From conformal to Einstein gravity

We provide a simple derivation of the equivalence between Einstein and conformal gravity (CG) with Neumann boundary conditions given by Maldacena. As Einstein spacetimes are Bach flat, a generic solution to CG would contain both Einstein and non-Einstein parts. Using this decomposition of the spacetime curvature in the Weyl tensor makes manifest the equivalence between the two theories, both at the level of the action and the variation of it. As a consequence, we show that the on-shell action for critical gravity in four dimensions is given uniquely in terms of the Bach tensor.