The discussion revolves around the relationship between the Riemann tensor, Stokes' theorem, and the concept of winding numbers in the context of parallel transport along closed curves, particularly in relation to closed timelike curves (CTCs). Participants explore the implications of these concepts on the multivalued nature of the Riemann tensor and the behavior of vectors under parallel transport.
Participants express differing views on the implications of winding numbers and the behavior of vectors under parallel transport, indicating that multiple competing perspectives remain unresolved.
The discussion includes assumptions about the nature of vectors in spacetime and the definitions of curvature, which may not be universally agreed upon. The implications of closed timelike curves on curvature and vector behavior are also not settled.
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 ?