In physics, we are learning about vectors, but not in much detail. I am hazy about many concepts. We have been taught that vectors can be parallelly shifted, provided that they maintain their length (magnitude) and their direction.
- Does this not mean that all vectors are co-initial since I can always just shift them?
There is an answer to a similar question on this site; the answerer says that on a manifold, two vectors may not be coinitial. But I'm confused because I don't see why his explanation (using example of a particle moving along the unit circle in the $x$–$y$ plane) is specific to manifolds and not just any 2D plane (I don't know what manifolds are; I'm just going by what the answerer said).
- Also, is it really true in the first place that I can parallelly shift any vector? It seems confusing to me that the starting point of a vector has no significance.
Best Answer
Try this: take a spherical object, or just imagine a sphere in front of you if you can't find one. Now place your index finger pointing north (up) right in the centre of the sphere, along the equator.
If you slide your finger forward to the north pole, your finger will be pointing straight ahead. But if you were to instead rotate 90 degrees along the equator and then slide your finger to the north pole, your finger is no longer pointing straight ahead, it's pointing 90 degrees away.
The name for this idea is Holonomy. Basically if you slide a vector around a loop on a curved surface, the vector can end up rotated. This does not happen when you slide vectors around on flat surfaces, however.
So you see that on a sphere, where your vector starts is important because if you try to slide your vector to a new starting point, it will depend on which path you take to the new point. There is no well-defined (i.e. independent of any choices) way to compare vectors on the equator of a sphere versus the north pole.
Part of the mathematical formalism of a manifold, is that you don't have just one set of vectors, each point in the manifold has its own set of vectors and we call that the tangent space at that point. For two tangent vectors to be considered equal, they need to share the same starting point. We also have ways of moving vectors from one tangent space (starting at p) to another (starting at q). But, as I said, that depends on the path from p to q.