A thought experiment. Imagine ants living on a merry-goround,
which is their two-dimensional world. From measurements
on small circles they are thoroughly familiar
with the number pi. When they measure the circumference
of their world, and divide it by the diameter, they
expect to calculate the number pi=3.14159. . . . We
see the merry-go-round turning at relativistic speed.
From our point of view, the ants’ measuring rods on the
circumference are experiencing Lorentz contraction in
the tangential direction; hence the ants will need some
extra rods to fill that entire distance. The rods measuring
the diameter, however, do not contract, because their
motion is perpendicular to their lengths. As a result, the
computed ratio does not agree with the number . If you
were an ant, you would say that the rest of the universe is
spinning in circles, and your disk is stationary. What possible
explanation can you then give for the discrepancy,
in view of the general theory of relativity?
The Attempt at a Solution
My friends and I have tried to answer this problem using the curvature of space. We tried to explain how the outer edges of the merry-go-round bend inward from an observer outside the merry-go-round and so the diameter calculated from the measure of the circumference will be different than measure the diameter directly. However, we think the entire outside of the merry-go-round will bend inwards equally.