Not Conformally-Einstein Metrics in Conformal Gravity

Abstract

The equations of motion of four-dimensional conformal gravity, whose Lagrangian is the square of the Weyl tensor, require that the Bach tensor $E_{\mu\nu}= (\nabla^\rho\nabla^\sigma + \ft12 R^{\rho\sigma})C_{\mu\rho\nu\sigma}$ vanishes. Since $E_{\mu\nu}$ is zero for any Einstein metric, and any conformal scaling of such a metric, it follows that large classes of solutions in four-dimensional conformal gravity are simply given by metrics that are conformal to Einstein metrics (including Ricci-flat). In fact it becomes more intriguing to find solutions that are {\it not} conformally Einstein. We obtain five new such vacua, which are homogeneous and have asymptotic generalized Lifshitz anisotropic scaling symmetry. Four of these solutions can be further generalized to metrics that are conformal to classes of pp-waves, with a covariantly-constant null vector. We also obtain large classes of generalized Lifshitz vacua in Einstein-Weyl gravity.