Plane waves in a relativistic, homogeneous and isotropic elastic continuum

The propagation of gravitational and acoustic plane waves in a flat universe filled with a general relativistic, homogeneous and isotropic, spatially flat continuum is studied. The continuum is described by analogues of nonrelativistic characteristics, namely energy per particle, pressure and Lamé coefficients, and considered in the comoving proper-time gauge. For all modes with the given wave covector, differential equations governing the time dependence of the amplitudes are derived. In particular, longitudinal acoustic waves are described, in analogy with the nonrelativistic theory, by two coupled first-order equations. As an example, plane waves in a stiff ultrarigid continuum are considered.