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Solving Einstein's field equations

Solving Einstein's field equations. for space-times with symmetries. Integrability structures and. nonlinear dynamics of. interacting fields. G.Alekseev. Many “languages” of integrability. Introduction. Gravitational and electromagnetic solitons. Lecture 1.

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Solving Einstein's field equations

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  1. Solving Einstein's field equations for space-times with symmetries Integrability structures and nonlinear dynamics of interacting fields G.Alekseev Many “languages” of integrability Introduction Gravitational and electromagnetic solitons Lecture 1 Monodromy transform approach Lecture 2 Solutions for black holes in the external fields Solution of the characteristic initial value problem; Colliding gravitational and electromagnetic waves Lecture 3

  2. Integrability of Einstein’s equations with symmetries Integrable cases: - Vacuum gravitational fields - Einstein – Maxwell - Weyl fields - Ideal fluid with - some string gravity models mathematical context: - infinite hierarchies of exact solutions, - initial and boundary value problems, - asymptotical behaviour physical context: - supeposition of stat. axisymm. fields, - nonlinear interacting waves, - inhomogeneous cosmological models

  3. Many "languages" of integrability • Associated linear systems and ``spectral’’ problems • Infinite-dimensional algebra of internal symmetries • Solution generating procedures (arbitrary seed): • -- Solitons, • -- Backlund transformations, • -- Symmetry transformations • Infinite hierarchies of exact solutions • -- Meromorfic on the Riemann sphere • -- Meromorfic on the Riemann surfaces (finite gap solutions) • Prolongation structures • Geroch conjecture • Riemann – Hielbert and Homogeneous Hilbert problems, • Various linear singular integral equation methods • Initial and boundary value problems • -- Characteristic initial value problems • -- Boundary value problems for stationary axisymmetric fields • Twistor theory of the Ernst equation

  4. Integrable reductions of the Einstein's field equations Gravitational fields in vacuum: Elektrovacuum Einstein - Maxwell fields: Gravity model with axion, dilaton and E-H fields Bosonic sector of heterotic string effective action:

  5. 1) Space-times with one Killing vector Einstein – Maxwell fields: the Ernst-like equations SU(2,1) – symmetric form of dynamical equations 1) W.Kinnersley, J. Math.Phys. (1973)

  6. 1) Space-times with two commuting Killing vectors Isometry group with 2-surface –orthogonal orbits: The Einstein’s field equations: -- the “constraint” equations -- the “dynamical” equations -- the “dynamical” equations

  7. Space-times with two commuting Killing vectors Generalized Weyl coordinates: Geometrically defined coordinates:

  8. Lecture 1 Integrable reductions of Einstein equations Belinski – Zakharov vacuum solitons Einstein – Maxwell solitons Examples of soliton solutions

  9. Ernst vacuum equation Kinnersley self-dual form of the reduced vacuum equations Belinski – Zakharov form of reduced vacuum equations 2x2-matrix form of self-dual reduced vacuum equations

  10. 1) Belinski - Zakharov inverse scattering approach Dynamical equations for vacuum Associated spectral problem “Dressing” method for constructing solutions 1) V.Belinski & V.Zakharov,, JETP 1978; 1979 ;

  11. 1) Formulation of the matrix Riemann problem Riemann problem for dressing matrix Linear singular integral equations Constraints for dressing matrix: 1) V.Belinski & V.Zakharov,, JETP 1978; 1979 ;

  12. 2N-soliton solution: 1) Vacuum solitons ( - solitons) Soliton ansatz for dressing matrix 1) V.Belinski & V.Zakharov,, JETP 1978; 1979 ;

  13. 1) Determinant form of Belinski – Zakharov vacuum solitons Stationary axisymmetric solitons on the Minkowski background: a set of 4 N arbitrary real or pairwise complex conjugated constants 1) GA, Sov.Phys.Dokl. (1981) ;

  14. Integrable reductions of Einstein-Maxwell equations Spacetime metric and electromagnetic potential:

  15. Electrovacuum fields Ernst potentials : Ernst equations:

  16. 3x3-matrix form of Einstein – Maxwell equaations

  17. 1) Spectral problem for Einstein – Maxwell fields For vacuum: 1) GA, JETP Lett.. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999)

  18. 1) (w - solitons) Einstein - Maxwell solitons Dressing matrix : Soliton ansatz for dressing matrix --- a set of 3 N arbitrary complex constants 1) GA, JETP Lett. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999)

  19. Example: one-soliton solution on the Minkowski background -- Superextreme part of the Kerr-Newman solution -- mass -- NUT-parameter -- angular momentum -- electric charge -- magnetic charge 1) Two-soliton solution on the Minkowski background -- Interaction of two superextreme Kerr-Newman sources 1) GA, Proc. Steklov Inst. Math. (1988); Physica D. (1999)

  20. Two-soliton solution on the Minkowski background -- Interaction of two superextreme Kerr-Newman sources

  21. Soliton gravitational and electromagnetic waves on the Minkowski background

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