Fine-Structure Constant for Gravitational and Scalar Interactions

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 alpha_G = G m_1 m_2/(hbar c), where G is Newton's gravitational constant, c is the speed of light and m_1 and m_2 >> m_1 are the masses of the two particles. The kinetic corrections due to space-time curvature are shown to lift the familiar (n,j) 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 m -> m (1 - \lambda/r), where r is the radial coordinate. We conclude with a few comments regarding the (n,j) degeneracy of the energy levels, where n is the principal quantum number, and j is the total angular momentum, and illustrate the calculations by way of a numerical example.