Inspired by: Angular deficit
The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward says it is possible to cut a wedge out of a flat spacetime and glue the edges together. So in my mind this looks like a paper cone.
I'm having trouble understanding why this is flat everywhere except at the 'tip' of the cone. Imagine a triangle on the original paper, and now after piecing it together, wouldn't an observer think the lines are now curved? And in piecing it together, wouldn't there now be another angle, so it is a 4 sided polygon, and the exterior angles won't correctly add up to 360 degrees anymore?
I'm just very confused because Edward and Lubos say the spacetime is flat everywhere except at the center, so the Riemann curvature tensor is zero everywhere except at the center, but Lubos says a parallel transported vector around a path on this flat spacetime can change angle!? Does this mean we can't describe the parallel transport of a vector with the local Riemann curvature?
Hopefully I've said enough that someone knowledgeable can see what is confusing me and can help me understand. If a clear question is needed then let it be this:
How can we explicitly calculate the curvature, and the effect this has on angles of paths or vectors, in conical spacetime?
The 'paste flat spacetime pieces together' process seems very fishy to me.
Okay, thanks to Ted and Edward I got most of it figured out (although my attempt couldn't say anything about the curvature 'spike' at the center)', but still can't figure out how to see the parallel transport of vector in a closed loop ala Lubos's comment. It would be neat to see this last part worked out for an arbitrary loop.
In particular Ted's comment "that (in some appropriately-defined sense) the average curvature inside the triangle is nonzero. In this particular case, that average comes entirely from a curvature "spike" at the origin." sounds like there may be an easy way to transfer the integral around a path to an integral over the area bounded by the path, ala Gauss-Bonnet but the integral I'm getting doesn't even look like a normal integral and I don't really understand what Gauss-Bonnet is saying physically.
Can someone work out this last little piece explicitly, and if you use something like Gauss-Bonnet maybe help explain what the math is telling us about the physics here?