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M. O. Katanaev Steklov Mathematical Institute, Moscow

Learn about the development, concepts, and applications of Hamiltonian formulation in General Relativity from historical perspectives to contemporary advancements with a focus on metrics, vielbein formulations, ADM parameterization, primary and secondary constraints, canonical transformations, and algebraic aspects. Explore the significance of Hamiltonian metric form, Lagrangian, phase space variables, Poisson brackets, and submanifold definition within the context of n-dimensional space-time.

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M. O. Katanaev Steklov Mathematical Institute, Moscow

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  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

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