Weakly Isolated horizons: first order actions and gauge symmetries

The notion of Isolated Horizons has played an important role in gravitational physics, being useful from the characterization of the endpoint of black hole mergers to (quantum) black hole entropy. With an eye towards a canonical formulation we consider general relativity in terms of connection and vierbein variables and their corresponding first order actions. We focus on two main issues: (i) The role of the internal gauge freedom that exists, in the consistent formulations of the action principle, and (ii) the role that a 3 + 1 canonical decomposition has in the allowed internal gauge freedom. More concretely, we clarify in detail how the requirement of having well posed variational principles compatible with general weakly isolated horizons (WIHs) as internal boundaries does lead to a partial gauge fixing in the first order descriptions used previously in the literature. We consider the standard Hilbert-Palatini action together with the Holst extension (needed for a consistent 3 + 1 decomposition), with and without boundary terms at the horizon. We show in detail that, for the complete configuration space-with no gauge fixing-, while the Palatini action is differentiable without additional surface terms at the inner WIH boundary, the more general Holst action is not. The introduction of a surface term at the horizon-that renders the action for asymptotically flat configurations differentiable-does make the Holst action differentiable, but only if one restricts the configuration space and partially reduces the internal Lorentz gauge. For the second issue at hand, we show that upon performing a 3 + 1 decomposition and imposing the time gauge, there is a further gauge reduction of the Hamiltonian theory in terms of Ashtekar-Barbero variables to a U(1)-gauge theory on the horizon. We also extend our analysis to the more restricted boundary conditions of (strongly) isolated horizons as inner boundary. We show that even when the Holst action is indeed differentiable without the need of additional surface terms or any gauge fixing for Type I spherically symmetric (strongly) isolated horizons-and a preferred foliation-, this result does not go through for more general isolated or weakly isolated horizons. Our results represent the first comprehensive study of these issues and clarify some contradictory statements found in the literature.