Eigenvalues of the static, spherically-symmetric Einstein-Proca equations

TL;DR

This paper derives static Proca potentials on a Schwarzschild background, revealing their distinct forms from flat space and showing how gravity can induce quantization of particle mass through the coupling of fields.

Contribution

It introduces the derivation of static Proca potentials in curved spacetime and uncovers a gravity-induced quantization condition on particle mass.

Findings

Proca potentials differ significantly in curved spacetime from flat space.

A quantization condition on particle mass is derived, inversely related to the range parameter.

Gravity may induce quantization in physical fields.

Abstract

The Proca potential has been used in many contexts in flat spacetime, but not so often in curved spacetime. Here, the static Proca potentials are derived on a Schwarzschild background, and are found to have very different forms from those in flat space. In addition, the coupling of the two fields leads to a quantum condition on the particle mass, which is found to be quantized and inversely proportional to the range parameter, $\mu$. This suggests that gravity may induce quantization in other physical fields.