(Generalized) quasi-topological gravities at all orders

Abstract

A new class of higher-curvature modifications of D(≥ 4)-dimensional Einstein gravity has been recently identified. Densities belonging to this "Generalized quasi-topological" class (GQTGs) are characterized by possessing non-hairy generalizations of the Schwarzschild black hole satisfying g tt g rr = −1 and by having second-order equations of motion when linearized around maximally symmetric backgrounds. GQTGs for which the equation of the metric function f (r) ≡ −g tt is algebraic are called "Quasi-topological" and only exist for D ≥ 5. In this paper we prove that GQTG and Quasi-topological densities exist in general dimensions and at arbitrarily high curvature orders. We present recursive formulas which allow for the systematic construction of n-th order densities of both types from lower order ones, as well as explicit expressions valid at any order. We also obtain the equation satisfied by f (r) for general D and n. Our results here tie up the remaining loose end in the proof presented in arXiv:1906.00987 that every gravitational effective action constructed from arbitrary contractions of the metric and the Riemann tensor is equivalent, through a metric redefinition, to some GQTG.

Partition functions on slightly squashed spheres and flux parameters

“…For general n, the densities R (n) are such that they allow for single-function Taub-NUT solutions, whose existence at arbitrary n is assumed. While we do not have a closed expression for them for general n, we know that when evaluated on a spherical/planar/hyperbolic black hole ansatz, they are equivalent to the densities constructed in [70]. Therefore, they produce the same on-shell actions, equations of motion, and so on.…”

“…Until recently, no additional AdS-Taub-NUT solutions were known in d+1 = 4 bulk dimensions for any other metric theories of gravity. However, new solutions of that kind have been recently constructed in [60] for cubic and quartic higher-order gravities of the so-called Generalized Quasi-topological (GQT) class [61][62][63][64][65][66][67][68][69][70][71][72][73][74]. Remarkably, the thermodynamic properties of such solutions can be obtained fully analytically -and nonperturbatively in the higher-order couplings-in all cases.…”

“…Let us now restrict ourselves to GQT theories. Their defining property is the following [64,69,70]. Consider a general static and spherically symmetric metric of the form 19) and let L N,V ≡ √ −gL| N,V be the effective Lagrangian resulting from the evaluation of L on (2.19).…”

“…in general dimensions and at arbitrarily high orders in curvature [70], and they have many interesting properties, such as possessing second-order equations of motion when linearized around maximally symmetric backgrounds, or the fact that the thermodynamic properties of their black hole solutions can be computed analytically -see e.g., [69] for a detailed summary. For the purposes of this work, the most relevant aspect is that a certain subset of GQT theories admit solutions of the AdS-Taub-NUT class which are also characterized by a single function and whose thermodynamic properties can be computed analytically [60,81,82].…”

“…represents the metric of the unit hyperbolic space, and where in principle N (r) and V (r) are two independent functions. The equations of motion of (3.1) evaluated on the metric (3.4) were computed in [70], where it was found that they are solved by N (r) = constant, while V (r) satisfies an equation which can be most conveniently written by defining…”