Klein-Gordon equation in curved space-time

R. D. Lehn, Sophia S Chabysheva, John R Hiller

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7 Scopus citations


We report the methods and results of a computational physics project on the solution of the relativistic Klein-Gordon equation for a light particle gravitationally bound to a heavy central mass. The gravitational interaction is 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. This project should be of interest to instructors and students of computational physics at the graduate and advanced undergraduate levels.

Original languageEnglish (US)
Article number045405
JournalEuropean Journal of Physics
Issue number4
StatePublished - May 17 2018

Bibliographical note

Funding Information:
RDL gratefully acknowledges the support of a Research Assistantship from the Department of Physics and Astronomy at the University of Minnesota-Duluth.

Publisher Copyright:
© 2018 European Physical Society.


  • Klein-Gordon equation
  • computational physics
  • gravitational binding
  • static spherical space-times


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