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arxiv: 2606.27433 · v1 · pith:YAISUEDHnew · submitted 2026-06-25 · 🌀 gr-qc · math-ph· math.MP

Tidal Forces in the Presence of Torsion and Nonmetricity

Armin van de Venn , Marcelo Netz-Marzola , David Vasak , J\"urgen Struckmeier This is my paper

Pith reviewed 2026-06-29 01:42 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.MP
keywords tidal forcestorsionnonmetricitymetric-affine gravitygeodesic deviationautoparallelsweak-field limit
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The pith

Torsion and nonmetricity add linear post-Riemannian corrections to tidal accelerations in the weak-field limit.

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

This paper extends the standard geodesic deviation equation to metric-affine geometries that incorporate torsion and nonmetricity. It derives a projected version of the equation and applies it to the relative acceleration between neighboring autoparallels. In the weak-field nonrelativistic regime, the tidal acceleration factors into the usual Newtonian term plus linear corrections sourced by torsion and nonmetricity. After decomposing these fields into irreducible Lorentz components, the analysis identifies distinct signatures in the tidal tensor and outlines how precise measurements could constrain the post-Riemannian contributions, provided the probes respond to the full affine connection.

Core claim

The tidal acceleration separates into the usual Newtonian contribution and linear post-Riemannian corrections sourced by torsion and non-metricity, after decomposing torsion and nonmetricity into their irreducible Lorentz components.

What carries the argument

The projected deviation equation that generalizes the geodesic deviation equation to metric-affine geometry and tracks relative acceleration of neighboring autoparallels.

If this is right

  • The tidal tensor encodes identifiable linear signatures from specific irreducible components of torsion and nonmetricity.
  • Future direct tidal measurements can be converted into benchmark upper limits on the strength of post-Riemannian contributions.
  • The separation between Newtonian and post-Riemannian terms holds only in the weak-field nonrelativistic limit for autoparallel motion.

Where Pith is reading between the lines

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

  • Laboratory or solar-system setups with controlled spin or matter distributions could supply the required independent check on torsion or nonmetricity.
  • The same projection technique may generalize to other deviation equations, such as those for spinning particles or light bundles.
  • If the corrections prove measurable, they would provide a new observable channel that distinguishes metric-affine models from purely metric extensions.

Load-bearing premise

Probe dynamics respond to the affine connection rather than the metric alone, so that tidal measurements can be turned into bounds on post-Riemannian effects.

What would settle it

A high-precision measurement of relative acceleration between nearby test masses that matches the Newtonian tidal field exactly, with no detectable linear corrections, in a regime where nonzero torsion or nonmetricity is independently established.

Figures

Figures reproduced from arXiv: 2606.27433 by Armin van de Venn, David Vasak, J\"urgen Struckmeier, Marcelo Netz-Marzola.

Figure 1
Figure 1. Figure 1: FIG. 1. The family of timelike curves [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of the rotational contribu [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

This work investigates how torsion and nonmetricity modify tidal accelerations in metric-affine gravity. We derive a projected deviation equation that generalizes the standard geodesic deviation equation to metric-affine geometry, and apply it to the relative acceleration of neighboring autoparallels in the weak-field, nonrelativistic limit. In this regime, the tidal acceleration separates into the usual Newtonian contribution and linear post-Riemannian corrections sourced by torsion and non-metricity. By decomposing torsion and nonmetricity into their irreducible Lorentz components, we identify the corresponding signatures in the tidal tensor and discuss to what extent these contributions can be distinguished. We then show how future direct tidal measurements could be translated into benchmark bounds on post-Riemannian tidal contributions, assuming probe dynamics sensitive to the affine connection. Our results suggest that the tidal acceleration may provide a systematic route toward probing post-Riemannian spacetime features in the future.

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

1 major / 2 minor

Summary. The paper derives a projected deviation equation generalizing the geodesic deviation equation to metric-affine geometry with torsion and nonmetricity. It applies the equation to the relative acceleration of neighboring autoparallels in the weak-field nonrelativistic limit, showing that the tidal acceleration separates into the standard Newtonian term plus linear corrections sourced by the irreducible components of torsion and nonmetricity. The work identifies corresponding signatures in the tidal tensor, discusses distinguishability, and outlines how future direct tidal measurements could yield benchmark bounds on post-Riemannian contributions, under the assumption that probe dynamics follow the full affine connection.

