Additional symmetry operation ensuring action–reaction for particles with different velocities

Identify a symmetry operation (coordinate transformation) that, within the time-symmetric action-at-a-distance electrodynamics or gravitational framework where the four-forces depend only on electric charges, rest masses, position four-vectors, and four-velocities of the two interacting particles, ensures that the principle of action and reaction holds for particles moving with different velocities by producing equal-magnitude and opposite-direction impulses of the advanced and retarded four-forces.

Background

The paper develops a symmetry-based derivation of the principle of action and reaction (PAR) in a time-symmetric action-at-a-distance framework where four-forces depend only on charges, masses, positions, and four-velocities, not accelerations. For particles at relative rest, the authors show that PAR emerges via a sequence of conformal inversion and reflection across the interparticle axis (supplemented to ensure a positive Jacobian), and by taking the limit to null separation of corresponding infinitesimal worldline segments.

However, conformal transformations preserve local angles and therefore cannot interchange differing four-velocities; as a result, the authors’ construction only establishes PAR for particles at relative rest (same velocity). They explicitly conjecture that an additional, yet-undetermined symmetry operation is required to extend the symmetry derivation so that PAR holds for particles moving with different velocities.