No Bel-Robinson Tensor for Quadratic Curvature Theories

We attempt to generalize the familiar covariantly conserved Bel-Robinson tensor Bμναβ ∼ RR of GR and its recent topologically massive third derivative order counterpart B ∼ RDR, to quadratic curvature actions. Two very different models of current interest are examined: fourth order D = 3 “new massive”, and second order D > 4 Lanczos-Lovelock, gravity. On dimensional grounds, the candidates here become B ∼ DRDR + RRR. For the D = 3 model, there indeed exist conserved B ∼ ∂R∂R in the linearized limit. However, despite a plethora of available cubic terms, B cannot be extended to the full theory. The D > 4 models are not even linearizable about flat space, since their field equations are quadratic in curvature; they also have no viable B, a fact that persists even if one includes cosmological or Einstein terms to allow linearization about the resulting dS vacua. These results are an unexpected, if hardly unique, example of linearization instability.