Propagation of gravitational waves through pressureless matter

The authors investigate the propagation of small-amplitude gravitational waves through pressureless matter ('dust'). After establishing the local linearisation stability of Einstein's equation for dust about any of its solutions they use the WKB method to study the locally plane, linearised perturbations of an arbitrary background dust spacetime asymptotically for small wavelengths. The dispersion relation exhibits two modes. One is simply degenerate and represents gravitational waves, whereas the other is doubly degenerate and describes density and vorticity perturbations. The waves are shown to propagate along the null geodesics of the background; in leading WKB order their amplitudes behave as in vacuo. The rays associated with the matter mode are the wordlines of the background dust. In leading order the perturbations of density and 4-velocity vanish for both modes