Gravitation-Induced Electric Field near a Metal

Abstract

A quantum-mechanical formalism is developed to calculate the electric field produced in the vicinity of a metallic object through the influence of the earth's gravitation. The field is proportional to the gradient of the ground-state energy eigenvalue of the object with respect to the position of a test charge located at the field point. This expression can be reduced to the solution of a problem in classical electrostatics, and is valid as well for a superconductor. Simple explicit results are obtained for the field within a closed metallic shell of arbitrary shape, and outside of a metallic sphere. In the former case, the field is uniform and equal to $\frac{\mathrm{mg}}{e}$, directed so as to exert an upward force on an electron; $m$ and $e$ are the electron mass and charge, and $g$ is the acceleration of gravity. This result is of importance in connection with current experiments on the free fall of electrons and positrons, and leads to the expectation that shielded electrons will not fall, while shielded positrons will fall with acceleration $2g$. Some comments are made on the gravitation-induced electric field near a nonconductor, and on the field near a rapidly rotating solid.