Klein-Gordon equation in curved space-time

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.

18 pages, 9 figures, RevTeX 4.1

References in corpus (1)

• Phase shift in atom interferometry due to spacetime curvature

Cited by in corpus (5)

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• Gravitational soliton solutions to self-coupled Klein-Gordon and Schrödinger equations
• Classification of quantum states based on the null energy condition
• Gravitational Waves beyond the Linear Approximation and Gravitational Wave Reflection
• What can we learn from the conformal noninvariance of the Klein-Gordon equation?