Weakly Isolated Horizons: $3+1$ decomposition and canonical formulations in self-dual variables

TL;DR

This paper develops a canonical formulation of general relativity with weakly isolated horizons using self-dual variables, revealing non-uniqueness in horizon degrees of freedom and energy definitions, thus generalizing prior isolated horizon models.

Contribution

It introduces a new canonical decomposition for spacetimes with WIHs in self-dual variables, highlighting the freedom in boundary variable choices and their impact on horizon properties.

Findings

Horizon degrees of freedom are not uniquely defined.

Horizon energy depends on boundary variable choices.

The formalism generalizes previous isolated horizon approaches.

Abstract

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. In particular, the definition of {\it weakly} isolated horizons (WIHs) as quasilocal generalizations of event horizons is purely geometrical, and is independent of the variables used in describing the gravitational field. Here we consider a canonical decomposition of general relativity in terms of connection and vierbein variables starting from a first order action. Within this approach, the information about the existence of a (weakly) isolated horizon is obtained through a set of boundary conditions on an internal boundary of the spacetime region under consideration. We employ, for the self-dual action, a generalization of the Dirac algorithm for regions with boundary. While the formalism for treating gauge theories with boundaries is unambiguous, the choice of dynamical variables on the boundary is not. We explore this freedom and consider different canonical formulations for non-rotating black holes as defined by WIHs. We show that both the notion of horizon degrees of freedom and energy associated to the horizon is not unique, even when the descriptions might be self-consistent. This represents a generalization of previous work on isolated horizons both in the exploration of this freedom and in the type of horizons considered. We comment on previous results found in the literature.