Is Special Relativity the Key to Understanding Shrinking Circles?

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The discussion centers on the implications of special relativity on a spinning circle, where the radius remains constant while the distance between points on the circumference appears to decrease due to gravitational effects and acceleration. The author questions the constancy of π in this scenario, suggesting that as the circle spins, it may not maintain its traditional geometric properties. They note that the equivalence principle does not apply in this case, leading to a paradox reminiscent of the historical context that influenced Einstein's exploration of non-Euclidean geometries. This inquiry highlights the complexities of applying special relativity to geometric shapes in motion. The conversation underscores the intriguing relationship between physics and geometry.
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am i missing something here?
we have a circle: (x-h)^2+(y-k)^2=r^2 where (h,k) is the center and r is the radius. we now spin the circle about an axis that is perpendicular to the plane on which the circle lies and it runs through the center of said circle. gravity contracts length (and my the equivelance principle, so does acceleration), so as the 1-sphere spins about the axis, the distance between any two points on it decreases while the radius stays the same. since \pi=\frac{c}{2r}, where c is circumference and r is radius, \pi no longer is a constant. the circle shrinks, but the radius stays the same. what is going on? does the circle turn into a cone?
 
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i just found out that the equivelence princliple doesn't apply here.
 
You have stumbled upon a fanatastic paradox of special relativity

As it appears in the history books, this is the very same case that lead Einstein to consider non-euclidean geometries in the physical universe.
 

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