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 Stationary axisymmetric solitons

soliton waves

Lecture 1

Monodromy transform approach Solutions for black

holes in the external fields

Lecture 2

Solving of the characteristic initial value

problems Colliding gravitational and

electromagnetic waves

Lecture 3

Monodromy tarnsform approach to solution of

integrable reductions of Einsteins field

equations.

Lecture 2

Monodromy data as coordinates in the space of

solutions

Direct and inverse problems of the monodromy

transform.

Integral equation form of the field equations

and infinite hierarchies of their solutions

Some applications solutions for black holes in

external gravitational and electromagnetic fields

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

Reduced dynamical equations generalized Ernst

eqs.

-- Vacuum

-- Electrovacuum

-- Einstein- Maxwell-

Weyl

Generalized (matrix) Ernst equations for D4

gravity model with axion, dilaton and one gauge

field

1)

Generalized (dxd-matrix) Ernst equations for

heterotic string gravity model in D dimensions

1)

1)

A.Kumar and K.Ray (1995)

D.Galtsov (1995), O.Kechkin, A.

Herrera-Aguilar, (1998),

Monodromy Transform approach to solving of

Einstein's equations

Free space of the mono- dromy data functions

The space of local solutions

(No constraints)

(Constraint field equations)

Direct problem

(linear ordinary differential equations)

Inverse problem

(linear integral equations)

NxN-matrix equations and associated linear

systems

Vacuum

Associated linear problem

Einstein-Maxwell-Weyl

String gravity models

NxN-matrix equations and associated linear

systems

Associated linear problem

Structure of the matrices U, V, W for

electrovacuum

NxN-matrix spectral problems

Analytical structure of on the

spectral plane

Monodromy matrices

1)

2)

Monodromy data of a given solution

Extended monodromy data

Monodromy data constraint

Monodromy data for solutions of the reduced

Einsteins field equations

Monodromy data of a given solution

Einstein Maxwell fields

Example for solution with none-matched monodromy

data

The symmetric vacuum Kazner solution is For

this solution the matrix

takes sthe form

The monodromy data functions

Examples for solutions with analytically matched

monodromy data

The simplest example of solutions arise for zero

monodromy data

This corresponds to the Minkowski space-time with

metrics

-- stationary axisymmetric or

cylindrical symmetry

-- Kazner form

-- accelerated frame (Rindler metric)

The matrix for these metrics

takes the following form (where

)

Generic and analytically matched monodromy data

Generic data

Analytically matched data

Unknowns

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

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

Explicit forms of soliton generating

transformations

-- 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

1)

Inverse problem of the monodromy transform

Free space of the monodromy 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

1)

GA, Sov.Phys.Dokl. 1985Proc. Steklov Inst. Math.

1988 Theor.Math.Phys. 2005

)

For any holomorphic local solution

near ,

Theorem 2.

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

)

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 N2d means a matrix product and

the scalar kernels (N2,3) or dxd-matrix (N2d)

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

)

Theorem 4.

For arbitrarily chosen extended monodromy data

the scalar functions and two pairs

of vector (N2,3) or only two pairs of dx2d and

2dxd matrix (N2d) 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.

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

"Direct" problem linear partial-diff.equations

Monodromy data as the coordinates in the space of

solutions

"Inverse" problem linear singular Integral

equations

Calculation of the metric components and

potentials

Analitically matched rational monodromy data

--- the solution can be found explicitly

Auxiliary polynomial

Auxiliary polynomial

Auxiliary functions

Solution of the integral equation and the matrix

Infinite hierarchies of exact solutions

- Analytically matched rational monodromy data

Hierarchies of explicit solutions

Some applications

Equilibrium configurations of two Reissner

Nordstrom sources

Schwarzschild black hole in a static position in

a homogeneous electromagnetic field

1)

Equilibrium configurations of two Reissner -

Nordstrom sources

In equilibrium

1)

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

Schwarzschild black hole in a static position in

a homogeneous electromagnetic field

The background space-time with homogeneous

electric field (Bertotti Robinson solution)

Schwarzschild black hole in a static position in

a homogeneous electromagnetic field

1)

Bipolar coordinates Metric components and

electromagnetic potential Weyl coordinates

1)

GA A.Garcia, PRD 1996

Global structure of a solution for a

Schwarzschild blck hole in the Bertotti

Robinson universe