Example of mass–energy relation: Classical hydrogen atom accelerated or supported in a gravitational field

A simple example of the mass–energy relation is given by using electromagnetic calculations for a classical hydrogen atom, a point mass electron moving under the Coulomb force in a uniform circular orbit around a heavy point mass proton when the radiation-reaction self-force is neglected. The system mass is determined by calculating the ratio of the total external force to the system acceleration in the limit of small acceleration, and is shown to be equal to $c−2$ times the total system energy including the particle rest-mass energies, the particle kinetic energies, and the electrostatic potential energy. From the equivalence principle, the external forces can be regarded either as accelerating the atom relative to an inertial frame or else as supporting the atom in a gravitational field. The calculations are carried out for two different models. In the first case, the atom is regarded as accelerated by a frictionless surface which applies forces to both the electron and the proton. In the second case, the atom is accelerated by a single external force applied to the proton, and the electron orbit is displaced relative to the proton. In both cases the accelerating forces are constant forces and there are no external forces stabilizing the system.