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arxiv: 2605.30221 · v1 · pith:RXY37EWBnew · submitted 2026-05-28 · 🌀 gr-qc · hep-th

Primary Constraints of Newer General Relativity

Carmen Ferrara , Alexey Golovnev , Mar\'ia Jos\'e Guzm\'an This is my paper

Pith reviewed 2026-06-29 06:26 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords primary constraintsNewer General Relativityteleparallel gravitynonmetricityscalar sectorcanonical momentasymmetric teleparallel gravityconstraint degeneracy
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0 comments X

The pith

Newer General Relativity produces either one or two scalar primary constraints depending on the choice of Lagrangian coefficients c_i.

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

The paper determines the full set of primary constraints in Newer General Relativity by checking when the map from metric velocities to canonical momenta loses invertibility. A complete nonlinear split into scalar, vector and tensor sectors shows that the tensor modes always supply five constraints and the vector modes three, while the scalar modes supply either one or two according to conditions on the c_i. This counting matters because the number of primary constraints fixes the propagating degrees of freedom before any secondary analysis is performed. A reader who accepts the result therefore obtains a parameter-dependent classification of the theory's constraint algebra that was missing from earlier literature.

Core claim

In Newer General Relativity the gravitational Lagrangian is a quadratic scalar built from the nonmetricity tensor with arbitrary coefficients c_i. After computing the canonical momenta conjugate to the metric and decomposing the velocity-momenta map into fully nonlinear scalar, vector and tensor sectors, the map is found to be non-invertible in the tensor sector for five independent reasons, in the vector sector for three independent reasons, and in the scalar sector for either one or two independent reasons according to the values of the c_i. The same non-invertibility conditions characterize the primary constraints of the covariant formulation of symmetric teleparallel gravity.

What carries the argument

Non-invertibility of the velocity-momenta map after a fully nonlinear scalar-vector-tensor decomposition of the Lagrangian.

If this is right

  • The tensor sector always contributes five primary constraints and the vector sector three, independent of the c_i.
  • The scalar sector contributes either one or two primary constraints according to algebraic conditions on the c_i.
  • The identical primary constraints appear in the covariant formulation of symmetric teleparallel gravity.
  • The total number of primary constraints is therefore either nine or ten according to the parameter region.

Where Pith is reading between the lines

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

  • Different numbers of scalar constraints will produce different final counts of physical degrees of freedom once secondary constraints are derived.
  • The parameter regions that yield two scalar constraints may admit additional scalar modes that are absent when only one scalar constraint appears.
  • The degeneracy condition on the c_i supplies an explicit algebraic criterion that can be used to partition the theory's parameter space before any Hamiltonian analysis begins.

Load-bearing premise

Non-invertibility of the velocity-momenta map after the nonlinear decomposition is enough to identify every primary constraint without further secondary-constraint or gauge-fixing analysis.

What would settle it

Direct computation of the rank of the Hessian matrix formed by second derivatives of the Lagrangian with respect to the metric velocities, for concrete numerical choices of the c_i that the paper predicts should produce one versus two independent scalar null eigenvectors.

read the original abstract

We study the primary constraint structure of Newer General Relativity, a gravity theory based on a torsionless teleparallel geometry. The gravitational action is built from a scalar formed by quadratic combinations of the nonmetricity tensor, with arbitrary coefficients $c_i$ in the Lagrangian. We decompose the Lagrangian and compute the canonical momenta conjugate to the metric. We characterize the primary constraints arising from these momenta by identifying when the map between velocities and momenta becomes non-invertible, and organize the outcome through a fully nonlinear decomposition into scalar, vector and tensor sectors. Comparing with previous results in the literature, we recover five and three primary constraints associated with the tensor and vector sectors, respectively. We also identify a previously unreported degeneracy in the scalar sector, which yields either one or two scalar primary constraints depending on the conditions imposed on the parameters $c_i$. Finally, we obtain the primary constraints associated with the covariant formulation of symmetric teleparallel gravity.

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

0 major / 2 minor

Summary. The paper analyzes the primary constraint structure of Newer General Relativity, a torsionless teleparallel theory whose action is quadratic in the nonmetricity tensor with free coefficients c_i. After a fully nonlinear SVT decomposition of the Lagrangian and computation of the canonical momenta, the authors identify non-invertibility of the velocity-momenta map to obtain five tensor primary constraints and three vector primary constraints; they further report a previously unreported degeneracy in the scalar sector that produces either one or two scalar primary constraints depending on the values of the c_i. The same procedure is applied to the covariant formulation of symmetric teleparallel gravity.

Significance. If the results hold, the work supplies a concrete count of primary constraints in a broad class of quadratic nonmetricity theories, which is a necessary step toward determining the physical degrees of freedom. The identification of a parameter-dependent scalar-sector degeneracy is a novel contribution that distinguishes the analysis from earlier literature on teleparallel gravity. The use of a fully nonlinear decomposition, rather than a linearized one, is a methodological strength that supports the robustness of the reported counts.

minor comments (2)
  1. [Abstract] The abstract states that the scalar degeneracy depends on conditions imposed on the c_i but does not list the explicit conditions; adding a brief parenthetical statement of those conditions would improve immediate readability.
  2. [Introduction] The comparison with previous literature is described qualitatively; a short table or paragraph explicitly mapping the recovered tensor and vector counts to the corresponding references would make the validation step more transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending acceptance. Their summary correctly reflects the scope and results of our analysis.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper follows the standard procedure for primary constraints in the Hamiltonian formalism: it decomposes the Lagrangian into SVT sectors, computes the canonical momenta from the quadratic nonmetricity action with parameters c_i, and identifies constraints from non-invertibility of the velocity-momenta map. It recovers the expected five tensor and three vector constraints from prior literature and reports a parameter-dependent degeneracy in the scalar sector. No derivation step reduces by construction to a self-defined quantity, fitted input, or load-bearing self-citation; the central claims rest on direct computation rather than renaming or ansatz smuggling. The analysis is self-contained against external benchmarks for counting primary constraints.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The c_i coefficients are free parameters in the quadratic nonmetricity action; the torsionless teleparallel geometry is a domain assumption. No invented entities appear. Full text unavailable for exhaustive audit.

free parameters (1)
  • c_i
    Arbitrary coefficients multiplying the quadratic nonmetricity terms in the Lagrangian, stated explicitly in the abstract.
axioms (1)
  • domain assumption The gravitational action is built from a scalar formed by quadratic combinations of the nonmetricity tensor
    This is the defining setup of Newer General Relativity as described in the abstract.

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discussion (0)

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