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An arbitrary vector field coupled to the Ricci scalar gradient yields Lagrangian formulations for Rastall and unimodular gravity.

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2026-06-29 04:12 UTC pith:D6SHTWHM

load-bearing objection The paper gives an auxiliary-vector construction that yields a Lagrangian for Rastall gravity and a variant for unimodular gravity, but the conditions on the vector field are imposed by hand. the 2 major comments →

arxiv 2606.27761 v1 pith:D6SHTWHM submitted 2026-06-26 gr-qc

A Lagrangian formulation for Rastall gravity and a covariant formulation for unimodular gravity

Juilson A. P. Paiva , Julio C. Fabris , Vilson T. Zanchin This is my paper
classification gr-qc
keywords Rastall gravityunimodular gravityLagrangian formulationPalatini variationvector fieldRicci scalarnon-conservative gravitymodified gravity
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.

The paper develops a Lagrangian for gravity models in which the energy-momentum tensor is not conserved. An arbitrary vector field is introduced that couples directly to the gradient of the Ricci curvature scalar. Variation of the resulting action in the Palatini formalism produces field equations that reduce to those of Rastall gravity when the vector field obeys certain conditions. Slightly altered conditions on the same vector field recover the equations of unimodular gravity, and the same outcomes appear in the metric variational approach.

Core claim

By introducing an arbitrary vector field that couples with the gradient of the Ricci curvature scalar, a Lagrangian formulation is obtained whose Palatini variation gives field equations that, under physically reasonable conditions on the vector field, reduce exactly to Rastall gravity, while slightly different conditions furnish unimodular gravity. The same results hold in the metric variational approach.

What carries the argument

Arbitrary vector field coupled to the gradient of the Ricci curvature scalar, which in the Palatini framework dictates the manifold geometry as Weyl or Riemannian and enforces the desired field equations.

Load-bearing premise

That there exist physically reasonable conditions on the auxiliary vector field such that the derived field equations reduce exactly to those of Rastall gravity and unimodular gravity.

What would settle it

A calculation showing that no choice of conditions on the vector field makes the Lagrangian-derived equations identical to the standard Rastall or unimodular field equations would falsify the claim.

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

If this is right

  • The derived field equations match Rastall gravity when the vector field satisfies specific conditions.
  • Slightly altered conditions on the vector field lead to unimodular gravity.
  • The metric variational approach also recovers the same models under appropriate conditions.
  • This establishes covariant Lagrangian formulations for these non-standard gravity theories.

Where Pith is reading between the lines

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

  • This method could be extended to other modified gravity models involving non-conservation laws.
  • The physical meaning of the auxiliary vector field might be explored in cosmological or astrophysical settings to test the models.
  • Further investigation could reveal whether this vector field has observable effects beyond the standard formulations.

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

2 major / 2 minor

Summary. The paper proposes a Lagrangian formulation for Rastall gravity (a non-conservative theory where the energy-momentum tensor divergence does not vanish) and a covariant formulation for unimodular gravity. This is achieved by introducing an arbitrary vector field that couples to the gradient of the Ricci scalar. Field equations are derived via both the Palatini variational principle (where the connection and metric are independent, allowing the vector field to dictate Weyl or Riemannian geometry) and the standard metric variation. Under specific physically reasonable conditions imposed on the auxiliary vector field, the equations reduce to those of Rastall gravity; slightly different conditions recover unimodular gravity.

Significance. If the reductions hold, the work supplies the first covariant Lagrangian for Rastall gravity and a new one for unimodular gravity. This could enable variational techniques, Hamiltonian analysis, or quantization attempts for these models. The Palatini treatment linking the auxiliary field to geometry choice is a potentially useful technical feature.

major comments (2)
  1. [Abstract and §3 (Palatini derivation)] The central claim rests on the statement that 'certain physically reasonable conditions' on the auxiliary vector field cause exact reduction to the Rastall and unimodular field equations. No derivation is visible showing these conditions emerge from the action principle rather than being imposed by hand after variation; this must be demonstrated explicitly (e.g., by showing the conditions follow from extremization or from a subsidiary equation derived from the action).
  2. [§2–3 (variational setup)] In the Palatini approach, the auxiliary vector field is said to 'dictate whether the manifold geometry is Weyl or Riemannian.' The precise mechanism (e.g., how the vector field modifies the connection or imposes the Weyl condition) needs to be spelled out with the resulting connection equation shown, because this step is load-bearing for the claim that the same action yields both geometries under different conditions.
minor comments (2)
  1. [§2] Clarify the precise coupling term in the action (e.g., is it ξ^μ abla_μ R or a more general contraction?) and state the full action before variation.
  2. [§4] Provide the explicit reduced field equations side-by-side with the standard Rastall and unimodular equations to allow direct verification of the match.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and §3 (Palatini derivation)] The central claim rests on the statement that 'certain physically reasonable conditions' on the auxiliary vector field cause exact reduction to the Rastall and unimodular field equations. No derivation is visible showing these conditions emerge from the action principle rather than being imposed by hand after variation; this must be demonstrated explicitly (e.g., by showing the conditions follow from extremization or from a subsidiary equation derived from the action).

