Weakly isolated horizons: 3 + 1 decomposition and canonical formulations in self-dual variables

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 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.

On Covariant and Canonical Hamiltonian Formalisms for Gauge Theories

“…The most salient feature appears within the theory in the vacuum. Here, one can see that the boundary contribution to the pre-symplectic structure in the covariant theory is entirely different from the (standard) contribution to the canonical description [8,9,21]. Moreover, when coupling to the Maxwell field, there is a contribution on the IH to the pre-symplectic structure in the covariant approach [11], while there is none in the canonical approach [8,9].…”

“…The most notable example is the treatment of isolated horizons (IH); that is, generalizations of Killing horizons used to model black holes in equilibrium. These systems have been treated both with canonical formalism [8,9,21] and with the covariant approach [11,22]. By comparing these results, we can immediately see that there is an important difference between them.…”

Canonical analysis of field theories in the presence of boundaries: Maxwell + Pontryagin

“…In a separate paper, that can be considered as a follow-up to the present manuscript, we have analysed a more general case of isolated horizons than in [17], and have performed a careful canonical analysis of the system. That manuscript shall be published soon [29].…”