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 flat 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 field theories, the Hamiltonian is bounded from above. This is a genuinely non-perturbative result. In the presence of a space-like Killing field, 3+1 dimensional vacuum general relativity is equivalent to 2+1 dimensional general relativity coupled to certain matter fields. 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 fields. In this case, our expression is related to the “c-energy” in a non-polynomial fashion. While in the weak field limit, the two agree, in the strong field regime they differ significantly. By construction, our expression yields the generator of the time-translation in the full theory, and therefore represents the physical energy in the gravitational field. 1