Dust matter and pure radiation in a plane gravitational wave

A non-perturbative model of a plane gravitational wave with dust matter and pure radiation is discussed. The form of the energy-momentum tensor of dust matter in a privileged coordinate system is obtained, where the model allows for an analytical integration of the equations of motion of test particles in the Hamilton-Jacobi formalism by the method of separation of variables. It is shown that this model leads to a contradiction and there is no solution to self-consistent field equations. This implies that the gravitational wave model should be more complex and include additional sources.