Teleparallel gravity as a gauge theory: coupling to matter with Cartan connection

TL;DR

This paper develops a consistent framework for coupling matter to teleparallel gravity using Cartan connections, linking it to gauge theories and ensuring compatibility with observations and the Equivalence Principle.

Contribution

It provides a complete description of matter coupling in TEGR via Cartan connections, connecting it to gauge theory structures and clarifying the role of translation gauge fields.

Findings

Coupling to matter in TEGR can be achieved with Cartan connections.

A theorem links Cartan and Ehresmann connections, ensuring proper transport.

The translation gauge field can be extracted if TEGR is viewed as a translation gauge theory.

Abstract

We present a consistent and complete description of the coupling to matter in the Teleparallel Equivalent to General Relativity (TEGR) theory built from a Cartan connection, as we proposed in previous works. A first theorem allows us to obtain parallel transport from the Cartan connection into a proper Ehresmann connection, while a second ensures to link the TEGR-Cartan connection to the Ehresmann one-form that contains the Levi-Civita connection. This yields a coupling to matter in agreement with observations and the Equivalence Principle. As the fundamental fields proceed from the Cartan connection, if one insists on interpreting TEGR as a gauge theory of translations, such translation gauge field can be extracted from the consistent theory presented. However, this would entail a fundamental change in the structures known for gauge theory and a split between gauge field and connection is imperative. The willingness to take such a step is left to the reader.