Abstract:
The fundamental character of Einstein's field equations is provided by not only beautiful physical content of these equations which the modern gravitational theory is based on, but also by their very rich mathematical structure which (been analyzed carefully and with minimal simplifying assumptions) can lead to discoveries of new interesting structures associated with these equations and even show some most appropriate ways for their analysis and solution. The illustration of this in the theory of integrable reductions of Einstein's field equations is the aim of the present talk.
During the last three and a half decades it was found that the Einstein's field equations in a number of physically important cases such as the Einstein equations for vacuum, electrovacuum Einstein–Maxwell equations, the Einstein–Maxwell–Weyl equations, as well as the equations, which govern the dynamics of the bosonic sectors of some string gravity and supergravity models, are completely integrable on the only assumption that in $D$-dimensional space-time all fields and their potentials depend only on two coordinates, i.e. the space-times and field configurations admit the Abelian isometry group of the dimension $D-2$. For solution of these (symmetry reduced) equations many authors developed and applied different methods such as the inverse scattering method and soliton generating techniques (V. A. Belinski and V. E. Zakharov), methods based on the theory of singular integral equations and on various formulations of the Riemann and Riemann–Hilbert problems (V. A. Belinski and V. E. Zakharov, I. Hauser and F. Ernst, N. R. Sibgatullin, G. Neugebauer and R. Meinel), the group-theoretic methods (W. Kinnersley and D. M. Citre, B. Julia, P. Breitenlohner and D. Maison), Bäcklund transformations (B. Harrison, G. Neugebauer), prolongation structures (F. Estabrook, H. Wahlquist, B. Harrison) and others.
In this talk, we describe a general and unified approach to solution of all fields equations mentioned above which was suggested by the author many years ago and which got recently its farther development. This approach called as the monodromy transform is based on a definition of a special set of coordinate-independent functional parameters — holomorphic functions of a free complex (“spectral”) parameter which characterize any local solution. These parameters arise as a complete set of the monodromy data for (normalized) fundamental solution of the associated linear system and play (similarly to the scattering data in the inverse scattering method) the role of “coordinates” in the infinite-dimensional space of all local solutions for each of the systems under consideration. The inverse problem of such mapping also possess always a unique solution. Namely, for any choice of functions which determine a set of monodromy data, the corresponding local solution always exists and it is unique. The system of the linear singular integral equations which solves this inverse problem of the monodromy transform admits explicit solutions for a large classes of analytically matched monodromy data represented by arbitrary rational functions of the spectral parameter. This allows to calculate explicitly infinite hierarchies of exact solutions with any number of free parameters. It is important that many known solutions enter these hierarchies and this allows to construct their superpositions and multi-parametric generalizations. We consider also the examples of different new solutions which describe e.g., the black holes in the external gravitational and electromagnetic fields, nonlinear interaction of strong gravitational and electromagnetic waves, the dynamics of simple inhomogeneous cosmological models which, though representing pure model situations, allow to study the not subject to our intuition details of the nonlinear character of strong gravitational fields and of their interaction with some other matter fields.