Coupled gravitational and electromagnetic waves in arbitrary space-times with magnetized plasmas and dust,I

We derive constraint-free, coupled wave equations for the propagation of coupled electromagnetic and gravitational waves traveling through a time-dependent inhomogeneous medium. The medium consists of an arbitrary gravitational field, dust, a cold two-fluid plasma, and an arbitrary magnetic field. In this first of two papers we apply a two-timing ansatz to the constraint-free system of wave equations. In the second paper, dispersion relation and transport equations are found by means of a WKB analysis.