1 Steklov Mathematical Institute RAS G. Alekseev G. Alekseev Cosmological solutions Dynamics of waves Fields of accelerated sources Stationary axisymmetric fields

2 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 = const z z e, m

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4 2-dimensional reduction of string effective action Generalized (matrix) Ernst equations: -- Vacuum -- Einstein-Maxwell -- String gravity models

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 for solutions of reduced Einstein’s field equations: Monodromy data constraint:

9 GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys )

10 Generic data:Analytically matched data: Unknowns: Rational, analytically matched data:

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13 Map of some known solutions Minkowski 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 Symmetric Kasner space-time Khan-Penrose and Nutku – Halil solutions for colliding plane gravitational waves

14 The laws of motion in Newton’s and Einstein’s theories Newton gravity: General Relativity: Geometry  the laws of motion t x t x

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 Bipolar coordinates: Metric components and electromagnetic potential Weyl coordinates: GA & A.Garcia, PRD )

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