General-relativistic effects in hydrogenic systems

We examine in detail the behavior of bound systems containing a spin-$\frac{1}{2}$ fermion (e.g., atomic hydrogen or positronium) in an external gravitational field. Starting with the generally covariant Dirac equation, we derive the effective Hamiltonian for a fermion in a weak gravitational field correct through order $\frac{v^2}{c^2}$, where $v$ is the fermion velocity. The resulting expression is then used to obtain the gravitational Hamiltonian for both hydrogen and positronium including relativistic effects. It is shown that the form of the Hamiltonian for the bound system depends on the choice of center-of-mass and relative coordinates, and several choices of these coordinates are considered. An extensive discussion is given of the relativistic variables used to describe the bound system and their physical significance. The principal focus of this paper is a relativistic gravitational analog of the Stark effect which arises from a set of post-Newtonian terms in the bound-state Hamiltonian. These are shown to mix opposite-parity states in hydrogen, such as $S_{\frac{1}{2}}$ and $P_{\frac{1}{2}}$, and lead to a correlation between the local acceleration of gravity $\overrightarrow{\mathrm{g}}$ and the photon polarization in electromagnetic decays. We discuss the possibility that a study of this polarization could be used to discriminate among different theories of gravity at the quantum level through its dependence on one of the parametrized post-Newtonian parameters. For a fermion-antifermion system such as positronium, the interaction Hamiltonian can admix states with opposite values of $P$ and $\mathrm{CP}$, as we illustrate with several example. Our results apply specifically to the case of a hydrogenic system supported in a gravitational field by nongravitational forces. The effects of these are not explicitly considered.