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Geometrodynamics regained: A Lagrangian approach
1.S. A. Hojman, K. Kuchař, and C. Teitelboim (unpublished).
2.The notation of this paper follows that of Ref. 1. The Greek indices range from 0 to 3, the lower case italic indices from 1 to 3, the range of the upper case italic indices is left unspecified. The Greek indices are raised by the space—time metric the italic indices by the intrinsic metric of a hypersurface. In general, space—time tensors carry an index4 indicating the dimension, hypersurface tensors are written without such an index (e.g., is the space—time curvature scalar, R is the curvature scalar of a hypersurface). Capital is used for the space—time coordinates, lower case (the indexi being often omitted) for the intrinsic coordinates of a hypersurface. Comma, denotes the partial differentiation, vertical stroke ∥ the covariant differentiation on a hypersurface, the nabla symbol ∇ the space—time covariant differentiation. The summation convention is extended to continuous spatial labels: whenever x is written as an index, the integration over a repeated x is implied. For example, All tensor densities used in the paper are tensor densities of weight 1. The δ‐functions are considered to be scalars in the first and scalar densities in the second argument. The Levi—Civita pseudotensors are written as and The symbol used in the formulas means “the same term with the spatial labels x and interchanged.” The pound symbol denotes the Lie derivative with respect to The square brackets indicate the functional dependence; e.g., means that the super‐Hamiltonian at the point x of a hypersurface is a functional of the metric and of the geometrodynamical momentum on that hypersurface. The Poisson brackets are denoted by the square brackets [,].
3.K. Kuchař, “Canonical Quantization of Gravity,” in Relativity, Astrophysics and Cosmology, edited by W. Israel (Reidel, Dordrecht, 1973).
4.P. A. M. Dirac, Lectures on Quantum Mechanics (Academic, New York, 1965).
5.Note that is not the extrinsic curvature under the standard convention, but rather the extrinsic curvature. This enables us to identify directly with the normal geometrodynamical velocity.
6.A preliminary account of this proof is given in S. A. Hojman, K. Kuchař, and C. Teitelboim, Nature (to be published).
7.C. W. Misner, Rev. Mod. Phys. 29, 497 (1957);
7.P. W. Higgs, Phys. Rev. Lett. 1, 373 (1958);
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8.J. W. York, Phys. Rev. Lett. 26, 1656 (1971);
8.J. W. York, J. Math. Phys. 13, 125 (1972).
9.C. Teitelboim, “The Hamiltonian Structure of Spacetime,” Ph.D. Thesis, Princeton University Preprint, May 1973 (unpublished).
10.See, e.g., M. von Laue, Die Relativiätstheorie II, Die Allgemeine Relativiätstheorie and Einsteins Lehre von der Schwerkraft (Vieweg, Braunschweig, 1922), 1st ed.
11.D. Lovelock, J. Math. Phys. 12, 498 (1971).
12.See, e.g., B. S. DeWitt, Phys. Rev. 160, 113 (1967).
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