Integrable Reductions of the Einsteins Field Equations

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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
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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
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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

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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
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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)

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Einsteins 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
Associated linear problem
Einstein-Maxwell-Weyl
String gravity models
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Associated NxN-matrix spectral problems
Vacuum
Einstein- Maxwell- Weyl
String gravity
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Associated 2dx2d-spectral problem for string
gravity
(a)
(b)
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Analytical structure of on the
spectral plane
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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)
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Monodromy matrices
1)
2)
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Monodromy data of a given solution
Extended monodromy data
Monodromy data constraint
Monodromy data for solutions of the reduced
Einsteins field equations
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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
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)
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
)
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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 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
)
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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.
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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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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)
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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)
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Characteristic initial value problem for the
hyperbolic Ernst equations
1)
Analytical data

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1)
Integral evolution'' equations
Boundary values for on the
characteristics

Scattering matrices and their
properties
1)
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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Characteristic initial value problem for
colliding plane gravitational and
electromagnetic waves
1)

GA J.B.Griffiths, PRL 2001 CQG 2004
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

Initial data on the right characteristic from the
right wave
-- v is chosen as the affine parameter
-- arbitrary functions, provided
and
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Irregular behaviour of Weyl coordinates on the
wavefronts
Generalized integral evolution equations
(decoupled form)

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

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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

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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
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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
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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
1)
Bipolar coordinates Metric components and
electromagnetic potential Weyl coordinates
1)
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 hole
test particle