No Bel-Robinson Tensor for Quadratic Curvature Theories

TL;DR

This paper investigates the possibility of generalizing the Bel-Robinson tensor to quadratic curvature theories in various dimensions, finding that such generalizations are generally not feasible due to linearization instability.

Contribution

It demonstrates the limitations of extending the Bel-Robinson tensor to quadratic curvature models, highlighting linearization instability in these theories.

Findings

Conserved Bel-Robinson tensor exists only in the linearized limit of D=3 models.

No viable Bel-Robinson tensor can be extended to full quadratic curvature theories.

Quadratic curvature models exhibit linearization instability even with cosmological or Einstein terms.

Abstract

We attempt to generalize the familiar covariantly conserved Bel-Robinson tensor B_{mnab} ~ R R 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 ~ dRdR 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.