Fourth order Weyl gravity

The fourth order Weyl gravity theory of Mannheim and Kazanas is based on replacing the Einstein-Hilbert action with the square of the Weyl tensor, and on modifying the matter action of the standard model of particle physics to make it conformally invariant. This theory has been suggested as a model of both dark matter and dark energy. We show that the theory contains two uncoupled sectors: a trivial sector containing no dynamics on which the conformal symmetry acts, and a dynamical sector in which all the fields are conformal invariants. We deduce that in the limit of weak fields and slow motions the theory does not agree with the predictions of general relativity, and is therefore ruled out by Solar System observations.