Integrable Reductions of the Einstein’s Field Equations

1. Integrable Reductions of the Einstein’s Field Equations Harry-Dym G. Alekseev Davey - Stewartson Monodromy transform approach and integral equation methods Kadomtsev-Petviashvili SU(2) YM Nonlinear Schrodinger Sine-Gordon Korteveg de Vries Steklov Mathematical Institute RAS

2. Integrable reductions of the Einstein’s field equations Hyperbolic reductions: (waves, cosmologocal models) Elliptic reductions (Stationary fields with spatial symmetry) Vacuum axion, dilaton,... Electrovacuum Weyl spinor field stiff matter

3. Many faces 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 • -- meromorphic on the Riemann sphere • -- meromorphic 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. Integrability and the solution space transforms Free space of func- tional parameters Space of solutions (Constraint: field equations) (No constraints) “Direct’’ problem: (linear ordinary differential equations) “Inverse’’ problem: (linear singular integral equations) • Applications: • Solution generating methods • Infinite hierarchies of exact solutions • ``Partial’’ superposition of fields • Initial/boundary value problems • Asymptotic behaviour Monodromy transform: Monodromy data

5. Monodromy transform approach and the integral equation methods Plan of the talk • Monodromy transform: • -- ``direct’’ and ``inverse’’ problems; • -- monodromy data and physical properties of solutions; • The integral equation methods: • -- the integral equations for solution of the inverse problem; • -- the integral ``evolution’’ equations; • -- particular reductions and relations with some other methods; • Applications: • -- characteristic initial value problem for colliding plane waves • -- Infinite hierarchies of solutions for rational monodromy data: • a) analytically matched data • b) analytically non-matched data • -- superposition of fields (examples)

6. Einstein’s equations with integrable reductions -- Vacuum -- Electrovacuum -- Einstein Maxwell Weyl Effective string gravity equations

7. Space-time symmetry ansatz Coordinates: Space-time metric: 2-surface-orthogonal orbits of isometry group: Generalized Weyl coordinates : Geometrically defined coordinates :

8. Reduced dynamical equations – generalized Ernst eqs. -- Vacuum -- Electrovacuum -- Einstein- Maxwell- Weyl Generalized dxd - matrix Ernst equations

9. NxN-matrix equations and associated linear systems Vacuum: Associated linear problem Einstein-Maxwell-Weyl: String gravity models:

10. Analytical structure of on the spectral plane

11. Monodromy matrices 1) 2)

12. Monodromy data of a given solution ``Extended’’ monodromy data: Monodromy data constraint: Monodromy data for solutions of the reduced Einstein’s field equations:

13. 1) Inverse problem of the monodromy transform Free space of the monodromy data Space of solutions Theorem 1. For any holomorphic local solution near , Is holomorphic on and the ``jumps’’ of on the cuts satisfy the H lder condition and are integrable near the endpoints. posess the same properties 1) GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005

14. *) Theorem 2. For any holomorphic local solution near , possess the local structures and where are holomorphic on respectively. Fragments of these structures satisfy in the algebraic constraints (for simplicity we put here ) and the relations in boxes give rise later to the linear singular integral equations. *) In the case N-2d we do not consider the spinor field and put

15. Theorem 3. *) For any local solution of the ``null curvature'' equations with the above Jordan conditions, the fragments of the local structures of and on the cuts should satisfy where the dot for N=2d means a matrix product and the scalar kernels (N=2,3) or dxd-matrix (N=2d) kernels and coefficients are where and each of the parameters and runs over the contour ; e.g.: In the case N-2d we do not consider the spinor field and put *)

16. Theorem 4. For arbitrarily chosen extended monodromy data – the scalar functions and two pairs of vector (N=2,3) or only two pairs of dx2d and 2dxd – matrix (N=2d) functions and holomorphic respectively in some neighbor-- hoods and of the points and on the spectral plane, there exists some neighborhood of the initial point such that the solutions and of the integral equations given in Theorem 3 exist and are unique in and respectively. The matrix functions and are defined as is a normalized fundamental solution of the associated linear system with the Jordan conditions.

