1994/06/22 by Abhay Ashtekar, Madhavan Varadarajan · 1 citation
Physics and Astronomy · #gr-qc
paper · pdf · doi:10.1103/physrevd.50.4944
published as Phys.Rev.D50:4944-4956,1994 · 22, CGPG-94/6-3
arxiv created 1994/06/22 · arxiv updated 2011/04/15
\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. \footnote1This is a detailed account of the results presented in the Brill-Misner symposium at the University of Maryland in May1993