Null canonical formulation and integrability of cylindrical gravitational waves

Abstract

The gravitational field in four dimensional spacetime may be described using free initial data on a pair of intersecting null hypersurfaces swept out by the future null normal geodesics to their two dimensional intersection surface. A Poisson bracket on such initial data was calculated by Michael Reisenberger. The expressions obtained are tractable but still rather intricate, and it is not at all obvious how this bracket might be quantized. A change of variables that simplifies the bracket would thus be desirable. The bracket does have the feature (reflecting causality) that it is non-zero only between data lying on the same generating null geodesic, and that it only depends on the data on this generator. That is, the data on each generator forms an essentially autonomous Poisson algebra. The limited role of the two transverse dimensions suggests that the Poisson algebra would remain substantially the same in a symmetry reduced model in which the transverse dimensions have been eliminated. Here this expectation is confirmed in the context of cylindrically symmetric gravitational waves. Specifically, the Poisson algebra of the metric variables in free null initial data for cylindrically symmetric gravitational waves is obtained, and it is found to be essentially identical to the bracket on the metric sector of the initial data found in by Reisenberger. Then, using the integrability of the dynamics of cylindrically symmetric gravitational waves an explicit transformation from metric data on a null hypersurface to so called "monodromy data", a one parameter family of unimodular matrices, is obtained. The Poisson brackets of the monodromy data are then obtained from that of the null data. These have been obtained earlier via another route in a slightly more restricted context. They are quite simple, and what is more, a unique preferred quantization is known. It is also demonstrated that the transformation to monodromy data is invertible. Aside from these original results extensive background material is presented, including a review of the Geroch group of symmetries in cylindrically symmetrical gravity. The original results presented here are joint work with Michael Reisenberger.