On the on-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries

TL;DR

This paper demonstrates that a generalized Holst action with nonmetricity and torsion yields the same solution space as General Relativity, despite differences in their Lagrangians and potential variations in their covariant phase space structures.

Contribution

It proves the on-shell equivalence of the generalized Holst and Palatini actions, clarifies their covariant phase space and charge relations, and addresses open issues in dual charges and formulation equivalences.

Findings

Solution spaces of Holst and Palatini actions are identical.

Holst and Palatini Lagrangians are not cohomologically equal.

Covariant phase spaces and charges are equivalent to GR.

Abstract

We study a generalization of the Holst action where we admit nonmetricity and torsion in manifolds with timelike boundaries (both in the metric and tetrad formalism). We prove that its space of solutions is equal to the one of the Palatini action. Therefore, we conclude that the metric sector is in fact identical to GR, which is defined by the Einstein-Hilbert action. We further prove that, despite defining the same space of solutions, the Palatini and (the generalized) Holst Lagrangians are not cohomologically equal. Thus, the presymplectic structure and charges provided by the Covariant Phase Space method might differ. However, using the relative bicomplex framework, we show the covariant phase spaces of both theories are equivalent (and in fact equivalent to GR), as well as their charges, clarifying some open problems regarding dual charges and their equivalence in different formulations.