We present two thought experiments. A) Consider an electron in a charged shell at 0.5 MeV. Moving this system from a gravitational field appears not to affect the total force on the electron. B) Consider an oriented diatomic molecule in a high resolution mass spectrometer. The orientation independence of the molecular axis relative to its acceleration is a test of covariance. These thought experiments are motivated by Boyer's selfforce picture AJP 46, 383 (1978), AJP 47, 129 (1979).
These thought experiments involve the gravitation and inertial mass including the electromagnetic self interaction, respectively. It is the self interaction that leads to the interesting properties of both systems.
To obtain a mass in an electromagnetically interacting system a distinction can be made between the energy derived mass, the momentum derived mass, and the self force derived mass, as beautifully reviewed by Griffiths AJP 51, 1120 (1983). These masses show discrepancies that have been discussed in the literature for many decades. The discrepancies are know to be resolved for the case of inertial mass by either using Poincare stresses or Rohrlich’s redefinition of the energy-momentum four-vector.