Matter waves in a gravitational field: An index of refraction for massive particles in general relativity

TL;DR

This paper introduces an index of refraction for matter waves of massive particles in a gravitational field within general relativity, linking wave propagation to classical optics concepts.

Contribution

It establishes a classical-like index of refraction for matter waves in curved spacetime, supported by a WKB analysis of the Klein-Gordon equation.

Findings

Existence of a classical-like index of refraction for matter waves

Validation through WKB solution of the Klein-Gordon equation

Insights into the optical action in gravitational fields

Abstract

We consider the propagation of massive-particle de Broglie waves in a static, isotropic metric in general relativity. We demonstrate the existence of an index of refraction that governs the waves and that has all the properties of a classical index of refraction. We confirm our interpretation with a WKB solution of the general-relativistic Klein-Gordon equation. Finally, we make some observations on the significance of the optical action.