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Local symmetries and covariant integration for algebraically special gravitational fields. (English. Russian original) Zbl 0834.53050
Russ. Math. 38, No. 2, 28-34 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No. 2 (381), 30-36 (1994).
The main purpose of this paper is the reduction of the Newman-Penrose (N.P.) equations for gravitational fields in general relativity under the assumptions that (i) the space-time manifold is an Einstein space, (ii) the Weyl tensor is algebraically special in the classification scheme of Petrov and (iii) that the so-called Sachs complex expansion scalar $$\rho \neq 0$$.
The paper begins with a brief review of the N.P. formalism [with notation taken from S. Chandrasekhar’s book “The mathematical theory of black holes.” Moskva: Mir (1986; Zbl 0671.53059)]. This is followed by a summary of the Petrov types, the restriction to the algebraically special types and the consequence that $$\chi = 0$$, $$\sigma = 0$$. The optical scalars associated with a null congruence are also discussed. The N.P. equations are then written down and simplified with coordinate transformations and some integrations are computed. The condition $$\rho \neq 0$$ is used and the author notes that the case $$\rho = 0$$ has been considered many years ago by Kundt. He does not point out that the case $$\rho \neq 0\;\text{Im} (\rho) = 0$$ was considered by I. Robinson and A. Trautman [Proc. R. Soc. Lond., Ser. A 405, 41–48 (1986; Zbl 0588.53018)].
##### MSC:
 53C80 Applications of global differential geometry to the sciences 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory