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Parallel Transport

by madness
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Feb21-09, 06:22 AM
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The covariant derivative again defines the full connection as the effect of holding coordinates constant (coordinate derivatives) and then adding the correction terms (Gammas) because this alone won't do. The fact that you can at a point choose coordinates so that the Gamma's are zero is the fact that you can choose special (geodesic) coordinates for which the parallel transport becomes the "coordinate transport".

The physical interpretation of this is "choosing a locally inertial frame".

The difficulty is that we need coordinates to express motion in a traditional format but we need to get at the geometry which is independent of the choice of coordinates. If you have the high level mathematics then you simply speak of [/i]General Covariance[/i] under the diffeomorphism group (group of arbitrary continuous changes of coordinates). But this gets farther from the intuitions about what "parallel transport" means (unless you have have a highly developed intuition from working in the higher level mathematics.)

I find it helps to start with say the gauge derivation of electromagnetism first see how these issues play out with the much simple connection between the little complex number space at each point where the real vs. imaginary directions are arbitrary and you need a connection between each complex space at each point. Generalize this to a bigger group representation i.e. (classical) Yang-Mills field theory, then attack GR first by thinking of the tangent space as abstract and independent of the actual manifold then understanding that there is an additional identification namely that the vectors in the tangent space are identified with motion in the manifold via parametric derivatives. Finally one chooses systems of parameters namely coordinates and speaks of a coordinate basis of the tangent space. (This last may make more sense when you've gone through the first steps.)

The hardest part I found with getting my head around GR (an on-going process) is unlearning the many implicit facts which no longer hold when we move away from flat euclidean space. It is a long process of retraining the intuition.

Examples: Null vectors are perpendicular to themselves! The visual size in space-time drawings are wrong since we are embedding non-euclidean space onto euclidean paper. etc.

There are no shortcuts. You have to dig through the sequence of topics and do the math to build up your understanding of what is going on in the examples...
...And as always ask lots of questions.
Mar9-09, 12:30 AM
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Can anyone explain geometrically the differences between Covariant derivative and Lie derivative ?
Mar9-09, 01:20 AM
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Quote Quote by nonexisting View Post
Can anyone explain geometrically the differences between Covariant derivative and Lie derivative ?
The Lie derivative is defined using a congruence of curves, so its value along one curve depends on the vector field on neighbouring curves. The covariant derivative only depends on the vector field on the particular curve.

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