Download presentation

Presentation is loading. Please wait.

Published byDeja Esco Modified over 2 years ago

1
Gravitational and electromagnetic solitons Monodromy transform approach Solution of the characteristic initial value problem; Colliding gravitational and electromagnetic waves Many “languages” of integrability Solutions for black holes in the external fields

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

3
3 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
4

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

6
6 Isometry group with 2-surface –orthogonal orbits: The Einstein’s field equations: -- the “constraint” equations -- the “dynamical” equations

7
7 Geometrically defined coordinates: Generalized Weyl coordinates:

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

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

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

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

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

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

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

15
15 Ernst potentials : Ernst equations:

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

17
17 1) GA, JETP Lett.. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999) 1) For vacuum:

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

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

20
20 -- Interaction of two superextreme Kerr-Newman sources

21
21

Similar presentations

OK

Gravitational Dirac Bubbles: Stability and Mass Splitting Speaker Shimon Rubin ( work with Aharon Davidson) Ben-Gurion University of the Negev Miami, 16-21.

Gravitational Dirac Bubbles: Stability and Mass Splitting Speaker Shimon Rubin ( work with Aharon Davidson) Ben-Gurion University of the Negev Miami, 16-21.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on animal farm themes Ppt on acute and chronic diseases for class 9 Ppt on google glasses technology Download ppt on environmental degradation Animated ppt on reflection of light Ppt on cross-sectional study in research Ppt on work power and energy class 11 Ppt on db2 introduction to management Ppt on marketing strategy of nokia and samsung Ppt on communication and networking concepts