No Bel-Robinson Tensor for Quadratic Curvature Theories

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.Comment: published versio

Circular Symmetry in Topologically Massive Gravity

We re-derive, compactly, a TMG decoupling theorem: source-free TMG separates into its Einstein and Cotton sectors for spaces with a hypersurface-orthogonal Killing vector, here concretely for circular symmetry. We can then generalize it to include matter, which is necessarily null.Comment: amplified published versio

Shortcuts to Spherically Symmetric Solutions: A Cautionary Note

Spherically symmetric solutions of generic gravitational models are optimally, and legitimately, obtained by expressing the action in terms of the two surviving metric components. This shortcut is not to be overdone, however: a one-function ansatz invalidates it, as illustrated by the incorrect solutions of [1].Comment: 2 pages. Amplified derivation, accepted for publication in Class Quant Gra

Birkhoff for Lovelock Redux

We show succinctly that all metric theories with second order field equations obey Birkhoff's theorem: their spherically symmetric solutions are static.Comment: Submitted to CQ

Time (in)dependence in general relativity

We clarify the conditions for Birkhoff's theorem, that is, time-independence in general relativity. We work primarily at the linearized level where guidance from electrodynamics is particularly useful. As a bonus, we also derive the equivalence principle. The basic time-independent solutions due to Schwarzschild and Kerr provide concrete illustrations of the theorem. Only familiarity with Maxwell's equations and tensor analysis is required.Comment: Revised version of originally titled "Kinder Kerr", to appear in American Journal of Physic

De/re-constructing the Kerr metric

We derive the Kerr solution in a pedagogically transparent way, using physical symmetry and gauge arguments to reduce the candidate metric to just two unknowns. The resulting field equations are then easy to obtain, and solve. Separately, we transform the Kerr metric to Schwarzschild frame to exhibit its limits in that familiar setting

Bel-Robinson as stress-tensor gradients and their extensions to massive matter

We show that the Bel–Robinson (BR) tensor is—generically, as well as in its original GR setting—an autonomously conserved part of the, manifestly conserved, double gradient of a system’s stress-tensor. This suggests its natural extension from GR to matter models, first to (known) massless scalars and vectors, then to massive ones, including tensors. These massive versions are to be expected, given that they arise upon KK reduction of massless D+1D+1 ones. We exhibit the resulting spin (0,1,2)(0,1,2) “massive” BR

The Bel-Robinson tensor for topologically massive gravity

We construct, and establish the (covariant) conservation of, a 4-index ‘super stress tensor’ for topologically massive gravity. Separately, we discuss its invalidity in quadratic curvature models and suggest a generalization

Is BTZ a separate superselection sector of CTMG?

We exhibit exact solutions of (positive) matter coupled to cosmological TMG; they necessarily evolve to conical singularity/negative mass, rather than physical black hole, BTZ. By providing evidence that the latter constitutes a separate, "superselection", sector not reachable from the physical one, they also provide justification for retaining TMG's original "wrong" G-sign to ensure excitation stability here as well.Comment: published versio