Aspects of three-dimensional higher curvature gravities

Abstract

We present new results involving general higher-curvature gravities in three dimensions. The most general Lagrangian can be written as a function of the Ricci scalar R, S2≡Rb aRa b and S3≡Rb aRc bRa c where Rab is the traceless part of the Ricci tensor. First, we provide a formula for the exact number of independent order-n densities, #(n). This satisfies the identity #(n−6)=#(n)−n. Then, we show that, linearized around a general Einstein solution, a generic order-n≥2 density can be written as a linear combination of Rn , which does not propagate the generic massive graviton, plus a density which does not propagate the generic scalar mode, Rn−12n(n−1)Rn−2S2 , plus #(n)−2 densities which contribute trivially to the linearized equations. Next, we obtain an analytic formula for the quasinormal frequencies of the BTZ black hole for a general theory. Then, we provide a recursive formula as well as a general closed expression for order-n densities which non-trivially satisfy an holographic c-theorem, clarify their relation with Born-Infeld gravities and prove that they never propagate the scalar mode. We show that at each order there exist #(n−6) densities which satisfy the holographic c-theorem trivially and that all of them are proportional to a single sextic density Ω(6)≡6S3 2−S2 3 . We prove that there are also #(n−6) order-n Generalized Quasi-topological densities in three dimensions, all of which are "trivial" in the sense of making no contribution to the metric function equation. The set of such densities turns out to coincide exactly with the one of theories trivially satisfying the holographic c-theorem. We comment on the relation of Ω(6) to the Segre classification of three-dimensional metrics.

Higher-curvature Gravities from Braneworlds and the Holographic c-theorem

“…for general n and d. The functional dependence of the c-function is then c(r) ∝ A (r) 2n−d . This sort of 'uniqueness' has been argued to hold for general curvature orders in d = 3 in [17], and has been recently proven in [46]. In the same references, one can find a characterization of all the densities of any curvature order in d = 3 that satisfy the c-theorem and which are constructed from general contractions of the metric and the Riemann tensor.…”

“…On the one hand, as observed in [49], the criterion that cubic extensions of NMG do not propagate a scalar mode usually present in the spectrum of higher-curvature gravities, and that they admit a Chern-Simons formulation, restricts them to a general linear combination of S 3 and T 3 . Hence, L3 satisfies these two requirements-the first one is in fact implied when the holographic c-theorem is required, as shown in [46].…”

“…As argued e.g., in [17,46,50], in d = 3 the most general higher-curvature density constructed from contractions of the metric and the Riemann tensor is a function of the three densities 12…”

“…At quadratic order, the resulting theory is the New Massive Gravity of [14]-more on this below. At higher curvature orders, theories of this kind arise from an order-by-order expansion [21,46,47] of a Born-Infeld-type gravity [21], which in turn satisfies the holographic c-theorem by itself [47,48]. In addition, it has been found that certain theories that satisfy the holographic c-theorem-some of which involve explicit covariant derivatives-are equivalent to Chern-Simons gravities [49].…”

“…More recently, theories of this kind have been related to truncations of certain infinitedimensional Lie algebras [45]. It has also been shown that theories of this kind never propagate the scalar mode that is present in the linearized spectrum of generic higher-curvature theories [46]. This feature is likely valid for general d.…”