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Imagine a space ship, loaded with all sorts of computer systems, traveling near the speed of light.

Electricity itself is very fast, and can reach speeds close the speed of light. (up to 99% according to wikipedia). So, what would happen to the electronic circuits in this spaceship?

Will the computers shut down, because electricity can't reach the components? Or are they just not related to each other and will the computers keep working perfectly?

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You're dealing with an incomplete form of relativity.

In the frame of the spaceship, nobody will notice anything different, since all inertial frames are equivalent

In the "ground" frame, electricity would be moving at a different speed, by the relation $$\rm v_{e,ground}=\frac{v_{ship}+v_{electricity}}{1+\frac{v_{ship}v_{electricity}}{c^2}}$$

We cannot simply use relative velocities, we need to use the above equation. If you compare this with the time dilation of the system, the computers will all seem to be working the same, albeit slower, from your POV.

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