Vacuum energy in Einstein-Gauss-Bonnet anti–de Sitter gravity

Abstract

A finite action principle for Einstein-Gauss-Bonnet anti−de Sitter gravity is achieved supplementing the bulk Lagrangian by a suitable boundary term, whose form substantially differs in odd and even dimensions. For even dimensions, this term is given by the boundary contribution in the Euler theorem with a coupling constant fixed demanding the spacetime to have constant (negative) curvature in the asymptotic region. For odd dimensions, the action is stationary under a boundary condition on the variation of the extrinsic curvature. A well-posed variational principle leads to an appropriate definition of energy and other conserved quantities using the Noether theorem, and to a correct description of black hole thermodynamics. In particular, this procedure assigns a nonzero energy to anti-de Sitter spacetime in all odd dimensions.In recent years, several experiments carried out suggest observational evidence for a positive value of the cosmological constant [1]. Nonetheless, from a theoretical point of view, the idea of existence of extra dimensions and alternative gravity theories does not rule out a negative cosmological constant in a higher-dimensional spacetime. On the contrary, for example, particular braneworld models induce a zero or positive cosmological constant on a four-dimensional Universe [2]. A negative cosmological constant is also appealing because of the possibility of a profound connection between anti-de Sitter gravity and a conformal field theory (CFT) living on its boundary, that has attracted a considerable attention in the literature [3]. Even though some remarkable progress has been achieved on a rather case-by-case basis, a general proof of this duality remains unknown. In that context, the existence of a nonzero energy for anti-de Sitter (AdS) vacuum spacetime in the gravity side may be helpful to identify the corresponding CFT at the boundary. Indeed, in five dimensions, the matching between the zero-point energy for Schwarzschild-AdS black hole and the induced (Casimir) energy of a precise boundary field theory is one of the best known examples that realizes this bulk/boundary correspondence [4]. General Relativity with AdS asymptotics requires a regularization procedure in order to define a finite energy for the solutions of the theory. Logically, a vacuum energy can appear only when this mechanism does not invoke the substraction of a background configuration as, e.g., in Hamiltonian formalism. That is the case of the counterterms method [5,6], where the finiteness of the action and its energy-momentum tensor is obtained by the addition of covariant functionals of the boundary metric, constructed by solving the Einstein equations in a given asymptotic form of the metric [7]. In spite this algorithm provides the correct counterterms for many cases, in high enough dimension it becomes rather cumbersome, what makes the full series still unknown. It is clear that the inclusion of quadratic curvature terms in the action will turn this method of regularization even more complex.In this artic...

Conserved charges of black holes in Weyl and Einstein–Gauss–Bonnet gravities

“…The mass (44) coincides with the one via ADT method [22,23] or other methods [34][35][36][37][38][39][40][41][42]. Such coincidences should be expected.…”

Counterterms in dimensionally continued AdS gravity

“…The construction of the boundary terms B d does not make use of the expansion in the metric (1.5). Therefore, for a given dimension, the Kounterterms expression remains the same regardless the particular Lovelock gravity considered, even for Einstein-Hilbert [21,22] and Einstein-Gauss-Bonnet theories [23]. Only the value of the coupling constant c d is consistently tuned to achieve a well-posed action principle in a given Lovelock-AdS theory.…”

“…As a consequence, this condition is suitable to treat the variational problem in a large set of gravity theories that support AAdS solutions. The corresponding boundary term B 2n that regulates the conserved quantities and Euclidean action in Chern-Simons-AdS gravity, provides also the correct answer for Einstein-Hilbert case [22,44], Einstein-Gauss-Bonnet gravity [23] and a generic Lovelock-AdS theory [20].…”

Universal Kounterterms in Lovelock AdS gravity

“…As the natural extension of what happens in Einstein-Gauss-Bonnet gravity [9], we show below that, whenever the effective cosmological constant is negative, the form of the Kounterterms is universal for all gravity theories of the Lovelock type [10].…”

“…[9] that the regularization prescription given above for Einstein-Hilbert gravity with AdS asymptotics is still valid for the action (12). This means that the form of the Kounterterms in even and odd dimensions (Eqs.…”