  Modeling the propagation of gravitational waves (GWs) through matter is
complicated by the gauge freedom of linearized gravity in that once
nonlinearities are taken into consideration, gauge artifacts can cause spurious
acceleration of the matter. To eliminate these artifacts, we propose how to
keep the theory of dispersive GWs gauge-invariant beyond the linear
approximation and, in particular, obtain an unambiguous gauge-invariant
expression for the energy--momentum of a GW in dispersive medium. Using
analytic tools from plasma physics, we propose an exactly gauge-invariant
``quasilinear'' theory, in which GWs are governed by linear equations and 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 formulated. As an example, we show how the well-known
properties of vacuum GWs are naturally and concisely yielded by our theory in a
manifestly gauge-invariant form. We also show how the gauge invariance can be
maintained within a given accuracy to an arbitrary order in the GW amplitude.
These results are intended to form a physically meaningful framework for
studying dispersive GWs in matter.
