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 $R$, ${\mathcal{S}}_2\equiv {\tilde{R}}_a^b{\tilde{R}}_b^a$ and ${\mathcal{S}}_3\equiv {\tilde{R}}_a^b{\tilde{R}}_b^c{\tilde{R}}_c^a$ where ${\tilde{R}}_{ab}$ 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-$n\ge 2$ density can be written as a linear combination of $R^n$, which does not propagate the generic massive graviton, plus a density which does not propagate the generic scalar mode, $R^n-12n(n-1)R^{n-2}{\mathcal{S}}_2$, 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 ${\Omega }_{(6)}\equiv 6{\mathcal{S}}_3^2-{\mathcal{S}}_2^3$. 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 ${\Omega }_{(6)}$ to the Segre classification of three-dimensional metrics.

arXiv comment41 pages; v2: typos fixed, references added

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