Derivation of the Equations of Motion of a Gyroscope from the Quantum Theory of Gravitation

Previous work on the gravitational two-body problem is surveyed. Next, we present a new approach, which we consider to be simpler and more transparent than the usual methods because it is based on a gravitational potential energy. This enables us to carry out our calculations using only the familiar tools of Newtonian mechanics and the Euler-Lagrange equations. Starting from a gravitational potential energy derived from Gupta's quantum theory of gravitation, the classical motion of a spherical gyroscope in the gravitational field of a much larger mass with a quadrupole moment is found. The results of the precession of the spin are compared with those of Schiff, and a detailed derivation of the results of O'Connell for the effect of a quadrupole moment (and higher moments) on the precession of the spin is presented. In addition, we present some new results. First, we show that the quadrupole moment manifests its presence in another way, which also contributes to the precession of the gyroscope a term that is about ten times larger than what could be detected. Second, with regard to the precession of the orbit, in addition to the usual contributions, our results include the effects of the spin of both particles (which enables us to calculate the effect of the rotation of Mercury on the precession of its perihelion).