About the (in)equivalence between holonomic versus non-holonomic theories of gravity

TL;DR

This paper explores the classical equivalence between holonomic and non-holonomic formulations of gravity, emphasizing the role of the equivalence principle and the impact of non-invertible vielbein configurations.

Contribution

It provides a bundle-theoretical analysis showing the conditions under which holonomic and non-holonomic gravity theories are equivalent, highlighting the importance of the equivalence principle.

Findings

Classical equivalence is independent of dynamics and spacetime dimension.

The equivalence principle is crucial for the equivalence to hold.

Non-invertible vielbein configurations break the equivalence, affecting the nature of gravity theories.

Abstract

We investigate the scenarios in which a holonomic versus a non-holonomic frame description of gravity theories are equivalent. It turns out that classically, the equivalence holds in a way that is independent of the particular dynamics and/or spacetime dimension. This includes general metric-affine dynamics. A global bundle-theoretical investigation is carried out, uncovering the equivalence principle as the culprit. The equivalence holds as long as the equivalence principle holds. This is not something to be expected when non-invertible configurations of the vielbein field are taken into account. In such case, the gauge-theoretical description of gravity unsolders from spacetime, and one has to decide if gravity is spacetime geometry or an internal gauge theory.