Force Transform Applying: Question on a Specific Case

[dan_b_]

When I try to apply the force transformation (the 3 vector one) to the describe following situation, I find a result that I can't make sense of. Hopefully someone can tell me what I'm doing wrong. Suppose observers A and B are in inertial frames, and B travels in the +x direction relative to A. Object C is initially at rest relative to A, and is immediately next to A. In B's frame object C initially travels in the -x direction along with A.

Now consider what happens when an unbalanced force is exerted on object C in the +y direction when the force is described in the reference frame of A.

The object will accelerate in the +y direction when it is described in A's frame. I would think that the object must accelerate only in the +y direction in B's frame also (it cannot accelerate in the +x or -x direction), otherwise we would have a paradox. But once the object is moving in the +y direction relative to A and B, the relativistic transformation for the x component of force in B's frame suggests that the x component of force in B's frame will not be zero - even though the x component of force is zero in A's frame! This arises because the "power term" (the dot product of the force on object C in A's frame and the velocity of C in A's frame) that is present in the transformation will not be equal to zero. But this doesn't make sense to me. You can't have an x component of force in B's frame in this situation when there is no x component of force in A's frame; the object can't have an
x- component of acceleration in B's frame when it has no x-component of acceleration in A's frame. I suspect the only way out of this mess is that the dot product term in the force transformation must be zero for some reason, but I don't see why from the way that this thought experiment has been described. Can anyone please help?

 

[Dale]

You can't have an x component of force in B's frame in this situation when there is no x component of force in A's frame; the object can't have an
x- component of acceleration in B's frame when it has no x-component of acceleration in A's frame.

I am not sure why you believe this. It is not true.

The reason is easier to see if you work with four-vectors instead of three-vectors. When C is moving in the frame of A then the four-force has a component in the time direction. This component is boosted into the x direction in the frame of B.