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

TL;DR

The paper shows that, classically, holonomic and non-holonomic gravity theories are on-shell equivalent across general metric-affine dynamics as long as the vielbein $e^A_\mu$ is invertible, an interpretation grounded in the equivalence principle and formalized through a bundle-theoretic framework. It develops a general argument for on-shell equivalence using field transformations and clarifies the roles of natural vs soldered bundles, revealing how a soldering map $e$ unifies spacetime and internal gauge structures. The work identifies degeneracy regions where $\det e^A_\mu=0$ as precise failures of the equivalence, implying a fundamental separation between spacetime geometry and internal gauge theory in those regions. It also notes that quantum effects and topology changes can further affect equivalence, and sketches how the framework generalizes to arbitrary metric-affine dynamics and spacetime dimensions. Overall, the results provide a geometrical criterion for when holonomic and non-holonomic formulations describe the same classical physics and when they must be treated as distinct.

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.