The force law for the dynamic two-body problem in the second post-Newtonian approximation of general relativity

Abstract

We calculate the force between two well-separated bodies in a fully dynamic system of two extended bodies up to and including the second post-Newtonian approximation (PNA). We use the iteration procedure as formulated by Anderson and Decanio in a version whose divergences have been pushed to the third PNA. We show the following, (i) The force law assumes the “Newtonian form” if a second approximation in 1/(separation of the bodies) is made; (ii) the mass terms appearing in the force law are the (Tolman) masses of the individual bodies expanded up the second PNA; the interal masses equal the (passive and active) gravitational masses of the bodies in order considered; they are all constants of the motion; (iii) the self-fields of the bodies vanish in the second PNA; hence there is no Nordvedt effect in the second PNA; (iv) the compactness of the bodies, i.e., (gravitational radius)/(body size), does not appear in the force law; only the relation between mass and the matter variables is changed in the PNA as compared with the corresponding Newtonian result.