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Assume that we have opposite charges initially static, like in the diagram. They pull on each other.
Let us then use a rocket to accelerate the positive charge upward very fast, so that it attains almost the speed of light.
The negative charge on the right still sees the positive charge at its original location for a while. The negative charge gains a momentum p to the left during that time.
We assume retardation. Signals do not travel faster than light. The negative charge does not know that the positive charge has moved.
Meanwhile, the positive charge gains a momentum q right, but |q| < |p| because the positive charge rapidly moves away.
Where did the difference of those momenta go? Could it be that the negative charge somehow emitted photons to the right, and those photons took the difference? That looks very implausible. The accelerating positive charge does emit energy, but that happens symmetrically and cannot take the extra momentum.
How can we ensure that momentum is conserved?
The rubber plate model comes to the rescue. The negative charge on the right pulls on the rubber plate of the positive charge. The negative charge gains a momentum p left, and the rubber plate gains a momentum p right.
The rubber plate delivers the momentum p later to the positive charge.
How can a massless rubber plate store momentum? Where is its inertia? Again, it is the finite speed of light which mimics inertia. When the negative charge pulls, the elastic rubber plate stretches a little bit. The longitudinal stretch wave travels at the speed of light to the positive charge and tugs on it.
We need to check if anyone has proved conservation of momentum under retarded forces. Gravity and the Coulomb force should honor conservation.
The Wheeler-Feynman (1949) absorber theory tries to tackle this issue.
S. Carlip (1999) writes that in General Relativity, a binary system conserves angular momentum (ignoring gravitational radiation). Retardation of forces would violate angular momentum conservation, but the effect is canceled by other factors.
Laplace in 1805 calculated that the "speed of newtonian gravity" has to be at least 7 million times the speed of light, so that retardation effects do not make the orbit of the Moon unstable.
Apparently, momentum conservation in classical electromagnetism is an open problem.
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