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Gravity World

View from the bulk…. Gravity World. GAA. Solitons, Collapses and Turbulence. Равнина облаков -- как океан, Когда зимой его недвижны льдины, Необозримей всех бескрайних стран Сменяет он застывшие картины. (В.Е.Захаров, "Перед небом"). GAA. V.E.Zakharov.

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Gravity World

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  1. View from the bulk… Gravity World GAA

  2. Solitons, Collapses and Turbulence Равнина облаков -- как океан, Когда зимой его недвижны льдины, Необозримей всех бескрайних стран Сменяет он застывшие картины ... (В.Е.Захаров, "Перед небом") GAA V.E.Zakharov

  3. Thirty years of solitons in General Relativity Zakharov-70 First integrability conjectures: -- R.Geroch –conjecture (1972) -- W.Kinnersley&D.Citre –inf. dim. algebra of symmetries (1977…) -- D.Maison - Lax pair +conjecture (1978) Vacuum: Symmetries: V.Belinski and V.Zakharov (1978) Integrability began to work: -- Inverse Scattering Method -- Soliton solutions on arbitrary backgrounds -- Riemann – Hilbert problem -- linear singular integral equations Many "faces" of integrability: -- Backlund and symm. transformations(K.Harrison 1978, G.Neugebauer 1979, HKX 1979) -- Homogeneous Hilbert problem (I.Hauser & F.J.Ernst, 1979 + N.Sibgatullin 1984) -- Monodromy transform + linear singular integral equations (GA 1985) -- Finite-gap solutions (D.Korotkin&V.Matveev 1987, G.Neugebauer&R.Meinel 1993) -- Boundary value problem for stationary fields (G.Neugebauer &R.Meinel 1996) -- Charateristic init. value probl.(I.Hauser &F.J.Ernst 1988;GA 2001; GA&J.Griffiths 2001) (M.Cosgrove 1980, D.Kramer 1981) Interrelations between different approaches

  4. Electrovacuum Einstein - Maxwell fields -- Infinite-dimensional algebra of symmetries (W.Kinnersley & D.Chitre 1977, …) -- Homogeneous Hilbert problem and singular integral equations for axisymm. stationary fields with regular axis (I.Hauser & F.J.Ernst 1979 + N.Sibgatullin 1984) -- Inverse scattering method and Einstein – Maxwell solitons (GA 1980) -- Backlund transformations (K.Harrison 1983) -- Monodromy Transform and linear singular integral equations (GA 1985) -- Charateristic initial value problem (GA 2001; GA & J.Griffiths 2001, 2003) Einstein - Maxwell + Weyl neutrino fields: -- Inverse scattering method (GA 1983) -- Generalization of the Hauser-Ernst approach (N.Sibgatullin 1984) -- Monodromy transform approach and linear singular integral equations (GA 1985) Gravity + stiff matter fluid -- Inverse scattering method (V.Belinski 1979) Integrable symmetry reduced string gravity models -- Vacuum equations in higher dimensions (V.Belinski & R.Ruffini 1980, A.Pomeranski 2006) -- D=4 gravity with axion and dilaton (Bakas 1996); D=4 EMDA (D.Gal’tsov, P.Letelier 1996) -- gravity coupled bosonic dynamics in D-dim. heterotic string effective action (GA 2009) -- D>4 generalized (matrix) Ernst equations in EMDA gravity model (GA 2005)

  5. Applications Solitons on arbitrary background: -- Colliding plane waves (Khan&Penrose 1972, Y.Nutku&Khalil) -- Inhomogeneous cosmologies (V.Belinski 1979) -- Interacting black holes: 2 x Kerr (D.Kramer&G.Neugebauer 1980), 2 x Kerr-Newman (GA 1986) 2 x Reisner-Nordstrom (GA&V.Belinski 2007) -- Black holes in external fields: in Melvin universe (F.Ernst 1975), in Bertotti-Robinson space-time (GA&A.Garcia 1996) -- … … … ? D=4 -- black holes with non-simple rotation (A.Pomeransky 2006) -- black rings (R.Emparan & H.S.Reall, A.Pomeransky & R.Sen’kov) -- black Saturn (H. Elvang & P. Figueras 2007) -- … … …? D=5 Algebro-geometrical methods (D=4): -- Finite-gap solutions for hyperelliptic curves (D.Korotkin & V.Matveev 1987) -- Solution for rigidly rotating thin disk of dust (G.Neugebauer & R.Meinel) Integral equation methods, boundary and initial value problems (D=4): -- Solutions with rational monodromy (GA 1988,1992; N.Sibgatullin 1993;GA & J.Griffiths 2000) -- Boundary value problems for stationary axisymm. fields (G.Neugebauer&R.Meinel 1996) -- Characteristic initial value problems (I.Hauser&F.Ernst 1987; GA & J.Griffiths 2001) -- Waves created by accelerated sources (C-metrics)

  6. Different types of field configurations G. Alekseev Steklov Mathematical Institute RAS, Moscow

  7. Integrability and Soliton Generating Transformations in the Low Energy String Gravity Theories G. Alekseev Steklov Mathematical Institute RAS, Moscow Bosonic sector of heterotic string effective action: The symmetry ansatz:

  8. Bosonic action in the Einstein frame: Field equations:

  9. Dynamical degrees of freedom: Conformal factor: Geometrically defined coordinates and :

  10. Matrix Ernst-like dynamical variables: Matrix Ernst-like form of the dynamical equations Examples for the choice of -- stationary axisymmetric fields -- colliding plane waves -- cosmological solutions

  11. 1) Belinski - Zakharov inverse scattering approach Dynamical equations for vacuum Associated spectral problem Matrix integral “Dressing” method for constructing of solitons: 1) V.Belinski & V.Zakharov,, JETP 1978; 1979 ;

  12. ISM for non-vacuum fields in D dimensions d x d - spectral problem for vacuum (Belinski & Zakharov ) 2d x 2d spectral problem for vacuum Non-vacuum spectral problem

  13. Transformation to self-dual form of the linear system 2d x 2d spectral problem (BZ-like) Transformation: 2d x 2d spectral problem (self-dual)

  14. (2d+n)x(2 d+n)-matrix equations

  15. Associated linear system for N x N matrices Coordinates:

  16. Spectral problem equivalent to dynamical equations

  17. Soliton generating transformations: "d-rank" solitons Dressing procedure Calculation of the field components in terms of te matrix Structure of the matrix "Constants of integration" and :

  18. Soliton generating transformations "confluent" rank=1 solitons Dressing procedure Calculation of the field components in terms of te matrix Structure of the matrix

  19. Spectral problem for N x N - matrices

  20. Inverse problem of the monodromy transform Monodromy matrices: Monodromy data: Linear singular integral equations: Integral representation of the solution:

  21. Equilibrium configurations of two charged masses In equilibrium: 1) GA and V.Belinski Phys.Rev. D (2007)

  22. Charged black hole accelerated by a homogeneous external electric field

  23. Thank you GAA

  24. Vacuum solitons of Belinski and Zakharov ( - solitons ): Electrovacuum solitons (w - solitons ): Comparison of - solitons and w-solitons for vacuum: 1) N w - solitons2 N- solitons with complex poles !? 2) w -solitons with real polesdo not arise in this technique

  25. 1) Inverse scattering approach to Einstein - Maxwell fields Dynamical equations Associated spectral problem Matrix integral “Dressing” method for constructing of solitons: 1) GA, JETP Lett 1980

  26. View from the bulk… Gravity World GAA

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