Linearized propagation equations for metric fluctuations in a general
  (non-vacuum) background geometry
  The linearized dynamical equation for metric perturbations in a fully
general, non-vacuum, background geometry is obtained from the Hamilton
variational principle applied to the action up to second order. We specialize
our results to the case of traceless and transverse metric fluctuations, and we
discuss how the intrinsic properties of the matter stress tensor can affect
(and modify) the process of gravity wave propagation even in most conventional
geometric scenarios, like (for instance) those described by a FLRW metric
background. We provide explicit examples for fluid, scalar field and
electromagnetic field sources.
