Conserved currents in the Palatini formulation of general relativity

TL;DR

The paper develops a systematic construction of local, on-shell co-dimension $2$ forms for conserved charges in the Palatini formulation of general relativity and proves their on-shell equivalence to the metric formulation for exact reducibility parameters. It identifies two Palatini-specific subtleties—off-shell non-metricity and the second-derivative action of diffeomorphisms on the connection—and resolves them using a contracting-homotopy approach to derive explicit expressions for the co-dimension $2$ forms and their breaking terms. The authors provide the explicit on-shell $k^{[\mu\nu]}_\boldsymbol{ξ}$ (Final expression Pal) and show how, upon eliminating the auxiliary connection, the Palatini result reduces to the metric-formulation case for exact Killing vectors, with caveats for asymptotic or non-Killing parameters. These results clarify the conserved charges in first-order formulations of GR and establish precise links between Palatini and metric approaches, including the role of non-metricity.

Abstract

We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Palatini formulation of general relativity and explicitly show their on-shell equivalence to those of the metric formulation. When compared to other first order formulations, two subtleties have to be addressed during the construction: off-shell non-metricity and the fact that the transformation of the connection under infinitesimal diffeomorphisms involves second order derivatives of the associated vector fields.