G R - Q C - 9 40 60 40 A Striking Property of the Gravitational Hamiltonian

A Hamiltonian framework for 2+1 dimensional gravity coupled with matter (satisfying positive energy conditions) is considered in the asymptotically at context. It is shown that the total energy of the system is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual eld theories, the Hamiltonian is bounded from above. This is a genuinely non-perturbative result. In the presence of a space-like Killing eld, 3+1 dimensional vacuum general relativity is equivalent to 2+1 dimensional general relativity coupled to certain matter elds. Therefore, our expression provides, in particular, a formula for energy per-unit length (along the symmetry direction) of gravitational waves with a space-like symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, space-like Killing elds. In this case, our expression is related to the \c-energy" in a non-polynomial fashion. While in the weak eld limit, the two agree, in the strong eld regime they di er signi cantly. By construction, our expression yields the generator of the time-translation in the full theory, and therefore represents the physical energy in the gravitational eld. 1