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

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

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

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

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

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

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

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

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

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

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