Abstract
Starting from the coupling of a relativistic quantum particle to the curved Schwarzschild space time, we show that the Dirac-Schwarzschild problem has bound states and calculate their energies including relativistic corrections. Relativistic effects are shown to be suppressed by the gravitational fine-structure constant , where is Newton's gravitational constant, is the speed of light, and and are the masses of the two particles. The kinetic corrections due to space-time curvature are shown to lift the familiar degeneracy of the energy levels of the hydrogen atom. We supplement the discussion by a consideration of an attractive scalar potential, which, in the fully relativistic Dirac formalism, modifies the mass of the particle according to the replacement , where is the radial coordinate. We conclude with a few comments regarding the degeneracy of the energy levels, where is the principal quantum number, and is the total angular momentum, and illustrate the calculations by way of a numerical example.
- Received 5 March 2014
DOI:https://doi.org/10.1103/PhysRevA.90.022112
©2014 American Physical Society