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Gravity World

Gravity World. GAA. Gallipoli 2008. GAA. Nonlinear phenomena in strong gravitational fields. with two-dimensional symmetries. Vacuum. Symmetries:. --- Non-vacuum integrable reductions of Einstein’s field equations. Thirty years of solitons in General Relativity:. Integrability - ?.

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Gravity World

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  1. Gravity World GAA

  2. Gallipoli 2008 GAA

  3. Nonlinear phenomena in strong gravitational fields with two-dimensional symmetries

  4. Vacuum Symmetries: --- Non-vacuum integrable reductions of Einstein’s field equations Thirty years of solitons in General Relativity: Integrability - ? -- R.Geroch –conjecture (1972) -- W.Kinnersley –inf.dim. algebra (1977…) -- D.Maison - Lax pair +conjecture (1978) Belinski and Zakharov (1978)-- Inverse Scattering Method -- Soliton solutions on arbit. backgr. -- Riemann – Hilbert problem + linear singular integral equations Later results: -- Backlund transformations (Harrison 1978, Neugebauer 1979) -- Homogeneous Hilbert problem (Hauser & Ernst, 1979+Sibgatullin 1984) -- Monodromy Transform + linera singular integral equations (GA 1985) -- Finite-gap solutions (Korotkin&Matveev 1987, Neugebauer&Meinel 1993) -- Charateristic initial value problem (GA & Griffiths 2001)

  5. Complete integrability and monodromy preserving structure of the symmetry reduced heterotic string effective equations in any dimensions Gallipoli 2008 G. Alekseev Bosonic sector of heterotic string effective action: The symmetry ansatz:

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

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

  8. Matrix Ernst-like dynamical variables: Matrix Ernst-like form of the dynamical equations Examples for the choice of

  9. Null-curvature representation for (2d+n)x(2 d+n)-matrices

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

  11. Spectral problem for N x N - matrices

  12. Equivalence of the spectral problem to dynamical equations

  13. Spectral problem for N x N - matrices

  14. 1) Monodromy transform approach Analytical structure of on w – plane: GA, Sov. Phys (1985) ; 1)

  15. Analytical structure of on the spectral plane

  16. Monodromy data for solutions: Monodromy data of a given solution

  17. 1) Inverse problem of the monodromy transform 1) GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005

  18. Solutions for analitically matched, rational monodromy data Generic data: Analytically matched data: Unknowns: Rational, analytically matched data:

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

  20. 1) Inverse problem of the monodromy transform Free space of the monodromy data Space of solutions Theorem 1. For any holomorphic local solution near , 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. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005

  21. *) Theorem 2. For any holomorphic local solution near , 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

  22. 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 N=2d means a matrix product and the scalar kernels (N=2,3) or dxd-matrix (N=2d) 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 *)

  23. Theorem 4. For arbitrarily chosen extended monodromy data – the scalar functions and two pairs of vector (N=2,3) or only two pairs of dx2d and 2dxd – matrix (N=2d) 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.

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