It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is either R x SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on 3 B, the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate 3 B. The integral of the energy density over such a two-surface B is the quasilocal energy associated with a spacelike threesurface Σ whose intersection with 3 B is the boundary B. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary B as a surface embedded in Σ. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on 3 B in the direction orthogonal to B. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on 3 B. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.

The coupling of the metric to an incoherent dust introduces into spacetime a privileged dynamical reference frame and time foliation. The comoving coordinates of the dust particles and the proper time along the dust worldlines become canonical coordinates in the phase space of the system. The Hamiltonian constraint can be resolved with respect to the momentum that is canonically conjugate to the dust time. Formal imposition of the resolved constraint as an operator restriction on the quantum states yields a functional Schrodinger equation. The ensuing Hamiltonian density has an extraordinary feature: it depends only on the geometric variables, not on the dust coordinates or time. This has three important consequences. First, the functional Schrodinger equation can be solved by separating the dust time from the geometric variables. Second. , disregarding the standard factor-ordering diKculties, the Hamiltonian densities strongly commute and therefore can be simultaneously defined by spectral analysis. Third, the standard constraint system of vacuum gravity is cast into a form in which it generates a true Lie algebra. The particles of dust introduce into space a privileged system of coordinates that allows the supermomentum constraint to be solved explicitly. The Schrodinger equation yields a formally conserved inner product that can be written in terms of either the instantaneous state functionals or the solutions of constraints. Gravitational observables admit a similar dual representation. Examples of observables are given, though neither the intrinsic metric nor the extrinsic curvature are observables. This comes as close as one can reasonably expect to a satisfactory phenomenological quantization scheme that is free of most of the problems of time. PACS number(s): 04.60.Ds, 04.20.Cv, 04.20.Fy I. INTKGDUCTIONThe Dirac constraint quantization of vacuum Einstein gravity yields the Wheeler-DeWitt equation for the quantum state of the intrinsic three geometry of space [1,2]. One can view this equation as a statement that only two out of three independent components of the intrinsic geometry are dynamical. The third component is an intrinsic time that specifies the location of space as a hypersurface in spacetime. The Wheeler-DeWitt equation is then interpreted as an evolution equation for the state in the intrinsic time.The Wheeler-DeWitt equation is a second-order variational differential equation. The space of its solutions carries no obvious Hilbert space structure [3,4]. This has prompted numerous attempts aimed at replacing the Wheeler-DeWitt equation by a first-order Schrodinger equation. In order to do that, one should identify the intrinsic time at the classical level, solve the Hamiltonian constraint for the momentum conjugate to time, and impose the resolved constraint as an operator restriction on the quantum states. Unfortunately, there is no natural candidate for the intrinsic time, and the procedure is beset by a number of conceptual and technical diKculties [51 Intrinsic clocks are strange contrap...

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We investigate the thermodynamical properties of black holes in (3+1) and (2+1) dimensional Einstein gravity with a negative cosmological constant. In each case, the thermodynamic internal energy is computed for a finite spatial region that contains the black hole. The temperature at the boundary of this region is defined by differentiating the energy with respect to entropy, and is equal to the product of the surface gravity (divided by 2π) and the Tolman redshift factor for temperature in a stationary gravitational field. We also compute the thermodynamic surface pressure and, in the case of the (2+1) black hole, show that the chemical potential conjugate to angular momentum is equal to the proper angular velocity of the black hole with respect to observers who are at rest in the stationary time slices. In (3+1) dimensions, a calculation of the heat capacity reveals the existence of a thermodynamically stable black hole solution and a negative heat capacity instanton. This result holds in the limit that the spatial boundary tends to infinity only if the comological constant is negative; if the cosmological constant vanishes, the stable black hole solution is lost. In (2+1) dimensions, a calculation of the heat capacity reveals the existence of a thermodynamically stable black hole solution, but no negative heat capacity instanton.