This problem can easily be understood by the following paradox.

Before stating this paradox, I want to derive a formula given in standard books on relativity. In Special Relativity, the four-force in the -direction for any inertial frame is:

- where is the Lorentz transformation.

-----(A)

--------(B)

- ------(1)

Now, Paradox:

An object is moving with constant velocity in the -direction on a frictionless platform. The only non-zero magnetic force is acting in -direction, and there is acceleration in the -direction only with velocity & .

If we apply Eq. (1) to this case then result will be

- ---------- as ,

This means that, if there is no magnetic force acting on object from outside in -direction and no , then the force will act on object in direction of -axis due to .

Important point (1):

This means that the applied magnetic force on object in the -direction is and that the acting force in the -direction is

- ,

- .

Step 2

Now, the force acting in the -direction is

- or .

- ,

So, Eq. (1) becomes

Now, see above equation carefully, it is of nature

This means that the applied magnetic force on the object in -direction is and the acting force in the -direction is , or .

Here, the resultant force in the -direction is zero but there is acceleration.

Step 3

Same things happen for force in the -direction (for less than or more).

Now, I am generalizing above result.

Steps 1&2 clearly show that when we apply any magnetic force in the -direction on the object, actual force acting on object is more & that quantity is .

Similarly, if we apply any magnetic force in the -direction on the object then actual force acting on object is more and that quantity is .

This is completely contradicts the results which say that the applied force and acting forces are different (larger) in Special Relativity.

Step 4

Force does work, consumes energy, gain energy, and we know that energy cannot be created. Energy can only be transferred.

It is clear that energy is transferred from the magnet to the object. But, if the applied force is less than the acting force, then the energy gain by the object will be more than the energy lose by the magnet. This means that more work is done by a larger force for the same displacement, so more energy is generated.

This excess energy generation is the problem. Where does this additional energy (or force) come from?