The phase transitions in gravitational and electromagnetic fields with relation petrov’s algebraic classification

This article shows the connection of Petrov’s algebraic classification of gravitational fields with the catastrophe theory as the generalised theory of phase transitions. The analogy between the transitions of the space-time algebraic types and the phase transitions at the curvature tensor level (Weyl’s matrices level) are demonstrated. There are the first and second order phase transitions. Algebraic types of space-times are "phases of a substance"(states of gravitational fields) and the order parameters of phase transitions are the curvature invariants. The most symmetric "phase"is the gravitational field with a conformally flat metric. The examples of the Petrov classification of gravitational fields, an algebraic classification of four-dimensional local Euclidean spaces, an algebraic classification of electromagnetic fields and of the deriving of the gravitational fields of the lightlike sources are presented in the connection with such analogy. In all cases the main catastrophe is the cusp catastrophe. It is the second order phase transition from point of view of the physics. It is shown that the algebraic classification of four-dimensional local Euclidean spaces is possible with using the theory of double variables as one sort of hypercomplex variables. In such classification there also are the catastrophe changes, i.e. the phase transitions between algebraic types. In the electromagnetic field the most symmetric "phase"is the lightlike state of the electromagnetic field (the plane electromagnetic wave). The particular demonstration of the theory of gravitational phase transitions of the second order is made under investigations of the lightlike limits of the particlelike solutions of Schwarzschild, Kerr and NUT. It is found that we have the changes not only the algebraic type of gravitational fields (from type D to wave types N and III) but and of their symmetry. The found lightlike particles under such limit are named as lightons and helixons depending on presence or absence of the helicity.