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.
Highlights
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
We recall the fact that, in the Euclidean sector, the action corresponds to the free-energy functional and depends on the canonical conjugates given by the black hole charges and corresponding chemical potentials
Summary
Conformal Gravity in four dimensions is constructed out of the unique conformal invariant allowed: the squared Weyl tensor. The conformally invariant theory possessing an Einstein-AdS sector was first obtained in ref. Are the Cotton and Schouten tensors, respectively This form of the LPP Conformal Gravity is convenient for the study of the Einstein sector of the theory. When the action (2.6) is evaluated in Einstein spacetimes, it reduces to the renormalized Einstein-AdS action up to a proportionality constant [8].
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