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 action up second order. We specialize our results case of traceless and transverse fluctuations, we discuss how intrinsic properties matter stress tensor can affect (and modify) process gravity wave propagation even most conventional geometric scenarios, like (for instance) those described by FLRW background. provide explicit examples fluid, scalar field electromagnetic sources.