What I am going to speak about may not be a paradox but i see a contradiction here so I used used the word "paradox". To begin with, let there be 2 charges A and B which are stationary with respect to each other and their separation is one light year in a frame which is stationary with respect to them. Then some change occur. For A , A is stationary and suddenly B made one oscillation (let it be in any direction and very small amplitude). At the end of oscillation , B is at its initial position WRT A. Thus according to A , a ripple in electric field will propagate away from B towards all directions which will reach A after 1 year from that change whereas no such ripple will propagate from A . Similarly for B , B is stationary and A oscillate. Thus from perspective of B , A is supposed to emit ripple and not B. If the ripple is emitted by A or B then both of them should see the ripple. Even every observer should see a ripple if it exists. So contradtion arises.
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2$\begingroup$ If one of the particles oscillate, they are accelerating. Acceleration is not relative, but can be measured with an accelerometer. So this situation is not symmetrical between A and B. $\endgroup$– Marius Ladegård MeyerCommented Feb 13 at 18:53
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2$\begingroup$ There is no equivalence between accelerating and non-accelerating observers in special relativity or in electrodynamics. $\endgroup$– Michael SeifertCommented Feb 13 at 18:54
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$\begingroup$ Thx for answering . Suppose A accelerated WRT inertial frame of reference then will a see the ripple around it? $\endgroup$– Sanjoy KunduCommented Feb 13 at 19:03
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Acceleration is a frame-independent phenomenon. All inertial observes agree on whether a given object is accelerating or not and what is its acceleration. Accelerating observers are not on the same ground as inertial observers, so the situation is not symmetric. In the description you gave, $B$ is accelerating, while $A$ is inertial. Hence, all inertial observers agree that radiation is emitted by $B$ and all inertial observers agree that no radiation is emitted by $A$.
answered Feb 13 at 19:05
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$\begingroup$ And what about non inertial observers like B? Will they know that they are radiating? $\endgroup$ Commented Feb 15 at 11:13
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$\begingroup$ @SanjoyKundu I'm not fully sure, but I believe so. I guess they should notice they are losing energy $\endgroup$ Commented Feb 17 at 21:37
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$\begingroup$ One think I notice is that if we treet electric field lines as ropes that are extending out from a charge at a speed of light instead of simple rigid rods or lines that never move then all questions regarding electric field gets answered. I even have an expression for line density at a point. That approach also proves that B will see waves around it. It also says that those lines emitted from B will travel a curved path from perspective Of B when it accelerates.@Níckolas Alves $\endgroup$ Commented Feb 19 at 21:13