Hamiltonian reduction of Einstein’s equations of general relativity
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A program is outlined which resolves the problem of the Hamiltonian reduction of Einstein's vacuum field equations in (3 + 1)-dimensions. The problem involves writing Einstein's vacuum field equations as an unconstrained Hamiltonian dynamical system where the variables of the unconstrained system are the true degrees of freedom of the gravitational field. Our analysis is applicable to vacuum spacetimes that admit constant mean curvature compact spacelike hypersurfaces M that satisfy certain topological restrictions. We find that for these spacetimes (3 + 1)-reduction can be completed much as in the (2 + 1)-dimensional case. In both cases, one gets as the reduced phase space the contangent bundle Tast;TM of the Teichmüller space TM=MPD0 of conformal structures on M and one gets reduction of the full classical Hamiltonian system with constraints to a non-local time-dependent reduced Hamiltonian system without constraints on the contact manifold R × T∗τM. For this reduced system, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact spacelike hypersurfaces in a neighborhood of the given initial one and the Hamiltonian is the volume functional of these hypersurfaces expressed in terms of the canonical variables of the hypersurface.
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The space of measured foliations MF is homeomorphic to the space of measured laminations ML.
“Given a measured foliation, one may obtain a measured lamination via a suitable straightening map [29, 33] (or collapsing a lamination yields a foliation). Therefore, there is a homeomorphism between MF and ML .”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [29]
The Introduction and Section 3 discuss Teichmüller space and conformal structures, but do not address measured foliations or measured laminations. Searches for those terms, straightening, and a homeomorphism between MF and ML found no relevant passage in our copy. Page numbers were not available. (LLM, conf=0.77)
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Cites
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On the global evolution problem in 2 + 1 gravity
Anderson, L., Moncrief, V., & Tromba, A. J. (1997) · Journal of Geometry and Physics · doi:10.1016/s0393-0440(97)87804-7 · Cited by 72 · see citing works
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Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space
Mondal, P. (2023) · Journal of Mathematical Physics · doi:10.1063/5.0136631 · Cited by 1 · see citing works
- Authors
- Fischer, A. E., & Moncrief, V.
- Affiliations
- University of California, Santa Cruz (US)
- Yale University (US)
- Corresponding author
- Fischer, A.E.
- Moncrief, V.
- ORCID
- Fischer, A.E.
- Venue
- Nuclear Physics B - Proceedings Supplements
- Publisher
- Elsevier BV
- Published
- 1997-08
- Volume
- 57
- Issue
- 1-3
- Pages
- 142-161
- ISSN
- 0920-5632 (ISSN-L)
- OpenAlex
- W1981809488
- Cited by
- 58
- References
- 2
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 1 · 2023: 2 · 2022: 1 · 2020: 3 · 2019: 6 · 2018: 5 · 2017: 5 · 2016: 3 · 2015: 3 · 2014: 1 · 2013: 2
- FWCI
- 1.58
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Nuclear Physics B - Proceedings Supplements (publishedVersion, cc-by-nc-nd)
- Funders
- National Science Foundation (9503133)
- Yale University
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- Hamiltonian reduction, vacuum Einstein equations, general relativity, constant mean curvature hypersurfaces, spacelike hypersurfaces, Teichmüller space, conformal structures, reduced phase space, volume functional, contact manifolds, three-dimensional gravity
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