I'm going to assume mass is small enough not to take GR into effect.
From the point of view of a particle on the toroid, the band it's in will extend to about 1.005 meters long. Due to Lorentz contraction, from the point of reference of someone in the center, it will appear one meter long.
The question is ONLY for the central observer. At first he sees 1 m long stripes, but when the whole thing reaches the speed of 0.1 c, how long is each stripe?
As per a recent comment this thread is meant to voice contrarian opinions, that is anything this community tends not to agree with. Thus I ask you to post your contrarian views and upvote anything you do not agree with based on personal beliefs. Spam and trolling still needs to be downvoted.