The Ehrenfest paradox breaks conservation of momentum in special relativity?

The Ehrenfest paradox concerns the measured circumference of a rigid rotating disk. The circumference appears lengthened in the co-rotating frame, whereas the circumference is 2 π r in an inertial frame.

https://en.wikipedia.org/wiki/Ehrenfest_paradox

Let us introduce a linear version of the paradox.

                      

                       movement ---->

                weights connected with springs

           ... ●/\/\/\●/\/\/\●/\/\/\●/\/\/\● ...

           ... ●      ●       ●       ●       ● ...

                                      weights

                       <----  movement

We have two very long rows of weights. Initially they are static in the laboratory frame. There is some energy stored in the lower row weights. We use that energy to push the upper row weights to the right. The lower row starts to move left. Then we stop the movement and return the energy to the lower row.

During the movement, the springs in the upper row are stretched in the co-moving frame of the upper row.

The upper row will move slower than the lower row because in the upper row, in the laboratory frame, some of the added energy becomes potential energy in the springs - it is not kinetic energy of the weights.

After the movement stops, the newtonian center of mass of the system has moved.

Should we give up the newtonian center of mass theorem? Or is there a way to save it?

Possible solution of the paradox. Let the springs be force fields. The fields cannot have zero mass. We need to make the weights lighter in the upper row, in order to have the same mass in both rows. When we use a force between the two rows to accelerate the two rows, the mass of the force field in the upper row appears as reduced, and we do more work to accelerate the upper row than the lower row. The extra work will eventually end up as potential energy in the force field.

If the solution is right, then the inertial mass of a force field must appear reduced when we accelerate the particle carrying that field.

Radiation from the accelerated force field complicates things.

Suppose that we try to minimize mass reduction by making the rows very long and by accelerating very slowly. But minimization probably fails, because, e.g., the electric field strength |E| for a long row of charges is

       ~ 1 / r,

where r is the distance from the row. Total field energy is

       ~ ∫ 1 / r²  *  r dr,

which diverges.