Canonical decomposition of Belinskii-Zakharov one-soliton solutions

Summary

Abstract

We perform the standard canonical (3+1) decomposition of the Belinskiĭ-Zakharov one-soliton solution. Our starting point is the general Bianchi I solution which we obtain by applying the symmetry transformation to the Kasner metric. We then construct the symplectic form for the Bianchi I model and on the way show that we have all the physical degrees of freedom for our starting solution. The Belinskiĭ-Zakharov soliton transfomation requires two degrees of freedom in the general Bianchi I solution to be frozen. Although integration of the linearized system, in the non-diagonal case, is a non-trivial step, the particular form of our starting solution simpifies the calculation and reduces the problem effectively to the diagonal case. Therefore, in our case, it is straightforward to obtain the one-soliton metric. Finally, we point out the problems related to the fact that the one-soliton solution is defined only in a certain region of the co-ordinate chart.

Introduction

The Einstein field equations for space-times that admit a two-dimensional Abelian group of isometries which acts orthogonally and transitively on non-null orbits are non-linear partial differential equations in two variables [Kramer et al. 1980]. Since the pioneering work of Geroch, it has been known that the field equations in the stationary axisymmetric case admit an infinite dimensional group of symmetry transformations [Geroch 1971, 1972]. This result has encouraged the research in solution-generating methods, the main idea being that the complete class of solutions can then be generated from a particular solution, such as flat space [Cosgrove, 1980-1982].