GENERAL TREATMENT OF GEODETIC AND LENSE–THIRRING EFFECTS ON AN ORBITING GYROSCOPE

We give a general derivation of the metric of a spinning body of any shape and composition using linearized general relativity theory (LGRT), and also obtain the same metric using a simple transformation argument. The latter derivation makes it clear that the linearized metric contains only the Eddington γ and α (≡ 1) parameters, so no new parameter is involved in any frame–dragging or Lense–Thirring (LT) effects. We then calculate the precession of an orbiting gyroscope in a general gravitational field, described by a Newtonian potential (gravito-electric field) and a vector potential (gravito-magnetic field). Finally we do a multipole analysis and give the general spherical harmonics expansion of the precession in terms of multipoles of the scalar and vector potentials, i.e., moments of the density distribution. In particular, in regard to the Gravity Probe B (GP-B) experiment, we find that the effect of the Earth's quadrupole moment J₂ on the geodetic precession is large enough to be measured by GP-B (a previously known result), but the effect on the LT precession is somewhat beyond the expected GP-B accuracy.