Vacuum spacetimes of embedding class two


Doubt is cast on the much quoted results of Yakupov that the torsion vector in embedding class two vacuum space-times is necessarily a gradient vector and that class 2 vacua of Petrov type III do not exist. The rst result is equivalent to the fact that the two second fundamental forms associated with the embedding necessarily commute and has been assumed in most later investigations of class 2 vacuum space-times. Yakupov stated the result without proof, but hinted that it followed purely algebraically from his identity: Rijkl Ckl = 0 where Cij is the commutator of the two second fundamental forms of the embedding.From Yakupov's identity, it is shown that the only class two vacua with non-zero commutator Cij must necessarily be of Petrov type III or N. Several examples are presented of non-commuting second fundamental forms that satisfy Yakupovs identity and the vacuum condition following from the Gauss equation; both Petrov type N and type III examples occur. Thus it appears unlikely that his results could follow purely algebraically. The results obtained so far do not constitute denite counter-examples to Yakupov's results as the non-commuting examples could turn out to be incompatible with the Codazzi and Ricci embedding equations. This question is currently being investigated.

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Additional Information: Creative Commons Attribution License 3.0.
Uncontrolled Keywords: General Physics and Astronomy
Publication ISSN: 1742-6596
Last Modified: 27 Jun 2024 07:53
Date Deposited: 09 Mar 2011 11:31
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Related URLs: http://www.scop ... tnerID=8YFLogxK (Scopus URL)
http://iopscien ... 4/1/012091/meta (Publisher URL)
PURE Output Type: Article
Published Date: 2011
Authors: Barnes, Alan



Version: Published Version

License: Creative Commons Attribution

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