian field equations via an asymptotic scheme and then integrating those equations inside the bodies, using a separation asymptotics framework previously developed for the first version. This reuses a methodological framework (asymptotic PN approximation and body integration) but does not reduce the target result to a fitted parameter, self-definition, or unverified self-citation chain. The local field equations of the theory serve as independent inputs, and the output EOM are obtained by explicit integration steps rather than by construction or renaming. No load-bearing premise collapses to a self-referential loop; the work is self-contained as a deductive extension.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit list of free parameters or axioms; the derivation implicitly rests on the validity of the post-Newtonian asymptotic scheme and the body-separation framework from the first version, which are treated as given.

pith-pipeline@v0.9.0 · 5416 in / 1075 out tokens · 35681 ms · 2026-05-10T09:37:53.014111+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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