Mass in Lovelock unique vacuum gravity theories

Abstract

We derive an expression for conserved charges in Lovelock AdS gravity for solutions having k-fold degenerate vacua, making manifest a link between the degeneracy of a given vacuum and the nonlinearity of the energy formula. We show for a black hole solution to the field equations on a branch of multiplicity k that its mass comes from an expression that contains the product of k Weyl tensors. We prove that all divergent contributions of the type (Weyl) q , with 1 ≤ q < k, are suppressed. Our conserved charge definition is a natural generalization of the Conformal Mass by Ashtekar, Magnon and Das to the cases when k > 1. Our results provide insight on the holographic properties of degenerate Lovelock theories.

Renormalized holographic entanglement entropy in Lovelock gravity

“…for some coefficients p (k+1,i) (given in eq. (C.8)), in agreement with the previous results for Lovelock gravity with k-fold degenerate vacua [45]. We can explicitly write the first three factors as…”

“…This procedure has been successfully applied to computations for holographic CFTs dual to Einstein-AdS gravity [41][42][43][44]. It was recently used to obtain an expression for the conserved charges of solutions having k-fold degenerate vacua in Lovelock AdS gravity, making manifest a link between the degeneracy of a given vacuum and the nonlinearity of the energy formula [45,46]. The prescription (1.2) exploits the replica formula, whose holographic implementation [47,48] generalized the minimal area prescription (1.1) beyond the sphericallysymmetric case [49].…”

“…Thus, the rewriting of the coefficients in terms of the degeneracy conditions indicates that the Noether prepotential of a k-degenerate theory is of order F k+1 at the normalizable order, in accordance to ref. [45]. For more information, see appendix D.…”

“…This form is convenient as the coefficients of the different powers of F are linear combinations of the degeneracy conditions (3.19), extending the analysis of ref. [45] for the Noether prepotential used in the computation of asymptotic charges. Indeed, writing the Lagrangian in terms of the degeneracy conditions makes it clear that any variation of L and, therefore, the Noether prepotential, will inherit the same property.…”

“…Consequently -for degenerate points-the Noether prepotential is not linear but of higher order in F , such that it is of normalizable order (in agreement with the results of ref. [45]. Now, we consider the coefficient of the F 3 term, which is given by…”