Many “languages” of integrability
Gravitational and electromagnetic solitons
Stationary axisymmetric solitons; soliton waves
Monodromy transform approach
Solutions for black holes in the external fields
+ Addendum to Lecture 2: How to calculate …
+ Solving of the characteristic initial value problems
Colliding gravitational and electromagnetic waves
How to calculate :
monodromy data  metric end potentials
How to calculate :
metric and potentials  monodromy data
1)
In equilibrium
1) GA and V.Belinski Phys.Rev. D (2007)
The space of
local solutions:
(Constraint: field equations)
“Direct’’ problem:
“Inverse’’ problem:
Free space of the monodromy data functions:
(No constraints)
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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Map of some known solutions
Minkowski
space-time
Rindler metric
Symmetric
Kasner
space-time
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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General structure of the matrices U, V, W
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The symmetric vacuum Kazner solution is
For this solution the matrix
The monodromy data functions
takes sthe form
The simplest example of solutions arise for zero monodromy data
This corresponds to the Minkowski space-time with metrics
-- stationary axisymmetric or
with cylindrical symmetry
-- Kazner form
-- accelerated frame (Rindler metric)
The matrix
(where
for these metrics takes the following form
):
Calculation of the metric components and potentials
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Infinite hierarchies of exact solutions
Analytically matched rational monodromy data:
Hierarchies of explicit solutions:
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Inversion formulae for the Cauchy type integrals:
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NxN-matrix spectral problems
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Solving of the characteristic initial value
problems for Einstein’s field equations with
symmetries
Characteristic initial value problem for colliding
plane gravitational, electromagnetic, etc. waves
Integral “evolution” equations as a new integral
equation form of integrable reductions of
Einstein’s field equations
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1)
Analytical data:
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Irregular behaviour of Weyl coordinates on the wavefronts
Generalized integral ``evolution’’ equations (decoupled form):
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1)
1) GA & J.B.Griffiths, PRL 2001; CQG 2004
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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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1)
Boundary values for
Scattering matrices
on the characteristics:
and their properties:
1) GA, Theor.Math.Phys. 2001; GA & J.B.Griffiths, PRL 2001; CQG 2004
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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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