Parallel transport to explain motion of light near black hole?

[bcrowell]

If you parallel transport a vector around a loop you can end up with a vector with a diffrent orientation. Then why can't you end up with a vector at 180 degrees angle with the original i.e. a scalor multiple, where the scalor is -1? It seems quit possible. Start at the north pole facing south, walk along a meridian till you reach the equator, move along the equator for half of the circumference, then go back north to the pole. You end up facing the other way compare to you start.

You're right, that's a clear counterexample. I think the problem is my assertion that the result generalizes from infinitesimal parallelograms to larger paths. Now that I think more carefully about it, it's clearly wrong. If you add a bunch of vectors that are not parallel to a given line, you can still get a sum that's parallel to the line.

 

[bcrowell]

After posting a bunch of wrong stuff about the rescaling of null vectors, I think I finally understand it better. The antisymmetry of the Riemann tensor on its first two indices is exactly the condition needed so that parallel transport around an infinitesimal closed loop doesn't change inner products. For such a loop lying in a fixed plane, the set of possible actions by Riemann tensors are therefore the six-dimensional set of linear transformations that preserve inner products. In other words, they are the set of possible Lorentz transformations. Such a transformation can certainly take a null vector and rescale it.