6 - Cylindrical symmetry

Summary

Cylindrically symmetric spacetimes also have the symmetries required to generate solutions by the ISM. In this chapter we review, briefly, the soliton solutions in the cylindrical context. The analytic expressions for such solutions can be obtained from the cosmological solutions of chapters 4 and 5 by a simple reinterpretation of the relevant coordinates. For this reason the sections in this chapter are considerably shorter. One of the main interesting features of these spacetimes is that a definition of energy, the so-called C-energy, can be given and, consequently, cylindrically symmetric waves can be understood as waves that carry energy. The study of the C-energy in the soliton solutions will play an important role in the interpretation of the cylindrically symmetric soliton waves. Some general properties are discussed in section 6.1. Diagonal metrics, i.e. one polarization waves, are described in section 6.2; these include all generalized soliton solutions of sections 4.4.1, 4.5 and 4.6 after appropriate transformations. Some attention is paid to solutions which have been used to describe the interaction of a straight cosmic string with gravitational radiation. In section 6.3 solutions with two polarizations are considered and the conversion of one of the modes of polarization into the other is described. This conversion is an effect of the nonlinear interaction between the two modes and is interpreted as the gravitational analogue of the Faraday rotation of electromagnetic waves by a magnetic field and plasma.