    Authors: The conditions on the auxiliary vector field are imposed after the variation to select the specific reductions to Rastall and unimodular gravity from the general equations obtained from the action. These are physically reasonable choices (e.g., the vector field being divergence-free or proportional to the gradient of the Ricci scalar) that recover the target theories. They do not arise as subsidiary equations from extremizing the action itself. In the revised version we will expand the discussion in §3 to clarify the physical motivation and consistency of these conditions within the variational setup. revision: yes

  2. Referee: [§2–3 (variational setup)] In the Palatini approach, the auxiliary vector field is said to 'dictate whether the manifold geometry is Weyl or Riemannian.' The precise mechanism (e.g., how the vector field modifies the connection or imposes the Weyl condition) needs to be spelled out with the resulting connection equation shown, because this step is load-bearing for the claim that the same action yields both geometries under different conditions.

    Authors: We agree that the mechanism requires explicit presentation. The auxiliary vector field enters the Palatini variation with respect to the independent connection, yielding a modified connection equation whose form depends on the vector field. In the revised manuscript we will derive and display this connection equation in §2–3, showing explicitly how different conditions on the vector field produce Weyl versus Riemannian geometry. revision: yes

standing simulated objections not resolved
  • The conditions on the auxiliary vector field are imposed after variation rather than emerging from the action principle, and we cannot demonstrate that they follow from extremization or a subsidiary equation derived from the action.

Circularity Check

0 steps flagged

No significant circularity; standard auxiliary-field construction

full rationale

The paper introduces an auxiliary vector field into a general action, performs Palatini and metric variations to obtain field equations, and then imposes physically motivated conditions on the vector field to recover the known Rastall and unimodular equations. This is a forward variational construction whose output is not equivalent to its inputs by definition; the target equations are external benchmarks, and the auxiliary field plus conditions constitute an independent ansatz whose validity can be checked against those benchmarks. No self-citation load-bearing steps, fitted inputs renamed as predictions, or uniqueness theorems imported from the authors' prior work are present in the derivation chain described.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The construction rests on the introduction of an auxiliary vector field whose properties are adjusted by hand to recover the target theories.

axioms (3)
  • standard math Metric and connection are varied independently in the Palatini formalism.
    Standard background assumption of Palatini variation.
  • domain assumption The vector field can be chosen so that the geometry is Riemannian or Weyl as needed.
    Invoked to link the vector field to the resulting geometry.
  • ad hoc to paper Physically reasonable conditions on the vector field recover Rastall and unimodular equations.
    The reduction step depends on these conditions.
invented entities (1)
  • arbitrary vector field no independent evidence
    purpose: Couples to the gradient of the Ricci scalar to allow non-vanishing divergence of the energy-momentum tensor.
    New auxiliary field introduced in the action to generate the desired models.

pith-pipeline@v0.9.1-grok · 5702 in / 1243 out tokens · 45204 ms · 2026-06-29T04:12:32.757836+00:00 · methodology

0 comments
read the original abstract

We propose a Lagrangian formulation for a non-conservative gravity model in which the divergence of the energy-momentum tensor in curved spacetime does not vanish. This is accomplished by introducing an arbitrary vector field that couples with the gradient of the Ricci curvature scalar. We first derive the field equations using the Palatini variational approach. Because the connection and the metric tensor are independent in the Palatini framework, the auxiliary vector field dictates whether the manifold geometry is Weyl or Riemannian. By assuming certain physically reasonable conditions on this vector field, the resulting field equations reduce to those of Rastall gravity. Furthermore, slightly different conditions on the vector field furnish unimodular gravity. For comparison, we also employ the standard metric variational approach to obtain the field equations, demonstrating that the same models can be recovered under appropriate conditions. Our key results are the derivation of a covariant Lagrangian formulation for Rastall gravity and a new Lagrangian formulation for unimodular gravity.

discussion (0)

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Forward citations

Cited by 1 Pith paper

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  1. Exact solutions for static spherically symmetric spacetime with a perfect fluid in Rastall theory

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    Exact Liouvillian solutions in Rastall gravity are derived for perfect fluids with p=wρ under special Rastall-parameter relations, reproducing GR solutions for w=0, -1/5, -1/3, -1 and yielding five additional solution...

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