So, I understand in order to evaluate the proper "derivative" of a vector valued function on a curved spacetime manifold, it is necessary to address the fact that the tangent space of the manifold changes as the function moves infinitesimally from one point to another. Therefore, you cannot just subtract the two vectors as you ordinarily would because they "live" in different tangent spaces, you need a "covariant" derivative. Correct so far?(adsbygoogle = window.adsbygoogle || []).push({});

Now, when Carroll addresses this in his notes he introduces the Christoffel symbols as a choice for the coefficient for the "correction" factor (i.e., the covariant derivative is the "standard" partial derivative plus the Christoffel symbol times the original tensor)

In his book Hartle introduces the Christoffel symbols much earlier, as the coefficients used when calculating geodesics. Later he covers the covariant derivative, but talks more about the concept of "parallel transport" which seems to have some connection to Christoffel symbols that I am unsure about.

So (in summary) how do these things work togther? It seems to me, the Christoffel symbols are the components of the original tensor that allow the "translation" to preserve parallelism, which allows definition of the covariant derivative, which then in turn finds it's way into geodesics. Am I on the right/wrong track?

Also, what is the signficance of the upper/lower indices on a Christoffel symbol? I see the Christoffel symbols are not tensors so obviously it is not a summation convention...or is it?

Thanks :)

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# Christoffel symbol+covariant derivative+parallel transport

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