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General Relativity and Pi



Please feel free to correct me if this question is WAAAAAY off base.

General Relativity equates 'gravity' with the curvature of space/time.

Does this imply that here, near the surface of the earth, the value of pi
is not exactly equal to the value currently calculated to a gazillion
decimal places?
(These calculations are, of course, based on Euclidian Flat space assumptions.)

How many decimal places out would we have to go to see the difference?

Couldn't this '# of decimal places' be a somewhat 'conceptual' measure of
the local curvature of space/time?

This idea just occured to me when a student asked a math-list how pi is
actually determined. (and the expected answer of various infinite
expansions was given)

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