Obtaining equations of motion for charged particles in the $(v/c)^3$-approximation by the Einstein–Infeld–Hoffmann method

We consider some principal methodological problems that appear when the Einstein-Infeld-Hoffmann method is used to find approximate solutions of the general relativity equations and to obtain information about the motion of particles whose interaction force is much greater than the gravitational attraction force. Among these problems are normalizing approximate expressions by expanding exact solutions written in the same coordinate conditions used in the Einstein-Infeld-Hoffmann method, assigning the smallness orders depending on relations between the smallness parameters in play, and verifying cancellations of divergent terms arising in surface integrals. Solving these questions in accordance with the internal logic of the Einstein–Infeld–Hoffmann method results in new tools and techniques for applying the method. We demonstrate these tools and techniques in the example of the problem of the motion of two electrically charged pointlike particles in the $(v/c)^3$-approximation.