Complex relativity and real solutions. II: Classification of complex bivectors and metric classes
type: Item
instance of: scholarly article
title: Complex relativity and real solutions. II: Classification of complex bivectors and metric classes
zbMATH Open document ID: 0564.53039
DOI: 10.1007/BF00761905
author: M. S. Hickman
author: G. S. Hall
author: Colin B. G. McIntosh
published in: General Relativity and Gravitation
publication date: 1985-01-01T00:00:00Z
review text: This paper continues the examination of real metrics and their properties from the viewpoint of complex relativity as initiated by the last two authors [ibid. 17, 111--132 (1985; Zbl 0561.53067)]. 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 \(P\mathbb C^ 3\) -- 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 subclasses are also discussed. The nonexpanding type-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.
Mathematics Subject Classification ID: 83C15
Mathematics Subject Classification ID: 83C05
Mathematics Subject Classification ID: 53B50
zbMATH DE Number: 3899707
zbMATH Keywords: complex spaces
zbMATH Keywords: complex relativity
zbMATH Keywords: null structure
zbMATH Keywords: complex bivectors
zbMATH Keywords: Kerr-Schild metrics
zbMATH Keywords: vacuum metric
MaRDI profile type: Publication
cites work: Complex relativity and real solutions. I: Introduction
cites work: Q5811706
cites work: Q3896182
cites work: Null strings and Bianchi identities
cites work: Null geodesic surfaces and Goldberg–Sachs theorem in complex Riemannian spaces
cites work: All algebraically degenerate ℋ spaces, via ℋℋ spaces
cites work: Type D Vacuum Metrics
description: scientific article; zbMATH DE number 3899707