Covariant phase space for gravity with boundaries: metric vs tetrad formulations

TL;DR

This paper compares metric and tetrad formulations of General Relativity with boundaries using covariant phase space methods, establishing their equivalence through a cohomological approach and computing consistent charges.

Contribution

It demonstrates the equivalence of metric and tetrad formulations in boundary contexts using a cohomological framework, clarifying solution spaces and charges.

Findings

Metric and tetrad formulations are equivalent with boundaries.

Cohomological approach resolves longstanding equivalence issues.

Computed charges are consistent across both formulations.

Abstract

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