Lightons and helixons as lightlike particles in general relativity

The exterior gravitational fields of the massive Schwarzschild, Kerr and NUT particles have an algebraic type $\mathbf{D}$ according to Petrov's algebraic classification of the gravitational fields. A lightlike limit is input for these particles. It is shown that under this limiting procedure the gravitational fields of the particles are transformed into the gravitational fields of the lightlike particles a lighton and a helixon. On the other hand this limit can be described as a cusp catastrophe on a level of Weyl's matrix with a change of a gravitational field symmetry of such sources. In considered cases we have a phase gravitational transition of second kind from $\mathbf{D}$ type into $\mathbf{N}$ type or $\mathbf{III}$ type. There is a transition of one “phase” to another. Petrov's algebraic types are the different “phases” of the gravitational field. It is shown that the lightlike sources in General Relativity “have no hairs”.