Energy functionals from Conformal Gravity

TL;DR

This paper derives the Hawking mass and Willmore energy functionals for asymptotically AdS spacetimes by embedding Einstein-AdS gravity into Conformal Gravity, revealing a link between conformal invariance and renormalization.

Contribution

It introduces a new derivation method for energy functionals using Conformal Gravity, highlighting their conformal invariance and finiteness in AdS spacetimes.

Findings

Energy functionals are finite for AdS asymptotics.

Hawking mass and Willmore energy are special cases of a conformal invariant functional.

The approach suggests a relation between conformal invariance and renormalization.

Abstract

We provide a new derivation of the Hawking mass and Willmore energy functionals for asymptotically AdS spacetimes, by embedding Einstein-AdS gravity in Conformal Gravity. By construction, the evaluation of the four-dimensional Conformal Gravity action in a manifold with a conical defect produces a codimension-2 conformal invariant functional $L_{\Sigma}$. The energy functionals are then particular cases of $L_{\Sigma}$ for Einstein-AdS and pure AdS ambient spaces, respectively. The bulk action is finite for AdS asymptotics and both Hawking mass and Willmore energy are finite as well. The result suggests a generic relation between conformal invariance and renormalization, where the codimension-2 properties are inherited from the bulk gravity action.