Not conformally Einstein metrics in conformal gravity (Q2851407)

It is a well-known fact, that all conformally Einstein metrics, i.e., all metrics conformally related to an Einstein space, are vacuum solutions of conformal gravity, i.e., of the Bach equation \(B_{ij} =0\). The Bach tensor \(B_{ij} \) is the variational derivative of the square of the Weyl tensor, and therefore, conformal gravity is also called Weyl gravity.NEWLINENEWLINEIt is the purpose of the present paper to direct the interest of researchers to the fact that by this procedure, not all solutions of Weyl gravity will be exhausted. This is the idea behind the headline: Not conformally Einstein metrics in conformal gravity. In fact, the authors find 5 classes of such solutions. They have the character of gravitational pp-waves, anisotropic Bianchi IX metrics, and others. In section 6, they discuss also solutions for Einstein-Weyl gravity, i.e., that case where the sum of the Bach tensor and the Einstein tensor appears in the vacuum equation. The appendix lists formulas for the Petrov type.