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Modeling the propagation of gravitational waves (GWs) through matter iscomplicated by the gauge freedom of linearized gravity in that oncenonlinearities are taken into consideration, gauge artifacts can cause spuriousacceleration of the matter. To eliminate these artifacts, we propose how tokeep the theory of dispersive GWs gauge-invariant beyond the linearapproximation and, in particular, obtain an unambiguous gauge-invariantexpression for the energy--momentum of a GW in dispersive medium. Usinganalytic tools from plasma physics, we propose an exactly gauge-invariant``quasilinear'' theory, in which GWs are governed by linear equations and alsoaffect the background metric on scales large compared to their wavelength. As acorollary, the gauge-invariant geometrical optics of linear dispersive GWs in ageneral background is formulated. As an example, we show how the well-knownproperties of vacuum GWs are naturally and concisely yielded by our theory in amanifestly gauge-invariant form. We also show how the gauge invariance can bemaintained within a given accuracy to an arbitrary order in the GW amplitude.These results are intended to form a physically meaningful framework forstudying dispersive GWs in matter.

Gauge-invariant gravitational waves in matter beyond linearized gravity