# Apparent violation of Newton's Third Law in relativistic force transformation

In special relativity, we know that, relativistic force is defined as F = dp/dt, where p = γmv. For forces perpendicular to the direction of relative motion, force transforms as F' = γF. Consider two particles approaching each other with equal speeds in an initial frame S. They collide at a single point, exerting instantaneous forces on each other in a direction perpendicular to their motion. These forces are equal and opposite in frame S. If we transform to the rest frame S' of one particle:

• Let F be the force on the stationary particle.
• Then force on the moving particle is γF.
• Assume the forces act at the same point in space and time in S' in direction perpendicular to v.
• The time interval dt for the instantaneous collision is the same for both particles in this frame.

This scenario seems to lead to unequal impulses (Fdt vs γFdt), apparently violating conservation of momentum and Newton's Third Law.

Given that the forces are applied at the same position and instant in the rest frame, they should be simultaneous in both frames.

How can we resolve this apparent paradox while preserving both the force transformation law and conservation of momentum? Does this scenario reveal a fundamental issue with the looking only at the mechanical forces without considering the origin of the forces and fields in special relativity?

• Isn't the force on the moving particle only changed in the frame of the moving particle? But in that frame we also have to change dt, don't we? Am I missing something? Commented Jul 10 at 12:26
• @FlatterMann, In the frame of the moving particle, the force should match the proper force on the other particle in the original frame by the symmetry of the problem though Commented Jul 10 at 12:30
• @FlatterMann I should also point out that if force was parallel to v, then F' = F and everything works nicely. If we said that we have to consider dt then we have a paradox in the parallel force case instead. Commented Jul 10 at 12:38
• Four-force is a tensor, so if two forces are equal and opposite in one frame, they will be equal and opposite in all frames. Commented Jul 10 at 13:13
• @Andrew but that doesn't mean the first three components will be lorentz invarient Commented Jul 10 at 13:23