Aspects of three-dimensional higher curvature gravities

Abstract

We present new results involving general higher-curvature gravities in three dimensions. The most general Lagrangian of that kind can be written as a function of R, S_2, S_3, where R is the Ricci scalar, S_2= R^a_b R^b_a, S_3=R^a_b R^b_c R^c_a, and R_ab is the traceless part of the Ricci tensor. First, we provide a general 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 R^n, which by itself would not propagate the generic massive graviton, plus a density which by itself would not propagate the generic scalar mode, R^n-12 n(n-1)R^{n-2} S_2, plus #(n) − 2 densities which contribute trivially to the linearized equations. Next, we obtain an analytic formula for the quasinormal modes and frequencies of the BTZ black hole as a function of the masses of the graviton and scalar modes 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 the scalar mode is always absent from their spectrum. We show that, at each order n ⩾ 6, there exist #(n − 6) densities which satisfy the holographic c-theorem in a trivial way and that all of them are proportional to a single sextic density Omega_6=6S_3^2-S_2^3. Next, we show 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. Remarkably, 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 meaning of Ω(6) and its relation to the Segre classification of three-dimensional metrics.