The discussion centers on the relationship between the Riemann tensor, Stokes' theorem, and the concept of winding numbers in the context of parallel transport along closed curves. It is established that parallel transport around a closed time-like curve (CTC) does not imply multivaluedness of the Riemann tensor, as each transported vector is distinct from the original vector field. Curvature is defined through limiting operations for circuits that return to the starting event once, irrespective of the nature of the curves involved.
PREREQUISITESThis discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students studying general relativity, particularly those interested in the implications of curvature and parallel transport in spacetime.
jk22 said:I saw briefly
No, because winding around a closed curve several times is a different closed curve.jk22 said:I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve.
If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
jk22 said:if a vector is parallel transported along a CTC
I am going to guess what might be confusing you (always dangerous, but I'll risk it). I'm thinking you have in mind a vector field with a value at every spacetime point (event). You take one of these and parallel transport around a circuit. You get a different vector. Around a circuit again, you get still a different vector. But each of these transported vectors is a different vector than the original, not part of the original vector field at all. There is no multi-valued anything. Curvature is defined via a limiting operation for circuits that return to the starting event exactly once. Note, CTC's are not special at all, in this. In defining curvature at a point, you will consider spacelike circuits and timelike circuits, even mixed. It doesn't matter for defining curvature.jk22 said:Then how is the coming back at the same time as starting, it should but start again or time were stopped ?
Namely if a vector is parallel transported along a CTC is it coming back to its original state ?
I think I'm locked since the two alternatives are for me : If it is yes then there is no curvature, if it is no then the vector at ##(x^\mu)## has several values, hence multivalued ?
What am I thinking wrong here ?