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Integrable Reductions of the Einsteins

Field Equations

- Monodromy transform approach
- and integral equation methods

Harry-Dym

G. Alekseev

Davey - Stewartson

Kadomtsev-Petviashvili

SU(2) YM

Nonlinear Schrodinger

Sine-Gordon

Korteveg de Vries

Steklov Mathematical Institute RAS

Integrable reductions of the Einsteins field

equations

Hyperbolic reductions (waves, cosmologocal

models)

Elliptic reductions (Stationary fields with

spatial symmetry)

Vacuum

axion, dilaton,...

Electrovacuum

Weyl spinor field

stiff matter

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

Integrability and the solution space transforms

Free space of func- tional parameters

Space of solutions

(No constraints)

(Constraint field equations)

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

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)

Einsteins equations with integrable reductions

-- Vacuum

-- Electrovacuum

-- Einstein Maxwell Weyl

Effective string gravity equations

Space-time symmetry ansatz

Coordinates

Space-time metric

2-surface-orthogonal orbits of isometry group

Generalized Weyl coordinates

Geometrically defined coordinates

Reduced dynamical equations generalized Ernst

eqs.

-- Vacuum

-- Electrovacuum

-- Einstein- Maxwell-

Weyl

Generalized dxd - matrix Ernst equations

NxN-matrix equations and associated linear

systems

Vacuum

Associated linear problem

Einstein-Maxwell-Weyl

String gravity models

Associated NxN-matrix spectral problems

Vacuum

Einstein- Maxwell- Weyl

String gravity

Associated 2dx2d-spectral problem for string

gravity

(a)

(b)

Analytical structure of on the

spectral plane

Some definitions used above

Local domains

Characteristic scalar functions

Weyl spinor field potentials

Algebraic constraints on the fragments of local

structure of on the cuts

(see the theorems below)

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

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

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)

Sibgatullin's integral equations in the monodromy

transform context

1)

The Sibgatullins reduction of the Hauser Ernst

matrix integral equations (vacuum case, for

simplicity)

To derive the Sibgatullins 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)

Characteristic initial value problem for the

hyperbolic Ernst equations

1)

Analytical data

1)

Integral evolution'' equations

Boundary values for on the

characteristics

Scattering matrices and their

properties

1)

GA, Theor.Math.Phys. 2001

Dynamical monodromy data and

Derivation of the integral evolution equations

Coupled system of the integral evolution

equations

Decoupled integral evolution equations

Characteristic initial value problem for

colliding plane gravitational and

electromagnetic waves

1)

GA J.B.Griffiths, PRL 2001 CQG 2004

1)

Space-time geometry and field equations

Matching conditions on the wavefronts

-- are continuous

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

Irregular behaviour of Weyl coordinates on the

wavefronts

Generalized integral evolution equations

(decoupled form)

Solution of the colliding plane wave problem in

terms of the initial data

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

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

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

Infinite hierarchies of exact solutions

- Analytically matched rational monodromy data

Hierarchies of explicit solutions

Schwarzschild black hole in a homogeneous

electromagnetic field

1)

Bipolar coordinates Metric components and

electromagnetic potential Weyl coordinates

1)

GA A.Garcia, PRD 1996

Reissner - Nordstrom black hole in a homogeneous

electric field

- Formal solution for metric and electromagnetic

potential

Auxiliary polynomials

Bertotti Robinson electromagnetic universe

- Metric components and electromagnetic potential
- Charged particle equations of motion
- Test charged particle at rest

Equilibrium of a black hole in the external field

- Balance of forces condition

Regularity of space- - in the Newtonian mechanics time

geometry in GR

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