M. O. Katanaev Steklov Mathematical Institute, Moscow

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

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

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

4. 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

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

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

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

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

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

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

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