New vacuum solutions of conformal Weyl gravity (Q2737987)

The Bach equations are fourth order conformally invariant field equations arising from a Lagrangian quadratic in the conformal Weyl curvature tensor. In this article the Bach equations are solved in the case that the metric is conformally related to the Cartesian product of two two-dimensional spaces. Hence, the solution is given for the special cases of spherically symmetric and plane symmetric space-times. The assumption on the metric allows results from two-dimensional gravity in \textit{H.-J. Schmidt} [Gen. Relativ. Gravitation 31, 1187-1210 (1999; Zbl 0937.83031)] to be used. Examples of cosmological solutions are provided in the case of spherical and plane symmetry and some general discussion of the role of the Bach tensor is included.