A spherical mass shell as a `total reflector' for gravitational waves

Abstract

For normal bulk material the reflection coefficient for gravitational waves is usually much smaller than 1. Therefore, a spherical mass shell (mass M, radius R) can serve as a `container' for gravitational radiation at most in the extreme case R = 2M. If, however, the shell material is especially adapted to the properties of the wave, total reflection can occur due to interference effects, i.e. suitable mass shells can totally reflect weak gravitational waves of arbitrary frequency and angular momentum l coming from the outside; for waves inside the shell total reflection is only possible for a series of discrete -values, dependent on R and l. This is proven by first-order perturbation theory on the basis of the Schwarzschild mass shell. Perturbation of the exterior Schwarzschild metric is treated in Regge - Wheeler gauge. Connection of this exterior metric to a (suitably transformed) flat interior metric at r = R according to Israel's method leads to a unique relation between the amplitudes of the ingoing and outgoing wave (total reflection!). For , the shell material fulfils the dominant energy condition. The logarithmic phase of the asymptotic gravitational wave (in radiation gauge) compares nicely with the phase in relativistic Coulomb scattering: different sign of the logarithmic term due to gravitational attraction, and a factor 2 due to the spin of the `graviton'.