Barbero recently suggested a modification of Ashtekar's choice of canonical variables for general relativity. Although leading to a more complicated Hamiltonian constraint this modified version, in which the configuration variable still is a connection, has the advantage of being real. In this article we derive Barbero's Hamiltonian formulation from an action, which can be considered as a generalization of the ordinary Hilbert-Palatini action.Comment: 6 page

A large variety of spacetimes - including the BTZ black holes - can be obtained by identifying points in (2 + 1)-dimensional anti-de Sitter space by means of a discrete group of isometries. We consider all such spacetimes that can be obtained under a restriction to time-symmetric initial data and one asymptotic region only. The resulting spacetimes are non-eternal black holes with collapsing wormhole topologies. Our approach is geometrical, and we discuss in detail the allowed topologies, the shape of the event horizons, topological censorship and trapped curves.

It is known from the work of Bañados et al. that a space-time with event horizons (much like the Schwarzschild black hole) can be obtained from 2+1 dimensional anti-de Sitter space through a suitable identification of points. We point out that this can be done in 3+1 dimensions as well. In this way we obtain black holes with event horizons that are tori or Riemann surfaces of genus higher than one. They can have either one or two asymptotic regions. Locally, the space-time is isometric to anti-de Sitter space.

We construct a 2+1 dimensional spacetime of constant curvature whose spatial topology is that of a torus with one asymptotic region attached. It is also a black hole whose event horizon spins with respect to infinity. An observer entering the hole necessarily ends up at a "singularity"; there are no inner horizons. In the construction we take the quotient of 2+1 dimensional anti-de Sitter space by a discrete group Γ. A key part of the analysis proceeds by studying the action of Γ on the boundary of the spacetime.

The observation that the 2+1 dimensional BTZ black hole can be obtained as a quotient space of anti-de Sitter space leads one to ask what causal behaviour other such quotient spaces can display. In this paper we answer this question in 2+1 and 3+1 dimensions when the identification group has one generator. Among other things we find that there does not exist any 3+1 generalization of the rotating BTZ hole. However, the non-rotating generalization exists and exhibits some unexpected properties. For example, it turns out to be non-static and to possess a non-trivial apparent horizon.