1 / 21

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.

billy
Download Presentation

Solving Einstein's field equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related