Series: The London Relativity and Cosmology Seminar
Speaker: Alan Barnes (Aston)
Abstract: Vacuum spacetimes of embedding class two (possibly with a non-zero Lambda term) are considered. Many years ago Yakupov stated that in such spacetimes the torsion vector of the embedding vanished (or equivalently the two second fundamental forms defining the embedding necessarily commuted) and that there were no such spacetimes of Petrov type III. However no proof was ever published. In this talk the existence of such spacetimes with torsion is reconsidered. Using an identity which is a purely algebraic consequence of the Gauss equations that Yakupov did prove, it is easy to see that when Lambda = 0, the only possible Petrov types are type N & III whereas if Lambda is non-zero, then only Petrov types I, II & D are allowed. It is also easy to construct many possible counterexamples in which the two fundamental forms do not commute, but which satisfy the vacuum conditions. In the pure vacuum case it turns out that all the type III cases are inconsistent with the Codazzi & Ricci (CR) equations. Together with a result of Van den Bergh for the zero torsion case this constitutes a proof of Yakupov's second result. For type N the CR equations imply that the only possibility is a rather special pp-wave. However, at the current time it is unclear whether or not the full CR equations can be satised by this spacetime. When Lambda is non-zero, the situation is more complicated and is 'work in progress'; one needs to check the many more possible algebraic counterexamples to see if they satisfy the CR equations. However all such candidates have a constant Weyl eigenvalue equal to 4 Lambda/3. This naturally leads one to consider vacuum spacetimes with constant non-zero Weyl eigenvalues. For type II & D this assumption alone is sufficient to allow one to prove that the non-repeated eigenvalue necessarily has the value 4 Lambda/3 and it turns out that the only possible spacetimes are some pp-waves considered by Lewandowski (which are type II) and a Robertson-Bertotti solution. For Petrov type I if we make the stronger assumption that all three Weyl eigenvalues are constant, the only solution turns out to be a homogeneous pure vacuum solution found long ago by Petrov using group theoretic methods. These results can be summarised by the statement that the only vacuum spacetimes with constant Weyl eigenvalues are either homogeneous or are pp-wave spacetimes. This result is similar to that of Coley et al. who proved the same result for general spacetimes under the assumption that all scalar invariants constructed from the curvature tensor and all its derivatives were constant.