On covariant and canonical Hamiltonian formalisms: weakly isolated horizons

The Hamiltonian description of classical gauge theories is a well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms, have received a lot of attention in the literature. The relation between them has also been extensively analyzed. However, a complete understanding of this relation is still not available, specially for gauge theories that are defined over regions with boundaries. Here we consider this issue, for spacetimes with isolated horizons as inner boundaries (representing black holes in equilibrium), and assess whether their corresponding descriptions can be seen as equivalent. First, we review and reanalyze both formalisms from anew and show that, if we compare them at face value, there are differences between them, for instance in the derivation of horizon energy, a physical observable of the theory, as well as in the number of boundary degrees of freedom. Here we construct and compare possible extensions of the corresponding phase spaces, in a way that has not been explored before. We analyze different possible interpretations in both approaches and show how one can achieve correspondence between results within covariant and canonical formalisms. Along the way, we shed light on the role of boundary terms in the symplectic structure in relation to boundary degrees of freedom.