Vector modes in Type 3 New GR

Abstract: Some time ago, we published the full count of degrees of freedom in the linearised weak gravity limit of arbitrary New GR models. We did it by considering the linear equations of motion and presented a thorough analysis with no ambiguity left. A bit later, we generalised it to linear cosmological perturbations and discussed the strong coupling issues that appear already at this level. Recently, there were claims that some dynamical modes had been missed in our work. However, the authors of the new claims did not look at the equations of motion and analysed the quadratic Lagrangian densities instead. In this paper, I take one of the most elementary cases, namely the vector modes in New GR of Type 3, and show what was their mistake that had led them to claiming that those were dynamical. The main message: Do not substitute constraint equations into a Lagrangian.

• The paper provides a rigorous analysis of vector modes in Type 3 New GR, demonstrating that these modes are non-propagating in weak-field scenarios.
• It critiques earlier analyses that improperly substituted constraint equations into the Lagrangian, leading to miscounting of degrees of freedom.
• The study underscores the need for precise gauge fixing in modified gravity models and reveals strong coupling effects under cosmological perturbations.

Vector Modes in Type 3 New General Relativity: A Rigorous Analysis

Overview

The paper "Vector modes in Type 3 New GR" (2605.22329) provides a systematic investigation of vector perturbations in Type 3 models within the New General Relativity (NGR) framework, focusing specifically on the linearized weak gravity limit and resolving recent disputes regarding the dynamical content of this sector. The author meticulously addresses errors in prior analyses where constraint equations were substituted directly into the quadratic Lagrangian, critiquing the validity of such procedures in degree-of-freedom counting.

Theoretical Framework and Perturbation Structure

NGR is formulated in the teleparallel paradigm, employing a quadratic action in terms of the torsion tensor $T^{\alpha}_{\hphantom{\alpha}\mu\nu}$ in a pure tetrad formalism. The Type 3 subclass is parameterized by $a = b \neq 0$ and $c \neq a$, ensuring deviation from TEGR and certain other NGR types. The vector perturbation sector is constructed via divergenceless vectors $u_i$, $v_i$, $c_i$, and $w_i$—each encoding different aspects of tetrad deviations from Minkowski spacetime. The diffeomorphism gauge is fixed by setting $c_i = 0$, and variables are reparametrized as $\mathcal{M}_i$, $\mathcal{L}_i$, and $a = b \neq 0$0.

Field Equations and Dynamical Structure

The core results hinge on the explicit derivation and analysis of the linearized field equations governing vector modes, demonstrating that for Type 3 NGR:

• $a = b \neq 0$1, i.e., one pure gauge vector.
• $a = b \neq 0$2, i.e., two constrained vectors.

Importantly, these equations do not admit any propagating dynamical degrees of freedom in the vector sector for the Type 3 model. The gauge structure is thoroughly clarified, revealing an extra gauge freedom in Minkowski background that disappears under cosmological perturbations, thus suggesting a strong coupling phenomenon indicative of qualitative background dependence in gauge behavior.

Critical Assessment of Prior Claims

The author identifies critical flaws in previous papers (Tomonari et al., 23 Sep 2025, Tomonari et al., 16 May 2026) that argued for missed dynamical modes. These works performed erroneous manipulations by inserting constraint equations directly into the action, inadvertently altering the variational structure and physical content. Such substitutions are shown to transform the model—introducing artificial dynamical modes that do not correspond to the original theory.

The pedagogical rigor extends to a toy model example, elucidating that substituting non-algebraic (nonholonomic) constraints into a Lagrangian can fundamentally change the system, breaking correspondence with original equations of motion and resulting in incorrect mode counting.

Implications and Future Directions

This analysis reinforces the necessity for careful distinction between constraint equations and dynamical equations in gauge theories, particularly when determining propagating degrees of freedom. The implications are profound for theory development in modified gravity: models must be scrutinized using proper gauge-fixed or gauge-invariant formulations at the level of field equations rather than solely at the Lagrangian density. Practically, the results imply that Type 3 NGR is not phenomenologically distinguishable from standard GR as regards vector perturbations in weak-field scenarios. Theoretical avenues for further investigation include:

• Extending similar analyses to other NGR types and to more general backgrounds, e.g., cosmological spacetimes with expansion.
• Investigating quantum aspects and path integral formulations, which are sensitive to the true dynamical content.
• Examining the robustness of gauge symmetries and emergence/breaking of strong coupling in cosmological dynamics.

Conclusion

The paper establishes that vector modes in Type 3 New General Relativity do not represent dynamical degrees of freedom, despite contrary claims in recent literature. The erroneous assessment in prior analyses stemmed from improper substitutive manipulation of constraint equations at the Lagrangian level. This outcome clarifies the theoretical structure of NGR, underscores the importance of rigorous degree-of-freedom counting in gauge theories, and motivates careful methodology in ongoing explorations of alternative gravitational models.