Propagation peculiarities of mean field massive gravity

Massive gravity (mGR) describes a dynamical "metric" on a fiducial, background one. We investigate fluctuations of the dynamics about mGR solutions, that is about its "mean field theory". Analyzing mean field massive gravity propagation characteristics is not only equivalent to studying those of the full non-linear theory, but also in direct correspondence with earlier analyses of charged higher spin systems, the oldest example being the charged, massive spin 3/2 Rarita--Schwinger (RS) theory. The fiducial and mGR mean field background metrics in the mean field mGR model correspond to the RS Minkowski metric and external EM field. The common implications in both systems are that hyperbolicity holds only in a weak background-mean-field limit, immediately ruling both theories out as fundamental theories. Although both can still be considered as predictive effective models in the weak regime, their lower helicities exhibit superluminal behavior: lower helicity gravitons are superluminal as compared to photons propagating on either the fiducial or background metric. This "crystal-like" phenomenon of differing helicities having differing propagation speeds in both mean field mGR and mGR is a peculiar feature of these models.