Index of refraction for scalar, electromagnetic, and gravitational waves in weak gravitational fields

Abstract

We consider the solutions of the scattering of scalar, electromagnetic, and gravitational waves by the gravitational field of a single particle, for the case of small wave amplitudes and weak gravitational fields. Scatterings are considered for both incident plane waves and incident spherical waves. For plane waves incident on a thin sheet of matter composed of free particles, the superimposed wave solutions give rise to a phase change arising from the coordinate dependence of the speed of light on the gravitational potential, focusing of the incident wave by the sheet, and, in some cases, a phase change due to dispersion of the wave by the matter. For gravitational waves, the index of refraction $n$ is given by $n\ensuremath{-}1=\frac{2\ensuremath{\pi}G\ensuremath{\rho}}{{\ensuremath{\omega}}^{2}}$, assuming $n\ensuremath{-}1$ is small, and for electromagnetic waves $n=1$ to the same order. The index of refraction for scalar waves depends on the form of the scalar-wave equation used. The generation of back-scattered waves is also treated. Calculations are repeated for spherical waves incident on a thin spherical shell of matter. The propagation of $\ensuremath{\delta}$-function wave packets is then treated in order to show that the solutions are consistent with causality, even though, in some cases, the group velocity exceeds the velocity of light.