Gravitational effects on geonium and free electron $\mathrm{g}_s$-factor measurements in a Penning trap

We present a theoretical analysis of an electron confined by a Penning trap, also known as geonium, that is affected by gravity. In particular, we investigate the gravitational influence on the electron dynamics and the electromagnetic field of the trap. We consider the special case of a homogeneous gravitational field, which is represented by Rindler spacetime. In this spacetime the Hamiltonian of an electron with anomalous magnetic moment is constructed. Based on this Hamiltonian and the exact solution to Maxwell equations for the field of a Penning trap in Rindler spacetime, we derived the transition energies of geonium up to the relativistic corrections of $1/\mathrm{c}^2$. These transition energies are used to obtain an extension of the well known $\mathrm{g}_s$-factor formula introduced by L. S. Brown and G. Gabrielse [Rev. Mod. Phys. 58, 233 1986].