17. General solution of the ``null-curvature’’ equations with the Jordan conditions in terms of 1) arbitrary chosen extended monodromy data and 2) corresponding solution of the master integral equations Reduction to the space of solutions of the (generalized) Ernst equations ( ) Calculation of (generalized) Ernst potentials

18. On some known integral equation methods Solution generating methods (arbitrary seed): Riemann – Hilbert problem (V.Belinskii & V.Zakharov) Homoheneous Hilbert problems (I.Hauser & F.Ernst) Direct methods (Minkowskii seed): Inverse scattering and discrete GLM (G.Neugebauer) Scalar singular equation in terms of the axis data (N.Sibgatullin) Scalar singular equations in terms the monodromy data (GA) ``Big’’ integral equation (G.Neugebauer & R. Meinel) Scalar integral ``evolution’’ equations (GA)

19. Sibgatullin's integral equations in the monodromy transform context 1) The Sibgatullin’s reduction of the Hauser & Ernst matrix integral equations (vacuum case, for simplicity): To derive the Sibgatullin’s equations from the monodromy transform ones 1) restrict the monodromy data by the regularity axis condition: 2) chose the first component of the monodromy transform equations for . In this case, the contour can be transform as shown below: (then we obtain just the above equation on the reduces contour and the pole at gives rise to the above normalization condition)

20. Characteristic initial value problem for the hyperbolic Ernst equations 1) Analytical data:

21. 1) Integral ``evolution'' equations Boundary values for on the characteristics: Scattering matrices and their properties: 1) GA, Theor.Math.Phys. 2001

22. Dynamical monodromy data and : Derivation of the integral ``evolution’’ equations

23. Coupled system of the integral ``evolution’’ equations: Decoupled integral ``evolution’’ equations:

24. Characteristic initial value problem for colliding plane gravitational and electromagnetic waves 1) 1) GA & J.B.Griffiths, PRL 2001; CQG 2004

25. Space-time geometry and field equations Matching conditions on the wavefronts: -- are continuous

26. Initial data on the left characteristic from the left wave -- u is chosen as the affine parameter -- arbitrary functions, provided and Initial data on the right characteristic from the right wave -- v is chosen as the affine parameter -- arbitrary functions, provided and

27. Irregular behaviour of Weyl coordinates on the wavefronts Generalized integral ``evolution’’ equations (decoupled form):

28. Solution of the colliding plane wave problem in terms of the initial data

29. Monodromy data map of some classes of solutions • Solutions with diagonal metrics: static fields, waves with linear polarization: • Stationary axisymmetric fields with the regular axis of symmetry are • described by analytically matched monodromy data:: • For asymptotically flat stationary axisymmetric fields • with the coefficients expressed in terms of the multipole moments. • For stationary axisymmetric fields with a regular axis of symmetry the • values of the Ernst potentials on the axis near the point • of normalization are • For arbitrary rational and analytically matched monodromy data the • solution can be found explicitly.

30. Explicit forms of solution generating methods -- the monodromy data of arbitrary seed solution. -- the monodromy data of N-soliton solution. Belinskii-Zakharov vacuum N-soliton solution: Electrovacuum N-soliton solution: (the number of solitons) -- polynomials in of the orders

31. Map of some known solutions Minkowski space-time Symmetric Kasner space-time Rindler metric Bertotti – Robinson solution for electromagnetic universe, Bell – Szekeres solution for colliding plane electromagnetic waves Melvin magnetic universe Kerr – Newman black hole Kerr – Newman black hole in the external electromagnetic field Khan-Penrose and Nutku – Halil solutions for colliding plane gravitational waves

32. Infinite hierarchies of exact solutions Analytically matched rational monodromy data: Hierarchies of explicit solutions:

33. Schwarzschild black hole in a homogeneous electromagnetic field 1) Bipolar coordinates: Metric components and electromagnetic potential Weyl coordinates: 1) GA & A.Garcia, PRD 1996

34. Reissner - Nordstrom black hole in a homogeneous electric field Formal solution for metric and electromagnetic potential: Auxiliary polynomials:

35. Bertotti – Robinson electromagnetic universe Metric components and electromagnetic potential: Charged particle equations of motion: Test charged particle at rest:

36. Equilibrium of a black hole in the external field Balance of forces condition Regularity of space- in the Newtonian mechanics time geometry in GR

37. Black hole vs test particle The location of equilibrium position of charged black hole / test particle In the external electric field: -- the mass and charge of a black hole / test particle -- determines the strength of electric field -- the distance from the origin of the rigid frame to the equilibrium position of a black hole / test particle black hole test particle