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arxiv: 2606.17580 · v1 · pith:5DAYIUHPnew · submitted 2026-06-16 · 🌀 gr-qc · hep-th

The Pre-geometric Origin of Geometric Trinity of Gravity

Salvatore Capozziello , Giuseppe Meluccio This is my paper

Pith reviewed 2026-06-27 00:29 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords pre-geometric gravitygeometric trinityspontaneous symmetry breakinggeneral relativityteleparallel gravitysymmetric teleparallel gravityaffine connectionmetric-affine theories
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The pith

Pre-geometric spontaneous symmetry breaking produces equivalent dynamics for curvature, torsion, and non-metricity formulations of gravity.

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

The paper establishes that a gauge theory formulated like Yang-Mills with a Higgs-like field can undergo spontaneous symmetry breaking to generate an effective metric and the classical gravitational dynamics. This breaking is shown to be consistent with the Geometric Trinity, meaning the same pre-geometric setup leads to General Relativity via curvature, the Teleparallel Equivalent via torsion, and the Symmetric Teleparallel Equivalent via non-metricity. The consistency is demonstrated by constructing specific pre-geometric actions and gauge-fixing conditions in the unbroken phase that reduce to each formulation. This matters because it provides a common origin for these apparently different geometric descriptions of gravity, suggesting they are different manifestations of the same underlying pre-geometric structure.

Core claim

Starting from a gauge formulation à la Yang-Mills with a Higgs-like field, a mechanism of spontaneous symmetry breaking gives rise to an effective metric as well as to the classical dynamics of the gravitational field, and this emergence is consistent with all different formulations of the Geometric Trinity of Gravity in terms of both actions and gauge choices for the affine connection, achieved by deriving suitable expressions in the unbroken phase for pre-geometric actions and pre-geometric gauge-fixing conditions.

What carries the argument

The spontaneous symmetry breaking of a Yang-Mills-like gauge theory with a Higgs-like field that generates the metric and selects equivalent affine connection gauges for curvature, torsion, or non-metricity.

If this is right

  • The Geometric Trinity formulations share a common pre-geometric origin rather than being merely equivalent at the level of equations of motion.
  • Different choices of gauge-fixing conditions in the unbroken phase correspond to the different geometric features (curvature, torsion, non-metricity).
  • All metric-affine theories of gravity can in principle emerge from this pre-geometric framework via appropriate symmetry breaking.
  • The dynamics of GR, TEGR, and STEGR are recovered exactly from the broken phase of the same pre-geometric actions.

Where Pith is reading between the lines

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

  • If the pre-geometric phase is realized in nature, it might leave imprints in early universe cosmology before the symmetry breaking completes.
  • This approach could be extended to include matter fields to see if standard model interactions also arise from the same breaking mechanism.
  • Connections to other unification attempts might be explored by identifying analogous pre-geometric structures.

Load-bearing premise

Suitable pre-geometric actions and gauge-fixing conditions exist in the unbroken phase that, upon spontaneous symmetry breaking, produce exactly the three equivalent dynamics of the Geometric Trinity.

What would settle it

A calculation showing that no pre-geometric action in the unbroken phase breaks to all three formulations simultaneously, or an explicit derivation where the broken phase dynamics differ for at least one of the Trinity members.

Figures

Figures reproduced from arXiv: 2606.17580 by Giuseppe Meluccio, Salvatore Capozziello.

Figure 1
Figure 1. Figure 1: The octet of gravity, i.e. a diagrammatic classification of metric-affine theories of gravity based on the number [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

The so-called Geometric Trinity of Gravity is based on three distinct geometric features of spacetime, i.e.\ curvature, torsion and non-metricity, which give rise to equivalent dynamics for General Relativity (GR), Teleparallel Equivalent of General Relativity (TEGR) and Symmetric Teleparallel Equivalent of General Relativity (STEGR). Pre-geometric gravity, on the other hand, offers a unifying framework from which all metric-affine theories can emerge. Starting from a gauge formulation \textit{\`a la} Yang--Mills with a Higgs-like field, a mechanism of spontaneous symmetry breaking can give rise to an effective metric as well as to the classical dynamics of the gravitational field. In particular, the emergence of gravity in the spontaneously broken phase is shown to be consistent with all the different formulations of the Geometric Trinity of Gravity, in terms both of actions and of gauge choices for the affine connection. This general result is achieved by deriving and analysing suitable expressions in the unbroken phase for pre-geometric actions and for pre-geometric gauge-fixing conditions respectively.

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 claims that a Yang-Mills-like gauge formulation supplemented by a Higgs-like field, upon spontaneous symmetry breaking, yields an effective metric and gravitational dynamics consistent with the full Geometric Trinity of Gravity. By constructing suitable pre-geometric actions and gauge-fixing conditions in the unbroken phase, the broken-phase limits are asserted to recover precisely the Einstein-Hilbert action (Levi-Civita connection), the teleparallel action (Weitzenböck connection), and the symmetric teleparallel action (symmetric teleparallel connection).

