Cosmological Perturbation in New General Relativity: Propagating mode from the violation of local Lorentz invariance

We investigate the propagating modes of New General Relativity (NGR) in second-order linear perturbations in the Lagrangian density (first-order in field equations). The Dirac-Bergmann analysis has revealed a violation of local Lorentz invariance in NGR. We review the recent status of NGR, considering the results of its Dirac-Bergmann analysis. We then reconsider the vierbein perturbation framework and identify the origin of each perturbation field in the vierbein field components. This identification is mandatory for adequately fixing gauges while guaranteeing consistency with the invariance ensured by the Dirac-Bergmann analysis. We find that the spatially flat gauge is adequate for analyzing a theory with the violation of local Lorentz invariance. Based on the established vierbein perturbative framework, introducing a real scalar field as matter, we perform a second-order perturbative analysis of NGR with respect to tensor, scalar, pseudo-scalar, and vector and pseudo-vector modes. We reveal the possible propagating modes of each type of NGR. In particular, we find that Type 3 has stable five propagating modes, \textit{i.e.}, tensor, scalar, and vector modes, compared to five non-linear degrees of freedom, which results in its Dirac-Bergmann analysis; the linear perturbation theory of Type 3 is preferable for applications to cosmology. Finally, we discuss our results in comparison to previous related work and conclude this study.

💡 Research Summary

This paper provides a comprehensive analysis of the propagating degrees of freedom (DOFs) in New General Relativity (NGR), an extension of the teleparallel equivalent of General Relativity (TEGR) characterized by three free parameters (c₁, c₂, c₃). The authors begin by reviewing the Dirac‑Bergmann (DB) constraint analysis performed in earlier works, which shows that, except for the TEGR limit (Type 6), NGR generically breaks local Lorentz invariance (LLI). The violation of LLI converts the six antisymmetric components of the vierbein into physical DOFs, raising the total number of non‑linear DOFs up to eight, depending on the parameter choice.

Recognizing that conventional metric‑only perturbation theory cannot capture these antisymmetric modes, the authors reconstruct the perturbation framework directly in terms of the vierbein. They decompose the vierbein into a symmetric part (¯e) that reproduces the metric and an antisymmetric part (˜e). The antisymmetric sector is further split into a pseudo‑scalar and a transverse pseudo‑vector, following the most recent treatment (Ref.