Lie Derivatives and Parallel Transport

[PhizzyQs]

Hi, I've begun learning about General Relativity, though I've already had some exposure to differential geometry.

In particular, I understand that Lie Differentiation is a more "primitive" process than Covariant Differentiation (in that the latter requires some sort of connection).

My question is this: parallel transport can be used to understand how a vector changes when you drag in along a curve on a certain surface. To be sure, you institute local coordinates, compute the metric, and then the connection (here, the connection being used, in this coordinate basis, are the Christoffel symbols), and then solve the differential equation.

In this way, you can find out, for instance, how much the vector changes its direction under a certain curve. But, is this information only encoded in the connection? That is to say, to find out how much the vector deviates, must I employ parallel transport, or is there some procedure, using only Lie Derivatives, to examine the change?

 

[atyy]

Lie derivatives also define a sort of transport. However, the covariant derivative does not depend on objects outside the curve, while the Lie derivative does. So in Lie transport, the curve must be specified as an integral curve of a vector field.

 

[Bill_K]

The absolute derivative of a vector u^μalong a curve with tangent vector v^μ is u^μ_;νv^ν (this is zero for parallel transport), whereas the Lie derivative along the same curve is u^μ_,νv^ν - v^μ_,νu^ν.

I think what you mean by "outside the curve" is that the Lie derivative depends on the gradient of v, not just v itself.

 

[atyy]

I think what you mean by "outside the curve" is that the Lie derivative depends on the gradient of v, not just v itself.

Yes, so v must be a vector field, and not just the tangent vector to a curve.

 

[PhizzyQs]

Oh, I know that much. My main concern is calculating angular deviation from using Lie derivatives.

I tried this: I begin with a vector A, and there are points P, and Q. They are connected by a curve Y, parametrized by an affine parameter t, whose tangent vector is u = dY(t)/dt. Using the pullback on the isomorphism generated by u, I take the vector from P to Q. Then, I use the metric at P to find <A(Q), u(Q)>. I compare this with <A(P), u(P)>. Given affine parametrization, u does not change under parallel transport, so I think this would be accurate.

EDIT: I am a little bit querulous about my last assumption there, and am examining it now.