We examine counterparts of the Reissner-Nordström-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface ⌺ of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S 2 . Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When ⌺ is closed, we derive the first law of thermodynamics using a Brown-York-type quasilocal energy at a finite boundary, and we identify the entropy as one-quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution. ͓S0556-2821͑97͒00818-7͔

The search for a theory of quantum gravity has for a long time been almost fruitless. A few years ago, however, Ashtekar found a reformulation of Hamiltonian gravity, which thereafter has given rise to a new promising quantization project; the canonical Dirac quantization of Einstein gravity in terms of Ahtekar's new variables. This project has already given interesting results, although many important ingredients are still missing before we can say that the quantization has been successful.Related to the classical Ashtekar Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1)-and (3+1)-dimensions, and also generalizations of Einstein's theory of gravity. In the first type of generalization, one introduces infinitely many new parameters, similar to the conventional Einstein cosmological constant, into the theory. These generalizations are called "neighbours of Einstein's theory" or "cosmological constants generalizations", and the theory has the same number of degrees of freedom, per point in spacetime, as the conventional Einstein theory. The second type is a gauge group generalization of Ashtekar's Hamiltonian, and this theory has the correct number of degrees of freedom to function as a theory for a unification of gravity and Yang-Mills theory. In both types of generalizations, there are still important problems that are unresolved: e.g the reality conditions, the metric-signature condition, the interpretation, etc.In this review, I will try to clarify the relations between the new and old actions for gravity, and also give a short introduction to the new generalizations. The new results/treatments in this review are: 1. A more detailed constraint analysis of the Hamiltonian formulation of the Hilbert-Palatini Lagrangian in (3+1)-dimensions. 2. The canonical transformation relating the Ashtekar-and the ADM-Hamiltonian in (2+1)-dimensions is given. 3. There is a discussion regarding the possibility of finding a higher dimensional Ashtekar formulation.There are also two clarifying figures (in the beginning of chapter 2 and 3, respectively) showing the relations between different action-formulations for Einstein gravity in (2+1)and (3+1)-dimensions.

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.

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.