Claudio Teitelboim

Black hole in three-dimensional spacetime.

The standard Einstein-Maxwell equations in 2+1 spacetime dimensions, with a negative cosmological constant, admit a black hole solution. The 2+1 black hole\char22{}characterized by mass, angular momentum, and charge, defined by flux integrals at infinity\char22{}is quite similar to its 3+1 counterpart. Anti\char21{}de Sitter space appears as a negative energy state separated by a mass gap from the continuous black hole spectrum. Evaluation of the partition function yields that the entropy is equal to twice the perimeter length of the horizon.

Geometry of the (2+1) black hole

The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant, and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti\char21{}de Sitter space by a discrete subgroup of SO(2,2). The generic black hole is a smooth manifold in the metric sense. The surface r=0 is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at r=0 to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti\char21{}de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum. A thorough classification of the elements of the Lie algebra of SO(2,2) is given in an appendix.

Role of Surface Integrals in the Hamiltonian Formulation of General Relativity

Abstract It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincare transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions. A Hamiltonian formalism which is manifestly covariant under Poincare transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g ij , π ij . In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincare transformations. In that case all ten generators of the Poincare group are obtained by inserting the solution of the constraints into corresponding surface integrals.

Asymptotically anti-De Sitter Spaces

Asymptotically anti-de Sitter spaces are defined by boundary conditions on the gravitational field which obey the following criteria: (i) they are O(3, 2) invariant; (ii) they make the O(3, 2) surface integral charges finite; (iii) they include the Kerr-anti-de Sitter metric. An explicit expression of the O(3, 2) charges in terms of the canonical variables is given. These charges are shown to close in the Dirac brackets according to the anti-de Sitter algebra. The results are extended to the case ofN=1 supergravity. The coupling to gravity of a third-rank, completely antisymmetric, abelian gauge field is also considered. That coupling makes it possible to vary the cosmological constant and to compare the various anti-de Sitter spaces which are shown to have the same energy.

Neutralization of the cosmological constant by membrane creation

The quantum creation of closed membranes by totally antisymmetric tensor and gravitational fields is considered in arbitrary space-time dimension. The creation event is described by instanton tunneling. As membranes are produced, the energy density associated with the antisymmetric tensor fielld decreases, reducing the effective value of the cosmological constant. For a wide range of parameters and initial conditions, this process will naturally stop as soon as the cosmological constant is near zero, even if the energy remaining in the antisymmetric tensor field is large. Among the instantons obtained, some are interpreted as representing a topology change, in which an open space spontaneously compactifies; however, the quantum probability for these processes vanishes.

Gravitation and hamiltonian structure in two spacetime dimensions

In two spacetime dimensions a c-number (“Schwinger term”, “central charge”) is allowed in the algebra of surface deformations. A non-trivial analog of gravitation theory in two dimensional spacetime is built upon this fact, with the inverse of the central charge playing the role of the gravitational constant. Classically the analog with gravitation theory is only partial in that the hamiltonian constraints cannot be imposed, but it becomes complete at the quantum level.

Dynamics of chiral (self-dual) p-forms

Abstract An action principle describing the dynamics of a p-form gauge field whose field strength is self-dual is given. The action is local, Lorentz invariant and also invariant under the standard gauge transformation of a p-form. The coupling to gravitation is described. The proposed action permits a consistent passage to quantum mechanics. The path integral is briefly discussed.

The Cosmological Constant and General Covariance

In the standard gravitational action without a cosmological constant, one may consider the determinant of the metric as an external field. One then extremizes the action only with respect to variations of the metric that do not change the local volume. The resulting field equations are Einsteins equations with a cosmological constant, which appears as a constant of integration. The dynamics of that theory is analyzed from the point of view of constrained hamiltonian systems. It is observed that contrary to what one might think, the theory is fully covariant and contains only one overall degree of freedom (the cosmological constant) in addition to the two degrees of freedom per point of ordinary Einsteins theory. In the hamiltonian formalism the missing coordinate invariance re-emerges through a tertiary constraint. A Yang-Mills analog is pointed out. The theory is then made manifestly generally covariant by introducing auxiliary fields that are pure gauge except for one overall zero mode. This global mode is a “cosmic time” canonically conjugate to the cosmological constant.

Dynamical neutralization of the cosmological constant

A dynamical process is described in which the cosmological constant is netralized through the quantum creation of closed membranes by totally antisymmetric tensor and gravitational fields.

How commutators of constraints reflect the spacetime structure

Abstract The structure constants of the “algebra” of constraints of a parametrized field theory are derived by a simple geometrical argument based exclusively on the path independence of the dynamical evolution; the change in the canonical variables during the evolution from a given initial surface to a given final surface must be independent of the particular sequence of intermediate surface used in the actual evaluation of this change. The requirement of path independence also implies that the theory will propagate consistently only initial data such that the Hamiltonian vanishes. The vanishing of the Hamiltonian arises because the metric of the surface is a canonical variable rather than a c-number. It is not assumed the constraints can be solved to express four of the momenta in terms of the remaining canonical variables. It is shown that the signature of spacetime can be read off from the commutator of two Hamiltonian constraints at different points. The analysis applies equally well irrespective of whether the spacetime is a prescribed Riemannian background or whether it is determined by the theory itself as in general relativity. In the former case the structure of the commutators imposes consistency conditions for a theory in which states are defined on arbitrary spacelike surfaces; whereas, in the later case it provides the conditions for the existence of spacetime— “embeddability” conditions which ensure that the evolution of a three-geometry can be viewed as the “motion” of a three-dimensional cut in a four-dimensional spacetime of hyperbolic signature.