Wave propagation through a spacetime containing thin concentric shells of matter

The interaction between waves and matter in the Universe is a fundamental but very challenging problem, since one needs to consistently solve the Einstein-matter equations in a dynamical regime. To shed some light on this problem, we investigate the transmission of scalar, electromagnetic, and linearized odd-parity gravitational waves in a toy model. This model consists of a static spacetime satisfying Einstein's field equations which is characterized by a spherical distribution of matter in the form of thin concentric equidistant shells of equal mass which is perturbed by a linear wave. More specifically, we assume that the central region has zero mass, and we verify that the resulting spacetime is stable with respect to small perturbations of the shell radii as long as the gravitational field is sufficiently weak. We focus on the transmission of monochromatic waves emitted from the center and propagating through a succession of $N$ shells. Analytical expressions for the transmission and reflection coefficients are obtained and their dependency on the frequency, the number of shells and their mutual distance is analyzed. In particular, in the high-frequency limit, we observe that the reflection coefficient decays with the fourth power of the frequency. Increasing the number of shells initially produces oscillations in the transmission coefficient; however, as $N$ grows, this coefficient rapidly stabilizes at a constant positive value. We attribute this property to the fact that reflections are mainly determined by the surface density of the shells, which decreases as the inverse square of their radii.