Solving Einstein's field equations

1
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
Addendum to Lecture 2 How to calculate
Solving of the characteristic initial value
problems Colliding gravitational and
electromagnetic waves
Lecture 3
2
Addendum to the Lecture 2
How to calculate monodromy data ?
metric end potentials
How to calculate metric and potentials ?
monodromy data
3
1)
Equilibrium configurations of two Reissner -
Nordstrom sources
In equilibrium
1)
GA and V.Belinski Phys.Rev. D (2007)
4
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)
5
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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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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Monodromy data as the coordinates in the space of
solutions
"Direct" problem linear partial-diff.equations
"Inverse" problem linear singular Integral
equations
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General structure of the matrices U, V, W
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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
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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 with cylindrical
symmetry
-- Kazner form
-- accelerated frame (Rindler metric)
The matrix for these metrics
takes the following form (where

)
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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
13
Inversion formulae for the Cauchy type integrals
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NxN-matrix spectral problems
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Lecture 3
Solving of the characteristic initial value
problems for Einsteins 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
Einsteins field equations
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Characteristic initial value problem for the
hyperbolic Ernst equations
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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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
21
1)
Integral evolution'' equations
Boundary values for on the
characteristics

Scattering matrices and their
properties
GA, Theor.Math.Phys. 2001 GA J.B.Griffiths,
PRL 2001 CQG 2004
1)
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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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Solution of the colliding plane wave problem in
terms of the initial data