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The Hamiltonian of Einstein affine-metric formulation of General Relativity

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Abstract

It is shown that the Hamiltonian of the Einstein affine-metric (first-order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as in the case of the second-order formulation. In the second-order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables (arXiv:0809.0097). For the first-order formulation, the necessity of such a redefinition “to correspond to diffeomorphism invariance” (reported by Ghalati, arXiv:0901.3344) is just an artifact of using the Henneaux–Teitelboim–Zanelli ansatz (Nucl. Phys. B 332:169, 1990), which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani (Ann. Phys. 143:357, 1982) is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second-order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second- and first-order formulations of metric GR. The first-order formulation of Einstein–Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.

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Authors and Affiliations

  1. Faculty of Arts and Social Science, Huron University College, N6G 1H3, London, Canada

    N. Kiriushcheva & S. V. Kuzmin

  2. Department of Applied Mathematics, University of Western Ontario, N6A 5B7, London, Canada

    N. Kiriushcheva & S. V. Kuzmin

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  1. N. Kiriushcheva
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Kiriushcheva, N., Kuzmin, S.V. The Hamiltonian of Einstein affine-metric formulation of General Relativity. Eur. Phys. J. C 70, 389–422 (2010). https://doi.org/10.1140/epjc/s10052-010-1458-4

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