"Flat" isn't really flat? How???? Why????

But the final coordinate transformation forced us to use the new angular coordinate thp = (1 - 4 G M) theta. Therefore, the new angular coordinate thp varies from 0 to 2 Pi (1 - 4 G M) . This means that a closed path around the origin with radius R has a circumference of 2 Pi (1 - 4 G M) R instead of the expected 2 Pi R. This is how curvature behaves! This is not how a flat spacetime behaves!

The way to resolve this apparent paradox is to say that this spacetime is "locally flat" but has curvature concentrated at the point in space (or the line in spacetime) where the mass is located. Circular paths that don't enclose the mass M have the normal flat relationship between radius and circumference, but any closed path around the mass M shows the influence of the curvature concentrated there. The point where this curvature is concentrated is called a "conical singularity".

Isn't that a clever way to have our flat spacetime and curve it, too?

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