Transcript Integrable Reductions of Einstein’s Field Equations

G. Alekseev
Dynamics of waves
Cosmological solutions
Fields of accelerated sources
Stationary axisymmetric fields
Steklov Mathematical Institute RAS
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Plan of the talk
Integrable reductions of Einstein’s 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
e, m
E = const
z
E = const
z
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2-dimensional reduction of string effective action
-- Vacuum
-- Einstein-Maxwell
-- String gravity models
Generalized (matrix) Ernst equations:
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NxN-matrix equations and associated linear systems
Associated linear problem
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NxN-matrix spectral problems
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Analytical structure of
on the spectral plane
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Monodromy data of a given solution
``Extended’’ monodromy data:
Monodromy data constraint:
Monodromy data
for solutions of reduced
Einstein’s field equations:
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1)
1)
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GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005
Generic data:
Analytically matched data:
Unknowns:
Rational, analytically matched data:
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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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The laws of motion in Newton’s and Einstein’s theories
Newton gravity:
General Relativity:
Geometry  the laws of motion
t
t
x
x
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Space-time with homogeneous electric field
(Bertotti – Robinson solution)
Metric components and electromagnetic potential:
Neutral test particle
Charged particle equations of motion:
Charged test particle
Test charged particle at rest:
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Schwarzschild black hole in a static position
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in a homogeneous electromagnetic
field
Bipolar coordinates:
Metric components and electromagnetic potential
Weyl coordinates:
1) GA & A.Garcia, PRD 1996
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Schwarzschild black hole
in a “geodesic motion”
1)
in a homogeneous electromagnetic field
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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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Equilibrium of a black hole in the external field
Balance of forces condition
in the Newtonian mechanics
Regularity of spacetime 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
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