Klein-Gordon equation in curved space-time

TL;DR

This paper numerically solves the Klein-Gordon equation in curved space-time for a particle bound to a massive object, analyzing relativistic effects and comparing them with nonrelativistic solutions across different mass density profiles.

Contribution

It introduces numerical solutions for the Klein-Gordon equation in various spherically symmetric space-times, exploring relativistic corrections to quantum bound states in curved space.

Findings

Relativistic corrections are quantified for different density profiles.

Solutions show deviations from Coulomb potential predictions due to space-time curvature.

Linear transition density profile avoids singularities in wave equations.

Abstract

We solve the relativistic Klein--Gordon equation for a light particle gravitationally bound to a heavy central mass, with the gravitational interaction prescribed by the metric of a spherically symmetric space-time. Metrics are considered for an impenetrable sphere, a soft sphere of uniform density, and a soft sphere with a linear transition from constant to zero density; in each case the radius of the central mass is chosen to be sufficient to avoid any event horizon. The solutions are obtained numerically and compared with nonrelativistic Coulomb-type solutions, both directly and in perturbation theory, to study the general-relativistic corrections to the quantum solutions for a $1/r$ potential. The density profile with a linear transition is chosen to avoid singularities in the wave equation that can be caused by a discontinuous derivative of the density.