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Reducing Einstein’s Equations to an Unconstrained Hamiltonian System on the Cotangent Bundle of Teichmüller Space

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Physics on Manifolds

Part of the book series: Mathematical Physics Studies ((MPST,volume 15))

Abstract

A program for reducing Einstein’s vacuum equations to an unconstrained dynamical system where the variables are the true degrees of freedom of the gravitational field is presented. The reduced phase space on which this dynamics occurs is the cotangent bundle T*Tof

$$ \mathcal{T} \equiv \frac{{\mathcal{M}/\mathcal{P}}}{{{{\mathcal{D}}_{0}}}} $$

the D o-restricted conformal superspace of a compact connected orientable 3-manifold M, which we also refer to as the Teichmüller space of conformal structures on M. From results regarding linearization stability of spacetimes, we argue that the program of reduction should be successful for spacetimes that admit constant mean curvature continuously non-symmetric (or deg(M) = 0) non-F6 compact spacelike Cauchy hypersurfaces, inasmuch as the space of solutions to Einstein’s equations is a manifold in a neighborhood of such spacetimes. Modulo the Poincaré conjecture and a conjecture regarding finite free actions on S3, the topological types of all such spacelike Cauchy hypersurfaces M are identified.

Restricting to the case of continuously non-symmetric Yamabe type —1 manifolds M, T can be represented as

$$ \mathcal{T} \approx {{\mathcal{M}}_{{ - 1}}}/{{\mathcal{D}}_{0}} $$

and the full Einstein dynamics can be reduced to an implicitly defined time-dependent Hamiltonian system on the cotangent bundle T*T. The parameter of evolution for this system is the parameter of a monotonically increasing constant mean curvature slicing by spacelike Cauchy hypersurfaces in a neighborhood of the given one, and the Hamiltonian is the volume functional of these hypersurfaces.

Some possible approaches to the more general case of continuously non-symmetric Yamabe type +1 manifolds are proposed. It is speculated that if conformal methods are not successful in treating this case, then, since linearization stability arguments imply that a reduction does exist (modulo discrete isometry groups of the spacetime), there must be another method of reduction that does work.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, California, 95064, USA

    Arthur E. Fischer

  2. Departments of Mathematics and Physics, Yale University, New Haven, Connecticut, 06511, USA

    Vincent Moncrief

Authors
  1. Arthur E. Fischer
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  2. Vincent Moncrief
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Editors and Affiliations

  1. Physique Mathématique, Université de Bourgogne, Dijon, France

    M. Flato

  2. Laboratoire de Gravitation et Cosmologie Relativistes, UPMC, Paris, France

    R. Kerner

  3. Collège de France, Paris, France

    A. Lichnerowicz

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Fischer, A.E., Moncrief, V. (1994). Reducing Einstein’s Equations to an Unconstrained Hamiltonian System on the Cotangent Bundle of Teichmüller Space. In: Flato, M., Kerner, R., Lichnerowicz, A. (eds) Physics on Manifolds. Mathematical Physics Studies, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1938-2_9

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  • DOI: https://doi.org/10.1007/978-94-011-1938-2_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4857-6

  • Online ISBN: 978-94-011-1938-2

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