### 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) |
---|---|

Article number | 045405 |

Journal | European Journal of Physics |

Volume | 39 |

Issue number | 4 |

DOIs | |

State | Published - May 17 2018 |

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### Keywords

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

### Cite this

*European Journal of Physics*,

*39*(4), [045405]. https://doi.org/10.1088/1361-6404/aabdde

**Klein-Gordon equation in curved space-time.** / Lehn, R. D.; Chabysheva, Sophia S; Hiller, John R.

Research output: Contribution to journal › Article

*European Journal of Physics*, vol. 39, no. 4, 045405. https://doi.org/10.1088/1361-6404/aabdde

}

TY - JOUR

T1 - Klein-Gordon equation in curved space-time

AU - Lehn, R. D.

AU - Chabysheva, Sophia S

AU - Hiller, John R

PY - 2018/5/17

Y1 - 2018/5/17

N2 - 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.

AB - 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.

KW - Klein-Gordon equation

KW - computational physics

KW - gravitational binding

KW - static spherical space-times

UR - http://www.scopus.com/inward/record.url?scp=85050069710&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050069710&partnerID=8YFLogxK

U2 - 10.1088/1361-6404/aabdde

DO - 10.1088/1361-6404/aabdde

M3 - Article

VL - 39

JO - European Journal of Physics

JF - European Journal of Physics

SN - 0143-0807

IS - 4

M1 - 045405

ER -