Metric for an oblate earth

In linearized general relativity the metric of a body is described by a scalar potential and a three-vector potential. We here present a simple transformation derivation of the linearized metric in terms of these potentials, and calculate the exact scalar and vector potentials fora field with oblate spheroidal symmetry. The results for the external potentials do not depend on details of the density distribution inside the earth; both the scalar and vector potentials are fully determined by the total mass, the total angular momentum, and a radial parameter, all of which are accurately known from observation. The scalar potential is accurate to roughly 10(-6) and the vector potential, which has never been accurately measured, should be accurate to about 10(-5). Applications include an accurate treatment of the details of the motion of satellites, and the precession of a gyroscope in earth orbit.