Noether-Wald charges in six-dimensional Critical 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.

Gravitational instantons with conformally coupled scalar fields

“…This case is interesting because the conformal symmetry allows one to renormalize the Euclidean on-shell action and conserved charges naturally without any reference to intrinsic boundary counterterms. This feature bears close resemblance to the pure metric formulation of conformal gravity [55,65,91,92]. We show this explicitly for a novel set of gravitational instantons we present next.…”

“…In the conformally invariant case, namely, when the Einstein sector decouples from the scalar-tensor one, the theory becomes a particular case of Brans-Dicke gravity enjoying conformal symmetry. Moreover, the Euclidean on-shell action, alongside the Noether-Wald charges, are finite without any reference to intrinsic boundary counterterms; similar to the metric formulation of conformal gravity [55,65,91,92]. We study topological properties of previously found solutions of the Taub-NUT family [77] and we show that higher-curvature corrections allows one to construct solutions of the Eguchi-Hanson type possessing a negative cosmological constant and a positive curvature Kähler base manifold, something forbidden in their absence.…”

“…In conformal gravity, the finiteness of Euclidean on-shell action and conserved charges for solutions with AdS asymptotics have been shown explicitly in four [65] and in six dimensions [91,92]. Nevertheless, a rigorous proof of the finiteness of the latter is still lacking; at least in the weakly modified asymptotically AdS sector.…”

Thermodynamics of non-linearly charged Anti-de Sitter black holes in four-dimensional Critical Gravity

“…In this work, we find that CG admits black hole solutions provided the existence of a non-linear electrodynamics described by eq. ( 6), and that these solutions are characterized by the gravitational potential (13), where, in our context, r > 0. However, the nature of these solutions and, in particular, the number of horizons that they will exhibit, will depend on the signs and relations between the constants α i 's.…”

“…However, as emphasized by the authors in [11], this solution is massless and has a vanishing entropy. This is also highlighted in [12,13] in four and six dimensions, where the entropy, as well as global conserved charges of their black holes solutions, vanish identically. Because of the interest of obtaining black hole solutions that can be framed in the context of the gauge/gravity correspondence, in this work we aim to find new AdS black hole configurations that can be supported by CG and exhibit non-vanishing thermodynamical properties.…”