One-electron atom as a probe of spacetime curvature

We consider a one-electron atom in an arbitrary curved spacetime. After reviewing the generalization of the Dirac equation to curved spacetime, we develop the perturbation theory of degenerate stationary states taking into account the Hermiticity properties appropriate to curved spacetime. We then calculate the Hamiltonian of the Dirac equation in Fermi normal coordinates to first order in the Riemann tensor, including the corrections to the electromagnetic field. As an application of these results, we obtain expressions in terms of the Riemann tensor for the shifts produced by the local curvature in the nonrelativistic $1S$, $2S$, and $2P$ energy levels, and in the relativistic $1S_{\frac{1}{2}}$, $2S_{\frac{1}{2}}$, and $2P_{\frac{1}{2}}$ energy levels.