On Vacuum Space-times of Embedding Class Two

Alan Barnes

Doubt is cast on the much quoted 1973 result of Yakupov that the torsion vector in embedding class two vacuum space-times is necessarily a gradient vector. The result is equivalent to the fact that the two second fundamental forms associated with the embedding necessarily commute. This result was used to show the non-existence of class 2 vacua of Petrov type III and was also assumed in later investigations of class 2 space-times by Hodgkinson and van den Bergh.
Yakupov stated the result without proof, but hinted that it followed purely algebraically from his 1968 identity: $R_{ijkl}C^{kl} = 0$ where $C_{ij}$ is the commutator of the two second fundamental forms of the embedding.
Here several examples are presented of non-commuting second fundamental forms that satisfy Yakupov’s identity and the vacuum condition following from the Gauss equation. Although Yakupov’s proof of his identity used the full Gauss, Codazzi and Ricci equations plus the vacuum condition, we show how it can be derived from the Gauss equation and the vacuum condition alone. Thus it appears unlikely that Yakupov’s result could follow purely algebraically as claimed.
The results obtained so far do not constitute a definite counter-example to Yakupov’s result as the non-commuting examples could turn out to be incompatible with the Codazzi and Ricci embedding equations. This question is currently being investigated.
From Yakupov’s identity, I show that the only class 2 vacua with non-zero commutator $C_{ij}$ must necessarily be of Petrov type III or N; the non-existence of such space-times of Petrov type I can be deduced by applying the 1975 result of Brans which showed the non-existence of any Petrov type I vacua in which one of the eigenvalues of the Weyl tensor vanishes.