Covariant phase space for gravity with boundaries: Metric versus tetrad formulations

We use covariant phase space methods to study the metric and tetrad formulations of general relativity in a manifold with boundary compare results obtained both approaches. Proving their equivalence has been long-lasting problem that we solve here by using cohomological approach provided relative bicomplex framework. This setting provides clean ambiguity-free way describe solution spaces associated symplectic structures. also compute several relevant charges schemes show they are equivalent, as expected.