Balancing of Electromagnetic and Gravitational Forces and Torques Between Spinning Particles at Rest

Abstract

The results reported here arise from a line of development that includes the following: (1) Schiff (1960), precession of a spinning object in a gravitational field endowed with "magnetic" components, (2) Oliveira and Tiomno (1962), same precession derived from the Dirac equation plus correspondence principle, (3) present work, determination of the force on a spinning object again by use of Dirac's equation plus the correspondence principle - in agreement with what Wald and what Wilkins and Ruffini find from Papapetrou's classical equations of motion for a spinning object. The same result also follows from Kramer's classical nonrelativistic Hamiltonian formalism applied to electromagnetic and gravitational interactions. We demand that any coupling of a spinning object with a "magnetic" type of gravitational field must satisfy the principle of equivalence of gravitational and inertial forces in the sense that the gravitational "gyromagnetic" factor must be exactly unity. The resulting equations of motion are used to prove that if a system of arbitrarily spinning charged particles, satisfying the conditions $q_i=m_i$ and with gyromagnetic factors $g_i=2\mathrm{}$, is initially at rest it will remain at rest and without precession of the spins. This gives an insight on how the solutions of the Einstein-Maxwell equations recently obtained by Perjés and by Israel and Wilson can be stationary in spite of a nonvanishing magnetic field.