A general linear gauge-invariant equation for dispersive gravitational waves (GWs) propagating in matter is derived. This equation describes, on the same footing, both the usual tensor modes and the gravitational modes strongly coupled with matter. It is shown that the effect of matter on the former is comparable to diffraction and therefore negligible within the geometrical-optics approximation. However, this approximation is applicable to modes strongly coupled with matter due to their large refractive index. GWs in ideal gas are studied using the kinetic average-Lagrangian approach and the gravitational polarizability of matter that we have introduced earlier. In particular, we show that this formulation subsumes the kinetic Jeans instability as a collective GW mode with a peculiar polarization, which is derived from the dispersion matrix rather than assumed a priori. This forms a foundation for systematically extending GW theory to GW interactions with plasmas, where symmetry considerations alone are insufficient to predict the wave polarization.

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Reduced theories of gravitational waves (GWs) often grapple with untangling the physical effects from coordinate artifacts. Here we show how to reinstate gauge invariance within a reduced theory of weakly nonlinear GWs in a general background metric and in the presence of matter. An exactly gauge-invariant "quasilinear" theory is proposed, in which GWs are governed by linear equations but also affect the background metric on scales large compared to their wavelength. As a corollary, the gauge-invariant geometrical optics of linear dispersive GWs in a general background is reported. We also show how gauge invariance can be maintained within a given accuracy if nonlinearities are included up to an arbitrary order in the GW amplitude.

A general method is proposed for identifying the gauge-invariant part of the metric perturbation within linearized gravity, and the six independent gauge invariants per se, for an arbitrary background metric. For the Minkowski background, the operator that projects the metric perturbation on the invariant subspace is proportional to the well-known dispersion operator of linear gravitational waves in vacuum.

In linearized gravity with distributed matter, the background metric has no generic symmetries, and decomposition of the metric perturbation into global normal modes is generally impractical. This complicates the identification of the gauge-invariant part of the perturbation, which is a concern, for example, in the theory of dispersive gravitational waves whose energy--momentum must be gauge-invariant. Here, we propose how to identify the gauge-invariant part of the metric perturbation and the six independent gauge invariants per se for an arbitrary background metric. For the Minkowski background, the operator that projects the metric perturbation on the invariant subspace is proportional to the well-known dispersion operator of linear gravitational waves in vacuum. For a general background, this operator is expressed in terms of the Green's operator of the vacuum wave equation. If the background is smooth, it can be found asymptotically using the inverse scale of the background metric as a small parameter.