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
Solving of the characteristic initial value
problems Colliding gravitational and
electromagnetic waves
Lecture 3
2
Monodromy tarnsform approach to solution of
integrable reductions of Einsteins field
equations.
Lecture 2
Monodromy data as coordinates in the space of
solutions
Direct and inverse problems of the monodromy
transform.
Integral equation form of the field equations
and infinite hierarchies of their solutions
Some applications solutions for black holes in
external gravitational and electromagnetic fields

3
Integrable reductions of the Einstein's field
equations
Gravitational fields in vacuum
Elektrovacuum Einstein - Maxwell fields
Gravity model with axion, dilaton and E-H fields
Bosonic sector of heterotic string effective
action
4
Reduced dynamical equations generalized Ernst
eqs.
-- Vacuum
-- Electrovacuum
-- Einstein- Maxwell-
Weyl
5
Generalized (matrix) Ernst equations for D4
gravity model with axion, dilaton and one gauge
field
1)
Generalized (dxd-matrix) Ernst equations for
heterotic string gravity model in D dimensions
1)
1)
A.Kumar and K.Ray (1995)
D.Galtsov (1995), O.Kechkin, A.
Herrera-Aguilar, (1998),
6
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)
7
NxN-matrix equations and associated linear
systems
Vacuum
Associated linear problem
Einstein-Maxwell-Weyl
String gravity models
8
NxN-matrix equations and associated linear
systems
Associated linear problem
9
Structure of the matrices U, V, W for
electrovacuum
10
NxN-matrix spectral problems
11
Analytical structure of on the
spectral plane
12
Monodromy matrices
1)
2)
13
Monodromy data of a given solution
Extended monodromy data
Monodromy data constraint
Monodromy data for solutions of the reduced
Einsteins field equations
14
Monodromy data of a given solution
Einstein Maxwell fields
15
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
16
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
cylindrical symmetry
-- Kazner form
-- accelerated frame (Rindler metric)
The matrix for these metrics
takes the following form (where

)
17
Generic and analytically matched monodromy data
Generic data
Analytically matched data
Unknowns
18
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

19
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
20
Explicit forms of soliton generating
transformations
-- 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
21
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
22
)
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
)
23
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
)
24
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.
25
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
26
"Direct" problem linear partial-diff.equations
Monodromy data as the coordinates in the space of
solutions
"Inverse" problem linear singular Integral
equations
27
Calculation of the metric components and
potentials
28
Analitically matched rational monodromy data
--- the solution can be found explicitly
29
Auxiliary polynomial
Auxiliary polynomial
30
Auxiliary functions
Solution of the integral equation and the matrix
31
Infinite hierarchies of exact solutions

• Analytically matched rational monodromy data

Hierarchies of explicit solutions
32
Some applications
Equilibrium configurations of two Reissner
Nordstrom sources
Schwarzschild black hole in a static position in
a homogeneous electromagnetic field
33
1)
Equilibrium configurations of two Reissner -
Nordstrom sources
In equilibrium
1)
GA and V.Belinski Phys.Rev. D (2007)
34
Schwarzschild black hole in a static position in
a homogeneous electromagnetic field
The background space-time with homogeneous
electric field (Bertotti Robinson solution)
35
Schwarzschild black hole in a static position in
a homogeneous electromagnetic field
1)
Bipolar coordinates Metric components and
electromagnetic potential Weyl coordinates
1)
GA A.Garcia, PRD 1996
36
Global structure of a solution for a
Schwarzschild blck hole in the Bertotti
Robinson universe