Local symmetries and covariant integration for algebraically special gravitational fields (Q1896717)

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 \textit{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 \textit{I. Robinson} and \textit{A. Trautman} [Proc. R. Soc. Lond., Ser. A 405, 41--48 (1986; Zbl 0588.53018)].