# Complex relativity and real solutions II: Classification of complex bivectors and metric classes

- Received:

DOI: 10.1007/BF00761905

- Cite this article as:
- Hall, G.S., Hickman, M.S. & McIntosh, C.B.G. Gen Relat Gravit (1985) 17: 475. doi:10.1007/BF00761905

## Abstract

This paper continues the examination of real metrics and their properties from the viewpoint of complex relativity as initiated by McIntosh and Hickman [1]. Tetrads of real metrics can be formally complexified by complex coordinate transformations and tetrad rotations and their properties investigated from the viewpoint of complex relativity. First, complex bivectors are examined and classified, partly by using the fundamental quadric surface of a metric in projective complex 3-space Pℂ^{3}-an elegant but not well-known method of investigating the null structure of a metric. A generalization of the Mariot-Robinson theorem from real relativity is then given and related to various canonical forms of complex bivectors. The second part of the paper discusses four classes of complex metrics. Real metrics of the first class are ones with a null congruence whose wave surfaces have equal curvature. The second class, a subcase of the first one, is the main one; it contains integrable double Kerr-Schild metrics. Different, but equivalent, definitions of such metrics are given from various viewpoints. Two other subcasses are also discussed. The nonexpanding typs-*D* vacuum metric is considered and it is shown how complex transformations may be made to write it (and subcases) in double (or single) Kerr-Schild form.