A WKB formalism for multicomponent fields and its application to gravitational and sound waves in perfect fluids

We review the WKB method for multicomponent fields obeying hyperbolic linear partial differential equations and derive a general necessary and sufficient condition for the formalism to provide transport equations. We apply the method to linearized perturbations of perfect fluid solutions to Einstein's equation and show that the gravitational and sound wave modes satisfy this condition, whereas a zero-frequency, non-propagating matter mode does not. We derive the transport equations for the wave amplitudes in leading order; they exhibit in particular the influence of background curvature on the propagation of gravitational waves