Significance. If the derivation is sound and the autoparallel assumption is justified in the relevant physical context, the results supply a concrete, linear-order framework for extracting post-Riemannian signatures from tidal observables. This is a useful addition to the literature on metric-affine gravity, as it moves beyond abstract field equations to a falsifiable observational channel. The explicit decomposition into Lorentz-irreducible pieces and the separation of corrections are technically valuable and could guide future experimental proposals.

major comments (1)
  1. [Abstract and discussion of observational implications] Abstract (final sentence) and the paragraph introducing the bound-translation step: the claim that the derived corrections can be translated into benchmark bounds on post-Riemannian tidal contributions rests on the assumption that 'probe dynamics [are] sensitive to the affine connection.' In standard metric-affine and Einstein-Cartan treatments, spinless test masses follow Levi-Civita geodesics of the metric; the autoparallel equation is typically reserved for particles with spin or for specific matter couplings. Without an explicit derivation or reference establishing that the probes considered here obey the autoparallel equation of the full connection (rather than the metric geodesic equation), the post-Riemannian terms drop out of the observable tidal acceleration and the proposed bounds cannot be obtained. This assumption is load-bearing for the central claim.
minor comments (2)
  1. [Abstract] The abstract refers to a 'projected deviation equation' without indicating the precise projection (e.g., onto the observer's rest space or orthogonal to the four-velocity). A brief clarification of the projection operator and its properties in the main text would improve readability.
  2. [Section on irreducible decomposition] Notation for the irreducible torsion and nonmetricity components is introduced but not cross-referenced to standard conventions (e.g., those of Hehl et al. or Obukhov). Adding one or two standard references would help readers map the signatures to existing literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this key point concerning the physical justification of the autoparallel assumption. We respond to the major comment below.

read point-by-point responses

  1. Referee: Abstract (final sentence) and the paragraph introducing the bound-translation step: the claim that the derived corrections can be translated into benchmark bounds on post-Riemannian tidal contributions rests on the assumption that 'probe dynamics [are] sensitive to the affine connection.' In standard metric-affine and Einstein-Cartan treatments, spinless test masses follow Levi-Civita geodesics of the metric; the autoparallel equation is typically reserved for particles with spin or for specific matter couplings. Without an explicit derivation or reference establishing that the probes considered here obey the autoparallel equation of the full connection (rather than the metric geodesic equation), the post-Riemannian terms drop out of the observable tidal acceleration and the proposed bounds cannot be obtained. This assumption is load-bearing for the central claim.

    Authors: We agree that the assumption requires explicit support to sustain the observational claims. The manuscript states the assumption clearly but does not supply a derivation or references. In the revised manuscript we will add a short discussion together with references to existing literature in metric-affine gravity where autoparallel motion is adopted for probes under specific matter couplings or non-minimal interactions. This will clarify the physical contexts in which the derived bounds apply and will not alter the mathematical derivation itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from affine geometry

full rationale

The derivation begins from the definition of the affine connection and autoparallels in metric-affine geometry, projects the deviation equation, and takes the weak-field nonrelativistic limit to separate Newtonian and linear torsion/nonmetricity terms. No quoted equation reduces a claimed prediction or correction to a fitted input, self-definition, or self-citation chain. The assumption that probes follow autoparallels is stated explicitly as a modeling choice rather than derived from prior results in the paper itself. The decomposition into irreducible components and discussion of distinguishability follow directly from the linearized expressions without tautological renaming or imported uniqueness theorems. This is the standard case of an independent geometric derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions of metric-affine geometry and the validity of the linear weak-field approximation; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Test particles follow autoparallels defined by the full affine connection.
    Invoked when applying the deviation equation to neighboring autoparallels rather than geodesics.
  • domain assumption The weak-field nonrelativistic limit permits clean separation of Newtonian and linear post-Riemannian terms.
    Used to isolate the tidal acceleration contributions in the regime of interest.

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Reference graph

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