Noether-Wald charges in six-dimensional Critical Gravity

TL;DR

This paper derives and analyzes Noether-Wald charges in six-dimensional Critical Gravity, demonstrating their triviality on Einstein manifolds and establishing an equivalence with Einstein-AdS gravity at the charge level.

Contribution

It provides a compact expression for Noether-Wald charges in six-dimensional Conformal Gravity and shows their triviality in Critical Gravity, revealing an equivalence with Einstein-AdS gravity.

Findings

Noether-Wald charges vanish for Einstein manifolds in Critical Gravity.

Critical Gravity's charges are equivalent to those of Einstein-AdS gravity.

Derived a compact form of the Noether prepotential in six-dimensional Conformal Gravity.

Abstract

It has been recently shown that there is a particular combination of conformal invariants in six dimensions which accepts a generic Einstein space as a solution. The Lagrangian of this Conformal Gravity theory -- originally found by Lu, Pang and Pope (LPP) -- can be conveniently rewritten in terms of products and covariant derivatives of the Weyl tensor. This allows one to derive the corresponding Noether prepotential and Noether-Wald charges in a compact form. Based on this expression, we calculate the Noether-Wald charges of six-dimensional Critical Gravity at the bicritical point, which is defined by the difference of the actions for Einstein-AdS gravity and the LPP Conformal Gravity. When considering Einstein manifolds, we show the vanishing of the Noether prepotential of Critical Gravity explicitly, which implies the triviality of the Noether-Wald charges. This result shows the equivalence between Einstein-AdS gravity and Conformal Gravity within its Einstein sector not only at the level of the action but also at the level of the charges.