Calculate Parallel Transport: Get Out of Logical Loop

[snoopies622]

I'm in a logical loop here:

1. A tensor undergoes parallel transport if, as it moves through a manifold, its covariant derivative is zero.

2. Covariant derivative describes how a tensor changes as it moves through a manifold.

3. A tensor undergoes change as it moves if it does not parallel transport.

So how do I get out of this loop? I have an intuitive sense of parallel transport, but I do not know how to describe it mathematically except by using the definition above (1). Perhaps the answer lies in the calculus of variations?

 

[pervect]

There are potentially a lot of ways. One way is to define the abstract properties a covariant derivative must have (things like obeying the chain rule, for instance, though that's not sufficient, there are 4-5 conditions needed), then finding the particular derivative operator or operators that are metric compatible.

You can start with the idea of a connection, which is a linear map from a tangent space at point p to the nearby point q as well, and look for a special connection called the Leva-Civita connection that's metric compatible and torsion free.

One rather offbeat way is to use Schild's ladder to define parallel transport. You need to have a definition of a geodesic for this to work as you must construct and extend geodesic segments for this approach to work. It's an approximate geometric construction that transports a vector by constructing parallelograms, and stating that the sides of a parallelogram are parallel. See for instance https://en.wikipedia.org/w/index.php?title=Schild's_ladder&oldid=636833569

You'll find a description of Schild's ladder in MTW"s Gravitation as well, but the approach seems to be rather offbeat and not particularly popular in textbooks. You'd need to define geodesics other than curves that parallel transport themselves for it not to be circular, this is possible by using variational principles.

 

[snoopies622]

Thanks, pervect. This stuff goes deep!