Hamiltonian Formulation of General Relativity M O Katanaev

ADM parameterization of the metric - n-dimensional space-time Pseudo-Riemannian manifold: - local coordinates -

ADM parameterization of the metric (continued) - time Theorem. The metric has Lorentzian signature

Hamiltonian metric form of General Relativity - ADM parameterization of the metric - the

Hamiltonian metric form of General Relativity (continued) - the Lagrangian - primary constraints -

Secondary constraints - the Hamiltonian - Poisson brackets - primary constraints secondary constraints 6

Algebra of secondary constraints - the Hamiltonian - phase space variables - Lagrange multipliers

where - irreducible decomposition The canonical transformation additional constraints: - generating functional depending on

The constraints - scalar curvature A. Peres, Nuovo Cimento (1963) - polynomial of degree

Algebra of the constraints - Poisson manifold Basic Poisson brackets: - degenerate - algebra

Four-dimensional General Relativity - Hamiltonian - independent variables - scalar curvature (fifth order) -

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Hamiltonian Formulation of General Relativity M. O. Katanaev Steklov Mathematical Institute, Moscow Short historical Notes Dirac (1958) Arnowitt, Deser, Misner (ADM) (1960) De. Witt (1967) Regge, Teitelboim (1974). . . . Dirac (1962) Schwinger (1963) - metric formulation - vielbein formulation (time gauge) . . . Deser, Isham (1976) Nelson, Teiltelboim (1978) Henneaux (1983) Charap, Nelson (1986) . . . - vielbein formulation 1

ADM parameterization of the metric - n-dimensional space-time Pseudo-Riemannian manifold: - local coordinates - metric The rule: - subsets - ADM parameterization - lapse function - shift function - the inverse to For - one-to-one correspondence 2

ADM parameterization of the metric (continued) - time Theorem. The metric has Lorentzian signature if and only if the metric is negative definite. Additional assumption: all sections are spacelike - is negative definite - the Hilbert – Einstein action 3

Hamiltonian metric form of General Relativity - ADM parameterization of the metric - the induced metric on hypersurfaces - the induced connection - the internal curvature - the extrinsic curvature - normal to a hypersurface - the trace of extrinsic curvature here and 4

Hamiltonian metric form of General Relativity (continued) - the Lagrangian - primary constraints - the canonical momenta - the Hamiltonian density where - the Hamiltonian 5

Secondary constraints - the Hamiltonian - Poisson brackets - primary constraints secondary constraints 6

Algebra of secondary constraints - the Hamiltonian - phase space variables - Lagrange multipliers - constraints - generator of space diffeomorphisms Dirac (1951) De. Witt (1967) where 7

where - irreducible decomposition The canonical transformation additional constraints: - generating functional depending on new coordinates and old momenta 8

The constraints - scalar curvature A. Peres, Nuovo Cimento (1963) - polynomial of degree - totally antisymmetric tensor density 9

Algebra of the constraints - Poisson manifold Basic Poisson brackets: - degenerate - algebra of the constraints Submanifold defined by the equations is the phase space 10

Four-dimensional General Relativity - Hamiltonian - independent variables - scalar curvature (fifth order) - quadratic polynomial 11