Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space
Abstract
In this paper the ADM (Arnowitt, Deser, and Misner) reduction of Einstein’s equations for three-dimensional ‘‘space-times’’ defined on manifolds of the form Σ×R, where Σ is a compact orientable surface, is discussed. When the genus g of Σ is greater than unity it is shown how the Einstein constraint equations can be solved and certain coordinate conditions imposed so as to reduce the dynamics to that of a (time-dependent) Hamiltonian system defined on the 12g−12-dimensional cotangent bundle, T*𝒯(Σ), of the Teichmüller space, 𝒯(Σ), of Σ. The Hamiltonian is only implicitly defined (in terms of the solution of an associated Lichnerowicz equation), but its existence, uniqueness, and smoothness are established by standard analytical methods. Similar results are obtained for the case of genus g=1, where, in fact, the Hamiltonian can be computed explicitly and Hamilton’s equations integrated exactly (as was found previously by Martinec). The results are relevant to the problem of the reduction of the 3+1-dimensional Einstein equations (formulated on circle bundles over Σ×R and with a spacelike Killing field tangent to the fibers of the chosen bundle) and to the recent discussion by Witten of the possible exact solvability of the ‘‘topological dynamics’’ associated with Einstein’s equations in 2+1 dimensions.
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The metric formalism can be used to give a full description of the solutions of the vacuum field equations, at least for spacetimes with topology R×Σ.
“As Moncrief [206] and Hosoya and Nakao [159] have shown, this metric formalism can also be used to give a full description of the solutions of the vacuum field equations, at least for spacetimes with the topology ℝ × Σ.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [206]
With the decomposition of [206], the momentum constraints become trivial, while the Hamiltonian constraint becomes an elliptic differential equation that determines the scale factor λ as a function of ḡij and pij.
“With the decomposition of [206], the momentum constraints ℌi = 0 become trivial, while the Hamiltonian constraint becomes an elliptic differential equation that determines the scale factor γ in Equation (31) as a function of ḡij and pij”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [206]
The momentum constraint fixes the undetermined vector field Yi in terms of the scale factor λ, but it does so nonlocally.
“The momentum constraint fixes in terms of the scale factor λ, but it does so nonlocally.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [206]
Existence and uniqueness of solutions to the Lichnerowicz equation (31) has been proved.
“Existence and uniqueness of solutions to (31) has been proved, see [4].”
From On the global evolution problem in 2 + 1 gravity (1997) · cites this paper as [4]
In the 2+1 dimensional case with Σ a compact oriented surface, the conformal method ideas can be carried out in full detail.
“In the 2+1 dimensional case with Σ a compact oriented surface, the above ideas can be carried out in full detail [4].”
From On the global evolution problem in 2 + 1 gravity (1997) · cites this paper as [4]
Cites
2 + 1 dimensional gravity as an exactly soluble system
Witten, E. (1988) · Nuclear Physics B · doi:10.1016/0550-3213(88)90143-5 · Cited by 2,562 · see citing works
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Lorentz spacetimes of constant curvature
Mess, G. (2007) · Geometriae Dedicata · doi:10.1007/s10711-007-9155-7 · Cited by 278 · see citing works
Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
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
Self-citationCosmological Time in (2+1) - Gravity
Benedetti, R., & Guadagnini, E. (2001) · Nuclear Physics B · doi:10.1016/s0550-3213(01)00386-8 · Cited by 43 · see citing works
Lorentz spacetimes of constant curvature
Mess, G. (2007) · arXiv · Cited by 5 · see citing works
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
- Moncrief, V.
- Affiliations
- Yale University (US)
- Corresponding author
- Moncrief, Vincent
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1989-12-01
- Volume
- 30
- Issue
- 12
- Pages
- 2907-2914
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2063562491
- Cited by
- 202
- References
- 1
- Citations by year
- 2025: 5 · 2024: 4 · 2023: 3 · 2022: 2 · 2021: 1 · 2020: 7 · 2019: 2 · 2018: 5 · 2017: 7 · 2016: 3 · 2015: 5 · 2014: 1 · 2013: 9 · 2012: 6
- FWCI
- 10.63
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Mathematical Physics (publishedVersion)
- Funders
- National Science Foundation
- 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 › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- Teichmüller space, Hamiltonian systems, cotangent bundle, Einstein constraint equations, Lichnerowicz equation, compact orientable surfaces, genus one, topological dynamics, exact solvability, circle bundles, Killing fields
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