A Striking Property of the Gravitational Hamiltonian

TL;DR

This paper establishes a non-perturbative Hamiltonian framework for 2+1 dimensional gravity coupled with matter, demonstrating that total energy is non-negative and providing a formula for energy per unit length in 3+1 dimensions with symmetry, with implications for gravitational wave energy.

Contribution

It introduces a Hamiltonian formulation for 2+1 gravity with matter that shows the total energy is non-negative and bounded from above, and relates 3+1 dimensional gravitational energy to 2+1 dimensional models with symmetry.

Findings

Total energy is non-negative and zero only for Minkowski space.

Hamiltonian is bounded from above, contrary to usual field theories.

Provides a formula for energy per unit length of symmetric gravitational waves.

Abstract

{\sl 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} {\rm bounded from above}. This is a genuinely non-perturbative result. {\sl 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.} \footnote{$^1$}{This is a detailed account of the results presented in the Brill-Misner symposium at the University of Maryland in May1993}