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 (mGR) 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 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; a situation in stark contrast with general relativity (GR) which is at least a consistent classical theory. Moreover, even though both mGR and RS theories can still in principle be considered as predictive effective models in the weak regime, their lower helicities then exhibit superluminal behavior: lower helicity gravitons are superluminal as compared to photons propagating on either the fiducial or background metric. Thus our approach has uncovered a novel, dispersive, "crystal-like" phenomenon of differing helicities having differing propagation speeds. This applies both to mGR and mGR, and is a peculiar feature that is problematic for consistent coupling to matter.