The equations of motion of slowly moving particles in the general theory of relativity

Abstract

The equations of motion of slowly moving particles in a harmonic co-ordinate system are obtained up to an accuracy of the 9th order inc-1, wherec is the velocity of light. This is done for the twoparticle case, and is an extension of the familiar Einstein, Infeld and Hoffmann (EIH) equations of motion, which hold up to an accuracy of the 6th order inc-1. However, our method of obtaining the equations of motion is essentially different from the EIH method, in that the retarded-potential solutions for the wave equations are chosen rather than the symmetric potentials; we thus obtain a radiative force term. The method is based on the assumption that the metric tensor of the space-time differs from the Minkowskian metric by perturbations, which can be expanded in a power series in the masses of the particles. The passage to the low-vlocity case is made by every function of the field being expanded in a power serie3 inc- 1. Throughout this work we have used the Infeld delta-function; it is shown that the field functions at the 8th order do not obey the rule for “ tweedling ” the products, as has so far been assumed by Infeld. The equations of motion so obtained are very complicated ; it is found that the choice of the potentials (retarded or advanced) affects them only in the 9th order, that the force term corresponding to the dipole radiation (namely the 7th order term) vanishes, and that the force terms up to the 6th order are identical with those of The EIH equations. It is remarkable that, by repeated use of the equations of motion, we can reduce them to a form containing derivatives (with respect to time) no higher than the second. A discussion on our method and comparisons with other methods are given.