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

James Evans, University of Puget Sound
Paul M. Alsing, Albuquerque High Performance Computing Center, University of New Mexico, Albuquerque, New Mexico?87131
Stefano Giorgetti, Department of Physics, University of Milan, 20133?Milano, Italy
Kamal Kanti Nandi, Department of Mathematics, University of North Bengal, Darjeeling (WB)?734430, India

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 Wentzel–Kramers–Brillouin solution of the general-relativistic Klein–Gordon equation. Finally, we make some observations on the significance of the optical action.