From conformal to Einstein Gravity

TL;DR

This paper demonstrates the equivalence between Einstein and Conformal Gravity with Neumann boundary conditions, clarifying their relationship through curvature decomposition and analyzing the on-shell action in four-dimensional Critical Gravity.

Contribution

It provides a straightforward derivation of Einstein and Conformal Gravity equivalence and links the on-shell action of Critical Gravity to the Bach tensor.

Findings

Einstein and Conformal Gravity are equivalent under certain boundary conditions

Decomposition of spacetime curvature reveals this equivalence

On-shell action in four-dimensional Critical Gravity is expressed via the Bach tensor

Abstract

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 part. 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.