freedom cannot be completely excluded by experiment, spin zero or one components of gravity are very strongly constrained, since, for example, they would imply violations of the principle of equivalence.

This aspect of diffeomorphism invariance presents a particularly strong challenge for emergent models, since an “approximate symmetry” can be qualitatively different from an exact one. Suppose, for instance, that an emergent weak gravitational field can be described at some length scale by a Lorentz invariant spin two field, but without the gauge invariance corresponding to linearized diffeomorphisms. The lowest order action is then the Fierz-Pauli action for a field of mass $m$. But although linear diffeomorphism invariance is restored in the $m\rightarrow0$ limit, that limit differs from weak field general relativity [@VDV; @Z], and gives incorrect predictions for Solar System tests. This van Dam-Veltman-Zakharov (vDVZ) discontinuity arises because an extra scalar mode fails to decouple even in the massless limit. Moreover, as noted earlier, nonlinear extensions of this model typically contain negative energy Boulware-Deser ghosts [@BD; @Hinterbichler].

As Vainshtein first pointed out [@Vainshtein], the vDVZ discontinuity may indicate a breakdown of weak field perturbation theory: nonlinear effects proportional to inverse powers of $m$ may appear, signaling the onset of a 