Significance. If the explicit reductions are verified without residual cross-terms or tuning, the result would supply a common pre-geometric origin for the three equivalent geometric formulations of GR, clarifying how curvature, torsion, and non-metricity can emerge from a single gauge-theoretic setup. The work would strengthen the case for pre-geometric approaches as a unifying framework for metric-affine theories.

major comments (2)
  1. [Abstract] Abstract and the final paragraph: the central claim that the derived unbroken-phase actions and gauge-fixing conditions reduce exactly to the three Trinity dynamics under SSB is asserted but not accompanied by explicit limit calculations, error estimates, or checks for vanishing cross terms (e.g., curvature-torsion or non-metricity residues). Without these, it is impossible to confirm that the construction is not tuned by design to the target theories.
  2. [Unbroken-phase construction (action and gauge-fixing sections)] The weakest assumption identified in the construction—that suitable pre-geometric actions and gauge conditions exist whose SSB produces exactly the Trinity—is load-bearing; the manuscript must demonstrate that the unbroken-phase expressions are selected by an independent principle rather than by requiring the broken-phase match a priori.
minor comments (2)
  1. [Notation and field content] Clarify the precise transformation properties of the Higgs-like field under the gauge group and its vacuum expectation value in the broken phase.
  2. [Results section] Add a table or explicit comparison showing the unbroken-phase action terms that map to each of the three Trinity actions after SSB.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and for highlighting the potential unifying value of the pre-geometric approach. We address each major comment below and will incorporate clarifications and explicit verifications to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the final paragraph: the central claim that the derived unbroken-phase actions and gauge-fixing conditions reduce exactly to the three Trinity dynamics under SSB is asserted but not accompanied by explicit limit calculations, error estimates, or checks for vanishing cross terms (e.g., curvature-torsion or non-metricity residues). Without these, it is impossible to confirm that the construction is not tuned by design to the target theories.

    Authors: We agree that the manuscript would benefit from explicit verification of the SSB limits. In the revised version we will add an appendix containing the detailed reduction calculations for each of the three cases (Levi-Civita, Weitzenböck, and symmetric teleparallel), including explicit checks that cross terms vanish and that no additional tuning is required beyond the gauge-invariant construction already presented. revision: yes

  2. Referee: [Unbroken-phase construction (action and gauge-fixing sections)] The weakest assumption identified in the construction—that suitable pre-geometric actions and gauge conditions exist whose SSB produces exactly the Trinity—is load-bearing; the manuscript must demonstrate that the unbroken-phase expressions are selected by an independent principle rather than by requiring the broken-phase match a priori.

    Authors: The unbroken-phase actions are the most general gauge-invariant Yang-Mills-type terms for the pre-geometric connection coupled to a Higgs-like field that admits a metric-inducing vacuum expectation value; the gauge-fixing conditions are likewise the standard covariant gauges compatible with the unbroken gauge group. These choices are dictated by the requirements of gauge invariance, renormalizability at the pre-geometric level, and the existence of a symmetry-breaking pattern that yields a metric, rather than by reverse-engineering the target Trinity actions. We will add a new subsection that explicitly separates the independent pre-geometric construction principles from the subsequent broken-phase analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a consistency construction within pre-geometric framework.

full rationale

The paper starts from a Yang-Mills-like gauge formulation plus Higgs-like field and derives unbroken-phase actions and gauge-fixing conditions whose spontaneous symmetry breaking is shown to reproduce the three equivalent dynamics of the Geometric Trinity (curvature, torsion, non-metricity formulations). The abstract explicitly states that 'suitable expressions' are derived and analysed to achieve this consistency result. No quoted equations or steps reduce the target Trinity actions or connection gauges to the inputs by definition or statistical fit; the construction is presented as an existence proof of emergence rather than a tautological renaming or self-referential fit. No self-citation chains or uniqueness theorems imported from the authors are invoked as load-bearing. The result is therefore self-contained against external benchmarks of the Trinity theories.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim rests on the existence of a Yang-Mills-like gauge formulation plus a Higgs-like field whose spontaneous symmetry breaking produces a metric; these are introduced without independent derivation in the abstract. No explicit free parameters, additional axioms, or new entities beyond the Higgs-like field are quantified.

invented entities (1)
  • Higgs-like field in pre-geometric gauge theory no independent evidence
    purpose: Triggers spontaneous symmetry breaking that generates an effective metric and gravitational dynamics
    Introduced in the abstract as the mechanism that produces the metric from the unbroken phase; no independent evidence or falsifiable prediction is stated.

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Forward citations

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