Integrable reductions of Einstein's field equations: monodromy transform and the linear integral equation methods
Seminar Room 1, Newton Institute
For each of the known today integrable reductions of Einstein's field equations for space-times with two commuting isometries, the monodromy transform (similarly to the well known Inverse Scattering Transform applied successfully for many other completely integrable equations) provides us with a unified and convenient mapping of the complete space of local solutions of the symmetry reduced field equations in terms of a finite set of unconstrained coordinate-independent functions of the spectral parameter (analogous to the scattering data). These set of functions arises as the monodromy data for the fundamental solution of associated linear systems (``spectral problems'') and they can serve as free independent ``coordinates'' in the infinite dimensional space of the local solutions. The direct and inverse problems of such ``coordinate transformation'', (monodromy transform), i.e. the problems of calculation of the monodromy data for given solution of the field equations and of calculation of the solution, corresponding to given monodromy data, possess unique solutions. In principle, the monodromy data functions can be calcul ated also from some boundary, or initial, or characteristic initial data for the fields, and many physical properties of solutions are simply ``encoded'' in the analytical structures of these functions. However, to find the solutions of the mentioned above direct and inverse problems, we have to solve explicitly the systems of ordinary differential and linear singular integral equations respectively, that can occur a difficult problems in many cases.
In the introduction we give a short survey of various integrable symmetry reductions of Einstein's field equations and mention some interrelationships between various developed linear integral equation methods. We describe also in a unified manner the common structure of various integrable reductions of Einstein's field equations -- the (generalized) hyperbolic and elliptic Ernst equations for vacuum and electrovacuum space-times, for Einstein - Maxwell - Weyl fields, for stiff matter fluids as well as their matrix generalizations for some string gravity models with coupled gravity and dilaton, axion and Abelian vector fields. The structure of the direct problem of the monodromy transform and general construction of the linear singular integral equation solving the inverse problem will be considered and some applications of this approach for construction of infinite hierarchies of exact solutions will be presented. In this context we present also another linear integral equation forms of integrable hyperbolic symmetry reductions of Einstein's field equations which provides a solution (viz. linearization) of the characteristic initial value problems for colliding waves and for evolution of inhomogeneous cosmological models.