Investigation of all Ricci semi-symmetric and all conformally semi-symmetric spacetimes

Jan E. Åman

Spacetimes are called conformally semi-symmetric if $\nabla_{[a}\nabla_{b]}C_{cdef} = 0$ and Ricci semi-symmetric if $\nabla_{[a}\nabla_{b]}R_{cd} = 0$. Spacetimes fulfilling both these conditions are semi-symmetric.

In this talk all Ricci semi-symmetric as well as all conformally semi-symmetric spacetimes will be presented. Neither of these properties will imply the other. However, only conformally flat spacetimes can be Ricci semi-symmetric without being conformally semi-symmetric while only vacuum spacetimes and spacetimes with just a $\Lambda$-term can be Ricci semi-symmetric without being conformally semi-symmetric.

The investigation has been made using spinor equivalents of the conditions above which are $\Box_{AB} \Psi_{CDEF} = 0$ and $\Box_{A'B'} \Psi_{CDEF} = 0$ for conformal semi-symmetry and $\Box_{AB} \Phi_{CDC'D'} = 0$ for Ricci semi-symmetry. They have been implemented in the computer algebra system Classi.

This talk will be dedicated to Brian Edgar (1945-2010) who in May 2010
introduced me to semi-symmetric spacetimes.