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Suppose there are two charged particles separated by some distance $d$ both with an equal positive charge of $q$. The particles also have equal masses of $m$. $m$ and $q$ are chosen so the electrostatic repulsive force between the particles exactly equals the gravitational attractive force between them and the particles are stationary. Now suppose the charges of the particles are increased by some small $\epsilon$ so the particles are repelled from each other with a very small force.

Suppose also that the particles are moving at a constant relativistic velocity $v$ at a right angle to their separation. In the stationary reference frame, the particles' masses will increase by some amount so that the gravitational force between them is larger than the electrostatic force and they accelerate towards each other. But in the reference frame of the particles, their masses do not increase and so they are accelerating away from each other. What is the resolution to this paradox?

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The resolution is that in the stationary frame, in addition to the Electric field present due to the charge of the particles, there will also be a magnetic field because the moving charges are a current.

Note that transforming from the particles' rest frame to the frame in which they move, not only do you have to transform their positions and velocities, but you also have to transform the electromagnetic field according to SR, see the Wikipedia article on the EM tensor.

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  • $\begingroup$ You should also transform the gravitational field! $\endgroup$ – xaxa Feb 13 '15 at 11:14
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The magnetic effect of the moving charge is trying to put them together instead of separating them. However, the charges will look closer in the stationary reference so the repel force should be bigger as well, which should overcome both gravity and magnetic field. The overall effect should be consistent in both stationary or the moving reference.

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