Integrable Reductions of Einstein

1
Integrable Reductions of Einsteins Equations
Hierarchies of solutions with rational monodromy
data and their applications for a black
hole dynamics
G. Alekseev
Stiff matter
Weyl spinor field
Axion - dilaton gravity
String gravity models
Dynamics of waves
Cosmological solutions
Einstein - Maxwell fields
Fields of accelerated sources

Stationary axisymmetric fields
Gravitational fields in vacuum
Steklov Mathematical Institute RAS
2
Plan of the talk
Integrable reductions of Einsteins field
equations
Monodromy transform (direct and inverse problems)
Hierarchies of solutions with analytically
matched, rational monodromy data
Application to a black hole dynamics Charged
black hole accelerated by an external electric
field
3
Gravitational fields in vacuum
Elektrovacuum Einstein - Maxwell fields
Bosonic sector of heterotic string effective
action
4
2-dimensional reduction of string effective action
-- Vacuum
-- Einstein-Maxwell
-- String gravity models
Generalized (matrix) Ernst equations
5
NxN-matrix equations and associated linear
systems
Associated linear problem
6
NxN-matrix spectral problems
7
Analytical structure of on the
spectral plane
8
Monodromy data of a given solution
Extended monodromy data
Monodromy data constraint
Monodromy data for solutions of reduced
Einsteins field equations
9
1)
Inverse problem of the monodromy transform
1)
GA, Sov.Phys.Dokl. 1985Proc. Steklov Inst. Math.
1988 Theor.Math.Phys. 2005
10
Solutions for analitically matched, rational
monodromy data
Generic data
Analytically matched data
Unknowns
Rational, analytically matched data
11
Modified integral equation
Auxiliary polynomials
Auxiliary functions
12
Solution of the integral equation
Calculation of the solution components
Determinant form of solution
13
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
14
The laws of motion in Newtons and Einsteins
theories
Newton gravity
Geometry ? the laws of motion
General Relativity
15
Space-time with homogeneous electric field
(Bertotti Robinson solution)

• Metric components and electromagnetic potential
• Charged particle equations of motion
• Test charged particle at rest

Neutral test particle

Charged test particle

16
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
17
Schwarzschild black hole in a geodesic motion
in a homogeneous electromagnetic field
1)
18
Reissner - Nordstrom black hole in a homogeneous
electric field

• Formal solution for metric and electromagnetic
  potential

Auxiliary polynomials
19
Equilibrium of a black hole in the external field

• Balance of forces condition
  Regularity of space-
• in the Newtonian mechanics time
  geometry in GR

20
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