We examine the behaviour of circular geodesics describing orbits of neutral test particles around an extreme Kerr-Newman black hole. It is well known that the radial Boyer-Lindquist coordinates of the prograde photon orbit r = r ph , marginally bound orbit r = r mb and innermost stable orbit r = r ms of the extreme Kerr black hole all coincide with the event horizon's value r = r + . We find that for the extreme Kerr-Newman black hole with mass M , angular momentum J and electric charge Q = ± M 2 − J 2 /M 2 (|J| ≤ M 2 ) the coordinate equalities r ph = r + , r mb = r + and r ms = r + hold if and only if |J| is greater than or equal to M 2 /2, M 2 / √ 3 and M 2 / √ 2, respectively.

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 derive the transition energies of geonium up to the relativistic corrections of 1/c 2 . These transition energies are used to obtain an extension of the well known gs-factor formula introduced by L. S. Brown and G. Gabrielse [Rev. Mod. Phys. 58, 233 1986].

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