Striking property of the gravitational Hamiltonian

Abhay Ashtekar, Madhavan Varadarajan

Research output: Contribution to journalArticle

66 Citations (Scopus)

Abstract

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 nonperturbative result. In the presence of a spacelike 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 spacelike symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, spacelike Killing fields. In this case, our expression is related to the "c energy" in a nonpolynomial 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.

Original languageEnglish (US)
Pages (from-to)4944-4956
Number of pages13
JournalPhysical Review D
Volume50
Issue number8
DOIs
StatePublished - Jan 1 1994

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relativity
energy
cylindrical waves
symmetry
gravitational waves
gravitational fields
generators
gravitation
vacuum

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics
  • Physics and Astronomy (miscellaneous)

Cite this

Ashtekar, Abhay ; Varadarajan, Madhavan. / Striking property of the gravitational Hamiltonian. In: Physical Review D. 1994 ; Vol. 50, No. 8. pp. 4944-4956.
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Striking property of the gravitational Hamiltonian. / Ashtekar, Abhay; Varadarajan, Madhavan.

In: Physical Review D, Vol. 50, No. 8, 01.01.1994, p. 4944-4956.

Research output: Contribution to journalArticle

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