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arxiv: 2606.04077 · v2 · pith:L3LSD3FInew · submitted 2026-06-02 · 🌀 gr-qc · physics.class-ph

Lagrangian Extensions of Newtonian Gravity constrained by Solar System tests

Pedro H. Dalpr\'a , J\'ulio C. Fabris , Hermano Velten , J\'unior D. Toniato This is my paper

Pith reviewed 2026-06-28 08:43 UTC · model grok-4.3

classification 🌀 gr-qc physics.class-ph
keywords Lagrangian gravity extensionscalar fieldNordtvedt effectMercury pericenterpost-Newtonian potentialSolar System testsvariable gravitational couplingN-body equations
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The pith

A generalized Lagrangian with a second scalar field produces an effective post-Newtonian potential whose free parameter is bounded by Solar System observations.

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

The paper constructs an extension of Newtonian gravity by adding a second dynamical scalar field to a generalized Lagrangian, then derives the full field equations and their weak-field limit. This limit yields an effective gravitational potential that incorporates post-Newtonian corrections and produces N-body equations in which inertial and gravitational masses are no longer identical. The resulting mass difference is confronted with the Nordtvedt effect, while the osculating-orbit method applied to a two-body system gives a secular pericenter advance that is compared with Mercury’s observed perihelion shift. Both comparisons supply numerical bounds on the single free parameter of the model.

Core claim

The introduction of a second dynamical scalar field into the generalized Lagrangian generates field equations whose weak-field reduction produces an effective post-Newtonian potential; the associated N-body equations of motion exhibit a difference between inertial and gravitational masses that is constrained by the Nordtvedt effect, while the osculating-orbit calculation for a two-body system yields a secular pericenter variation that is matched to Mercury’s observed perihelion advance, thereby bounding the model’s free parameter.

What carries the argument

Generalized Lagrangian with a second dynamical scalar field, whose field equations reduce in the weak-field limit to an effective post-Newtonian potential and mass-difference terms used for Solar-System constraints.

If this is right

  • The model predicts a nonzero difference between inertial and gravitational masses that must satisfy existing Nordtvedt bounds.
  • The two-body osculating-orbit equations produce a calculable secular pericenter drift whose magnitude is fixed by the same free parameter.
  • Any concrete Lagrangian of the proposed class inherits the same observational upper and lower limits on its free parameter.
  • The weak-field N-body equations can be applied directly to other Solar-System bodies once the parameter is fixed.

Where Pith is reading between the lines

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

  • Tighter future bounds on the parameter would restrict the model’s viability on galactic or cosmological scales where the weak-field assumption no longer holds.
  • The same Lagrangian construction could be tested against lunar laser ranging or binary-pulsar timing once the parameter range is narrowed.
  • If the scalar-field coupling is universal, the derived mass-difference effect should appear in any precision test of the equivalence principle.

Load-bearing premise

The specific functional form chosen for the generalized Lagrangian, together with the validity of the weak-field approximation at Solar-System scales, is taken as given.

What would settle it

A future measurement of the Nordtvedt parameter or of Mercury’s secular pericenter advance lying outside the interval permitted by the model’s allowed range for its free parameter would rule out that range or the assumed Lagrangian form.

read the original abstract

We explore an extension to Newtonian gravity through a generalised Lagrangian function with the introduction of a second dynamical scalar field. Building on previous research into gravity with variable gravitational coupling, the work derives the complete field equations and applies a weak-field approximation. This leads to an effective post-Newtonian gravitational potential that includes key aspects of relativistic theories. The resulting N-body equations of motion highlight differences among inertial and gravitational masses, which can constrain the theory's free parameter through data from the Nordtvedt effect. By employing the method of osculating orbits for a two-body system, the study calculates the secular variation of the orbital pericenter and aligns this with the latest data on Mercury's perihelion shift, for another observational constraint on the model. Furthermore, a few examples of theories are discussed.

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 proposes an extension of Newtonian gravity via a generalized Lagrangian incorporating a second dynamical scalar field. It derives the complete field equations, applies a weak-field approximation to obtain an effective post-Newtonian gravitational potential, and extracts N-body equations of motion that exhibit differences between inertial and gravitational masses. These differences are used to constrain the model's single free parameter via the Nordtvedt effect. The secular variation of the orbital pericenter is computed for a two-body system using the osculating-orbit method and compared to Mercury perihelion data for a second constraint. Examples of specific theories fitting the framework are discussed.

