STRIKING PROPERTY OF THE GRAVITATIONAL HAMILTONIAN

The Hamiltonian framework for (2+1)-dimensional gravity coupled with m atter (satisfying positive energy conditions) is considered in the asy mptotically hat context. It is shown that the total energy of the syst em is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual held theories, the Hamiltonian is bounded from above. This is a genuinely nonperturbative result. In the presence of a spacelike Killing field, (3+1)-dimensional vacuum general relativity is equivalent to (2+1)-dim ensional general relativity coupled to certain matter fields. Therefor e, our expression provides, in particular, a formula for energy per-un it length (along the symmetry direction) of gravitational waves with a spacelike symmetry in 3+1 dimensions. A special case is that of cylin drical waves which have two hypersurface orthogonal, spacelike Killing fields. In this case, our expression is related to the ''c energy'' i n a nonpolynomial fashion. While in the weak field limit the two agree , in the strong field regime they differ significantly. By constructio n, our expression yields the generator of the time translation in the full theory, and therefore represents the physical energy in the gravi tational field.