Gravitational and electromagnetic solitonsStationary axisymmetric solitons; soliton waves

Monodromy transform approach Solutions for black holes in the external fields

Solving of the characteristic initial value problemsColliding gravitational and electromagnetic waves

Many “languages” of integrability

2

Monodromy tarnsform approach to solution of integrable reductions of Einstein’s field equations.

Direct and inverse problems of the monodromy transform.

Monodromy data as coordinates in the space of solutions

Some applications: solutions for black holes in external gravitational and electromagnetic fields

Integral equation form of the field equations and infinite hierarchies of their solutions

3

4

Reduced dynamical equations – generalized Ernst eqs.

-- Vacuum

-- Electrovacuum

-- Einstein- Maxwell- Weyl

5

Generalized (dxd-matrix) Ernst equations for heterotic string gravity model in D dimensions:

Generalized (matrix) Ernst equations for D4 gravity model with axion, dilaton and one gauge field:

A.Kumar and K.Ray (1995); 1)

1)

D.Gal’tsov (1995), O.Kechkin, A. Herrera-Aguilar, (1998),;

1)

(No constraints)

(Constraint: field equations)

The space of local solutions:

Free space of the mono- dromy data functions:

“Direct’’ problem:

“Inverse’’ problem:

7

NxN-matrix equations and associated linear systems

Vacuum:

Einstein-Maxwell-Weyl:

String gravity models:

Associated linear problem

8

NxN-matrix equations and associated linear systems

Associated linear problem

9

Structure of the matrices U, V, W for electrovacuum

10

NxN-matrix spectral problems

11

Analytical structure of on the spectral plane

12

Monodromy matrices

1)

2)

13

Monodromy data of a given solution

``Extended’’ monodromy data:

Monodromy data for solutions of thereduced Einstein’s field equations:

Monodromy data constraint:

Monodromy data of a given solution

Monodromy data for solutions of reduced Einstein’s field equations:

Einstein – Maxwell fields:

The symmetric vacuum Kazner solution is

For this solution the matrix takes sthe form

The monodromy data functions

The simplest example of solutions arise for zero monodromy data

-- stationary axisymmetric or cylindrical symmetry

-- Kazner form

-- accelerated frame (Rindler metric)

This corresponds to the Minkowski space-time with metrics

The matrix for these metrics takes the following form (where ):

Generic data: Analytically matched data:

Unknowns:

18

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.

19

Map of some known solutions

Minkowski 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

SymmetricKasner space-time

Khan-Penrose and Nutku – Halil solutions for colliding plane gravitational waves

20

Explicit forms of soliton generating transformations

-- the monodromy data of arbitrary seed solution.

Belinskii-Zakharov vacuum N-soliton solution:

Electrovacuum N-soliton solution:

-- polynomials in of the orders (the number of solitons)

-- the monodromy data of N-soliton solution.

21

Free space of themonodromy data

Space of solutions

For any holomorphic local solution near ,Theorem 1.

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

GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005 1)

1)

22

*) For any holomorphic local solution near ,Theorem 2.possess the local structures

Fragments of these structures satisfy in the algebraic constraints

and

and the relations in boxes give rise later to the linear singular integral equations.

(for simplicitywe put here

)

where are holomorphic on respectively.

In the case N-2d we do not consider the spinor field and put *)

23

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 *)

*)

24

Theorem 4. For arbitrarily chosen extended monodromy data – the scalar functions and two pairs of vector (N=2,3) or only twopairs 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.

25

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

27

Calculation of the metric components and potentials

--- the solution can be found explicitly

Auxiliary polynomial

Auxiliary polynomial

Auxiliary functions

Solution of the integral equation and the matrix

31

Infinite hierarchies of exact solutions

Analytically matched rational monodromy data:

Hierarchies of explicit solutions:

Equilibrium configurations of two Reissner – Nordstrom sources

Schwarzschild black hole in a static position in a homogeneous electromagnetic field

GA and V.Belinski Phys.Rev. D (2007)1)

In equilibrium

1)

34

The background space-time with homogeneous electric field (Bertotti – Robinson solution):

Schwarzschild black hole in a static position in a homogeneous electromagnetic field

Schwarzschild black hole in a static position in a homogeneous electromagnetic field

Bipolar coordinates:

Metric components and electromagnetic potential

Weyl coordinates:

GA & A.Garcia, PRD 1996

1)

1)

36

Global structure of a solution for a Schwarzschild blck hole in the Bertotti – Robinson universe