Relativistic Casimir Contradiction

Question

If i have two very large parallel rectangular metal plates separated at a distance $d$ , I can measure the casimir force between them as

$$ \frac{hc\pi^2}{240d^4} $$

Now suppose I give the contraption with plates a boost, to a velocity $v$ orthogonal to the plates, Length contraction suggests that the apparent distance between the plates will be

$$ d \sqrt{1 - \frac{v^2}{c^2}}$$

And therefore the expected force between them should be

$$ \frac{hc \pi^2}{240d^2(1 - \frac{v^2}{c^2})^2}$$

So consider this thought experiment:

I have 2 plates, I boost each, now the expected force is a lot higher and they start coming together.

Vs. I'm in the same frame of reference, and notice nothing odd going on.

I feel like theres a problem here, since different inertial frames of reference can disagree on what goes on.

Answer 1

Be careful how forces transform. Here is another thought experiment: 2 + charges and 2 - charges are arranged in a square. If the square is perfect, the forces balance. But it is unstable - the slightest inequality in the lengths of the arms will make + and - snap together. If you run by at relativistic speeds, two arms are foreshortened and two are not. You might think it is no longer balanced. But it is. Now there are magnetic forces. Forces transform in more complex ways than you might expect.