Parallel transport and geodecics

[Austin0]

I don't think so; being static is a very restrictive condition on manifolds. The paper doesn't seem to talk about this, as far as I can tell.



No. Parallel transport does imply "moving" vectors from one event to another; that's what it's for, to allow you to "compare" vectors at different events by moving one of them from one event to the other.

OK I'll take another try. A particle can travel on a closed geodesic but it's world line will of course be unbounded.
Then wrt closed geodesic:
a) it's world line would be a set of point worldlines together comprising a tube.
b) it is a single entitiy whose world line is a tube.
c) It is an abstraction that can't really be said to have a worldline.

so does parallel transport include a velocity term?

 

[Dale]

When you talk about geodesics on a sphere aren't you talking about the surface as a 2 d topology?

Yes. The surface of a sphere is an orientable, curved 2D manifold.

My recent comments are about a Mobius strip which is a non-orientable, flat 2D manifold.

AN early post mentioned a geodesic as a helical world line which I get. The path of an inertial particle.
Then PeterDonis seemed to indicate you were not talking about this but some other concept in the context of geometric topology . SO ?

A geodesic is the generalization of a "straight line" into arbitrary manifolds, including curved manifolds. So, if you are on the surface of a sphere and you start walking and you never turn even slightly left or right then you will walk along a great circle. Great circles are geodesics on a sphere.

Helical worldlines are geodesics in the Schwarzschild spacetime manifold, not in all manifolds.

 

[yuiop]

I don't understand why you say it is an ANALOG to a geodesic. My understanding of the word agrees w/ the first definition I found on-line just now:


Of, relating to, or denoting the shortest possible line between two points on a sphere or other curved surface.

So how is a great circle not a geodesic but just an "analog" of a geodesic? What am I missing?

I was thinking in terms of a timelike manifold and I could not think of an example in nature, so I called it an an analogue. The use of that word was just a reflection of my uncertainty rather than a statement of fact. As others have pointed out, the surface of a sphere is a spacelike manifold rather than the analogue of one. I am not very good with manifolds so I probably still have the terminology wrong.

I was also wondering about timelike paths return to the same point in space and time, i.e. a CTC. Is it possible for a CTC to be a geodesic? I imagine all CTCs are the worldlines of particles with proper acceleration, rather than the worldline of a particle in freefall, but I may be wrong.