The Hamiltonian formulation of General Relativity: myths and reality

Abstract

A conventional wisdom often perpetuated in the literature states that: (i) a 3+1 decomposition of space-time into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [Proc. Roy. Soc. A 246 (1958) 333] and of ADM [Arnowitt, Deser and Misner, in "Gravitation: An Introduction to Current Research" (1962) 227] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i-iii) have been shown to be incorrect in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. We show that points (i-iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric $g_{\mu \nu }$ to lapse and shift functions and the three-metric $g_{km}$, which is not canonical. This proves that point (iv) is incorrect. Points (i-iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein's theory itself.

References are added and updated, Introduction is extended, Subsection 3.5 is added, 83 pages; corresponds to the published version

References in corpus (7)

• Hyperbolicity of scalar-tensor theories of gravity
• Diffeomorphism Invariance in the Hamiltonian formulation of General Relativity
• First-order Formalism and Odd-derivative Actions
• Gauge invariances vis-{á}-vis Diffeomorphisms in second order metric gravity: A new Hamiltonian approach
• Symmetries of topological gravity with torsion in the hamiltonian and lagrangian formalisms
• A Canonical Analysis of the Einstein-Hilbert Action in First Order Form
• Hamiltonian formulation of tetrad gravity: three dimensional case

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