Translational invariance of the Einstein-Cartan action in any dimension

TL;DR

This paper proves that the Einstein-Cartan action in any dimension exhibits translational invariance in tangent space, derived from its first order formulation, revealing a gauge symmetry accessible through Lagrangian and Hamiltonian methods.

Contribution

It demonstrates translational invariance of the Einstein-Cartan action in all dimensions higher than two using differential identities and explicit field transformations.

Findings

Translational invariance exists in Einstein-Cartan action in higher dimensions.

Lagrangian methods can identify gauge symmetries from differential identities.

The invariance complements known rotational invariance in these theories.

Abstract

We demonstrate that from the first order formulation of the Einstein-Cartan action it is possible to derive the basic differential identity that leads to translational invariance of the action in the tangent space. The transformations of fields is written explicitly for both the first and second order formulations and the group properties of transformations are studied. This, combined with the preliminary results from the Hamiltonian formulation (arXiv:0907.1553 [gr-qc]), allows us to conclude that without any modification, the Einstein-Cartan action in any dimension higher than two possesses not only rotational invariance but also a form of \textit{translational invariance in the tangent space}. We argue that \textit{not} only a complete Hamiltonian analysis can unambiguously give an answer to the question of what a gauge symmetry is, but also the pure Lagrangian methods allow us to find the same gauge symmetry from the \textit{basic} differential identities.