Complex relativity and real solutions II: Classification of complex bivectors and metric classes
 G. S. Hall,
 M. S. Hickman,
 C. B. G. McIntosh
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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 3space Pℂ^{3}an elegant but not wellknown method of investigating the null structure of a metric. A generalization of the MariotRobinson 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 KerrSchild metrics. Different, but equivalent, definitions of such metrics are given from various viewpoints. Two other subcasses are also discussed. The nonexpanding typsD vacuum metric is considered and it is shown how complex transformations may be made to write it (and subcases) in double (or single) KerrSchild form.
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 Title
 Complex relativity and real solutions II: Classification of complex bivectors and metric classes
 Journal

General Relativity and Gravitation
Volume 17, Issue 5 , pp 475491
 Cover Date
 19850501
 DOI
 10.1007/BF00761905
 Print ISSN
 00017701
 Online ISSN
 15729532
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
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 Authors

 G. S. Hall ^{(1)}
 M. S. Hickman ^{(2)}
 C. B. G. McIntosh ^{(2)}
 Author Affiliations

 1. Department of Mathematics, University of Aberdeen, Edward Wright Building, Dunbar Street, AB9 2TY, Aberdeen, Scotland
 2. Mathematics Department, Monash University, Wellington Road, 3168, Clayton, Victoria, Australia