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Integrable Reductions of the Einstein’s Field Equations

Monodromy transform approach and integral equation methods

G. G. AlekseevAlekseev

Steklov Mathematical Institute RAS

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Hyperbolic reductions:(waves, cosmologocal models)

Elliptic reductions(Stationary fields with

spatial symmetry)

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• 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

Many faces of integrability

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Integrability and the solution space transforms

Free space of func-tional parameters

(No constraints)

(Constraint: field equations)

Applications:• Solution generating methods• Infinite hierarchies of exact solutions• ``Partial’’ superposition of fields• Initial/boundary value problems• Asymptotic behaviour

“Direct’’ problem:

Space of solutions

“Inverse’’ problem:

(linear ordinary differential equations)

(linear singular integral equations)

Monodromy transform:

Monodromy data

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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)

Plan of the talk

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Einstein’s equations with integrable reductions

-- Vacuum

-- Electrovacuum

-- Einstein Maxwell Weyl

Effective string gravity

equations

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Space-time symmetry ansatz

Coordinates:

Space-time metric:

2-surface-orthogonal orbits of isometry group:

Generalized Weyl coordinates :

Geometrically defined coordinates :

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Reduced dynamical equations – generalized Ernst eqs.

-- Vacuum

-- Electrovacuum

-- Einstein- Maxwell- Weyl

Generalized dxd - matrix

Ernst equations

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NxN-matrix equations and associated linear systems

Vacuum:

Einstein-Maxwell-Weyl:

String gravity models:

Associated linear problem

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Analytical structure of on the spectral plane

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Monodromy matrices

1)

2)

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Monodromy data of a given solution

``Extended’’ monodromy data:

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

Monodromy data constraint:

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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)

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*) 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 *)

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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 *)

*)

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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.

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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

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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)

Scalar integral ``evolution’’ equations (GA)

``Big’’ integral equation (G.Neugebauer & R. Meinel)

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1)

To derive the Sibgatullin’s equations from the monodromy transform ones

The Sibgatullin’s reduction of the Hauser & Ernst matrix integral equations (vacuum case, for simplicity):

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)

Sibgatullin's integral equations in the monodromy transform context

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1)

Analytical data:

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1)

1)

Boundary values for on the characteristics:

Scattering matrices and their properties:

GA, Theor.Math.Phys. 2001

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Dynamical monodromy data and :

Derivation of the integral ``evolution’’ equations

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Coupled system of the integral ``evolution’’ equations:

Decoupled integral ``evolution’’ equations:

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GA & J.B.Griffiths, PRL 2001; CQG 2004 1)

1)

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Space-time geometry and field equations

Matching conditions on the wavefronts:

-- are continuous

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Initial data on the left characteristic from the left wave

-- u is chosen as the affine parameter

-- arbitrary functions, provided and

-- v is chosen as the affine parameter

-- arbitrary functions, provided

and

Initial data on the right characteristic from the right wave

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Irregular behaviour of Weyl coordinates on the wavefronts

Generalized integral ``evolution’’ equations (decoupled form):

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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.

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Explicit forms of solution generating methods

-- 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.

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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

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Infinite hierarchies of exact solutions

Analytically matched rational monodromy data:

Hierarchies of explicit solutions:

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Schwarzschild black hole in a homogeneous electromagnetic field

Bipolar coordinates:

Metric components and electromagnetic potential

Weyl coordinates:

GA & A.Garcia, PRD 1996

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Reissner - Nordstrom black hole in a homogeneous electric field

Formal solution for metric and electromagnetic potential:

Auxiliary polynomials:

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Bertotti – Robinson electromagnetic universe

Metric components and electromagnetic potential:

Charged particle equations of motion:

Test charged particle at rest:

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Equilibrium of a black hole in the external field

Balance of forces condition Regularity of space-

in the Newtonian mechanics time geometry in GR

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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 holetest particle