## Abstract

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 language | English (US) |
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Article number | 045405 |

Journal | European Journal of Physics |

Volume | 39 |

Issue number | 4 |

DOIs | |

State | Published - 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.

## Keywords

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