Hyperfine splitting in Kerr spacetime

Neutral hydrogen hyperfine splitting of the atomic energy levels originates from the interaction between the electron and proton spins. Such a splitting is responsible for the 21-cm emission line of hydrogen during a transition between the hyperfine states. The 21-cm emission line represents a fundamental tool in astrophysics for studying the distribution of neutral hydrogen gas in galaxies and in the early Universe, e.g., during the Dark Ages and Cosmic Dawn. In the present paper we investigate the influence of gravity on the 21-cm emission line considering, in particular, the case of a cloud of hydrogen orbiting in the equatorial plane of a Kerr black hole. Using a Fermi-Walker tetrad, adapted to an orbiting hydrogen atom, we solve the related two-body Dirac equation in the Kerr background, analyzing the resulting hyperfine structure, as regards the energy spectrum and the polarization of the photons resulting from the allowed transitions. We show that the combined gravito-inertial effects of the orbital motion and of the background geometry, due to the Kerr black hole, cooperate to remove the residual three-fold degeneracy of the higher energy level in the hyperfine splitting. The studied effect can be considered as the gravitational analog of the anomalous Zeeman effect, with respect to which a peculiar property of innermost stable circular orbits is also highlighted.