Significance. If the central derivations hold, the work supplies a concrete route for bounding a class of scalar-tensor extensions with existing Solar-System data, complementing post-Newtonian tests of GR. The dual use of the Nordtvedt effect and Mercury's pericenter precession, together with the explicit osculating-orbit calculation, constitutes a strength; the approach is falsifiable with current observational precision and could guide future mission design.

major comments (2)
  1. [weak-field approximation and N-body equations] Weak-field approximation and N-body equations section: the central claim that the second scalar field produces a composition-independent inertial-gravitational mass difference controlled by exactly one free parameter requires that the expansion isolates this parameter without residual scalar-mediated forces or higher-order terms that would modify the osculating-orbit pericenter rate at Solar-System precision. No explicit verification or term-by-term cancellation is shown that the chosen functional form guarantees this isolation.
  2. [Nordtvedt-effect and Mercury-pericenter sections] Nordtvedt-effect and Mercury-pericenter sections: the bounds on the free parameter are obtained by fitting the same class of observations that motivate the Lagrangian form; the manuscript does not demonstrate that the derived constraints remain robust under small deformations of the scalar-field coupling function.
minor comments (2)
  1. [Lagrangian definition] Notation for the second scalar field and its coupling constants is introduced without a consolidated table; a single reference table would improve readability.
  2. [examples section] The abstract states that 'a few examples of theories are discussed' but the manuscript does not list the explicit Lagrangians or parameter values for those examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: Weak-field approximation and N-body equations section: the central claim that the second scalar field produces a composition-independent inertial-gravitational mass difference controlled by exactly one free parameter requires that the expansion isolates this parameter without residual scalar-mediated forces or higher-order terms that would modify the osculating-orbit pericenter rate at Solar-System precision. No explicit verification or term-by-term cancellation is shown that the chosen functional form guarantees this isolation.

    Authors: The functional form of the generalized Lagrangian is chosen so that, in the weak-field limit, scalar contributions to the effective potential produce the inertial-gravitational mass difference at post-Newtonian order while residual scalar-mediated forces and higher-order terms are either suppressed or cancel by construction. We acknowledge that an explicit term-by-term verification of these cancellations was not included in the original text. In the revised manuscript we will add a dedicated subsection performing this expansion to confirm isolation of the single free parameter at the precision relevant for the osculating-orbit calculation. revision: yes

  2. Referee: Nordtvedt-effect and Mercury-pericenter sections: the bounds on the free parameter are obtained by fitting the same class of observations that motivate the Lagrangian form; the manuscript does not demonstrate that the derived constraints remain robust under small deformations of the scalar-field coupling function.

    Authors: The Lagrangian is introduced as a specific one-parameter extension motivated by prior work on variable gravitational coupling. The Nordtvedt and pericenter constraints are applied directly to this class using independent Solar-System data sets. Small deformations of the coupling function would in general introduce additional free parameters and therefore define a different model class outside the scope of the present analysis. The bounds we report are therefore valid for all theories that fit the single-parameter framework, as illustrated by the explicit examples discussed in the manuscript. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation from Lagrangian to constraints is self-contained

full rationale

The paper introduces a generalized Lagrangian with a second dynamical scalar field, derives the complete field equations, performs a weak-field approximation to obtain an effective post-Newtonian potential, extracts N-body equations of motion that exhibit inertial-gravitational mass differences, and then applies external Solar-System data (Nordtvedt effect and Mercury pericenter precession) to bound the single free parameter. None of the enumerated circularity patterns apply: the parameter is not defined in terms of the output constraints, no fitted input is relabeled as a prediction, and no load-bearing self-citation or uniqueness theorem is invoked. The constraints are applications of the derived equations to independent observations, leaving the central derivation chain independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on one ad-hoc Lagrangian extension and one free parameter whose value is fixed only by the Solar-System data themselves; no independent evidence is supplied for either.

free parameters (1)
  • free parameter of the generalized Lagrangian
    Introduced to generalize Newtonian gravity and later bounded by Nordtvedt and perihelion data.
axioms (1)
  • ad hoc to paper The chosen functional form of the generalized Lagrangian with second scalar field
    The specific extension is postulated without derivation from a more fundamental principle.
invented entities (1)
  • second dynamical scalar field no independent evidence
    purpose: To extend Newtonian gravity beyond the standard formulation
    Postulated in the Lagrangian; no independent falsifiable signature outside the fitted constraints is provided.

pith-pipeline@v0.9.1-grok · 5676 in / 1131 out tokens · 20204 ms · 2026-06-28T08:43:17.231678+00:00 · methodology

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