One-Electron Atom in Curved Space-Time

A one-electron atom in a general curved space-time is considered. The Hamiltonian of the Dirac equation is found in Fermi normal coordinates, and expressions are obtained, to first order in the Riemann tensor, for the shifts in energy of the $1S_{\frac{1}{2}}$, $2S_{\frac{1}{2}}$, and $2P_{\frac{1}{2}}$ levels.