Abelian symmetry and the Palatini variation

TL;DR

Allowing abelian symmetry in the Palatini variation transforms the resulting geometry from Riemannian to integrable Weyl, revealing a deeper connection between gauge symmetries and geometric structures in gravity theories.

Contribution

This work demonstrates that incorporating abelian symmetry into the Palatini variation naturally leads to integrable Weyl geometry instead of Riemannian geometry, extending the understanding of geometric structures in gravity.

Findings

Palatini variation with abelian symmetry yields integrable Weyl geometry.

Both metric/connection and solder form/spin connection approaches agree.

Relates to and clarifies existing literature on geometric formulations of gravity.

Abstract

Independent variation of the metric and connection in the Einstein-Hilbert action, called the Palatini variation, is generally taken to be equivalent to the usual formulation of general relativity in which only the metric is varied. However, when an abelian symmetry is allowed for the connection, the Palatini variation leads to an integrable Weyl geometry, not Riemannian. We derive this result using two possible metric/connection pairs: (1) the metric and general coordinate connection and (2) the solder form and local Lorentz spin connection of Poincar\`e gauge theory. Both lead to the same conclusion. Finally, we relate our work to other treatments in the literature.