Wave propagation through a spacetime containing thin concentric shells of matter

TL;DR

This paper analyzes how scalar, electromagnetic, and gravitational waves transmit through a static spacetime with multiple thin matter shells, deriving analytical expressions for transmission and reflection coefficients and exploring their dependence on frequency and shell configuration.

Contribution

It provides the first analytical study of wave transmission through multiple concentric matter shells in a static spacetime, including explicit formulas and stability analysis.

Findings

Reflection coefficient decays as the inverse fourth power of frequency at high frequencies.

Increasing the number of shells causes initial oscillations in transmission, which stabilize as shells grow.

Reflection mainly depends on surface density, decreasing with the inverse square of shell radii.

Abstract

We investigate the transmission of scalar, electromagnetic, and linearized odd-parity gravitational waves in a static spacetime characterized by a spherical distribution of matter in the form of thin concentric equidistant shells of equal mass. These shells connect Schwarzschild spacetimes of different masses between themselves, and they satisfy the Israel junction conditions with a polytropic-type equation of state for the surface energy-momentum tensor. 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. To this purpose, we neglect the self-gravity of the waves and solve the Regge-Wheeler equation in the weak field limit of the background field. 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.