Gravitational Solitons and Monodromy Transform Approach to Solution of Integrable Reductions of Einstein Equations

In this paper the well known Belinskii and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so called "monodromy transform approach", which provides some general base for the study of various integrable space - time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for solution of associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskii and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the background solution) as simple (linear-fractional) transformations of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. The similar characterization for electrovacuum soliton generating transformations is also presented.Comment: 8 pages, 1 figure, LaTeX2e; based on a talk given at the International Conference 'Solitons, Collapses and Turbulence: Achievements, Developments and Perspectives', (Landau Institute for Theoretical Physics, Chernogolovka, Moscow region, Russia, August 3 -- 10, 1999); as submitted to Physica

Einstein billiards and overextensions of finite-dimensional simple Lie algebras

In recent papers, it has been shown that (i) the dynamics of theories involving gravity can be described, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space bounded by hyperplanes; and (ii) that the relevant billiard has remarkable symmetry properties in the case of pure gravity in d + 1 spacetime dimensions, or supergravity theories in 10 or 11 spacetime dimensions, for which it turns out to be the fundamental Weyl chamber of the Kac-Moody algebras AEd, E10, BE 10 or DE10 (depending on the model). We analyse in this paper the billiards associated to other theories containing gravity, whose toroidal reduction to three dimensions involves coset models G/H (with G maximally non compact). We show that in each case, the billiard is the fundamental Weyl chamber of the (indefinite) Kac-Moody "overextension" (or "canonical lorentzian extension") of the finite-dimensional Lie algebra that appears in the toroidal compactification to 3 spacetime dimensions. A remarkable feature of the billiard properties, however, is that they do not depend on the spacetime dimension in which the theory is analyzed and hence are rather robust, while the symmetry algebra that emerges in the toroidal dimensional reduction is dimension-dependent. © SISSA/ISAS 2002.info:eu-repo/semantics/publishe