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I'm trying to figure out the link between the connection coefficients (Christoffel symbols), the propagator, and the coordinate description of the covariant derivative with the connection coefficients.

As in http://en.wikipedia.org/wiki/Parall...ng_the_connection_from_the_parallel_transport one can write

[tex]

\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h}= \frac{d}{dt}\left. \Gamma(\gamma)_t^0V_{\gamma(t)}\right|_{t=0}[/tex]

However, we also know that

[tex]

\nabla_b V^a = \partial_b V^a + {\Gamma^a}_{cb}V^c.[/tex]

I understand how, in some loose sense, one can think of the connection coefficients as the derivative of the parallel propagator:

[tex]

{\Gamma^a}_{cb} = \left.\frac{\partial}{\partial y^c}{[\Gamma(\gamma)^x_y]^a}_b\right|_{y \to x}.[/tex]

However, I cannot see how to link the three together, and formalise things. Carrol's notes (http://preposterousuniverse.com/grnotes/grnotes-three.pdf [Broken]) give a hint, but without using the first sort of equation I gave--- he just jumps in with the second equation as an "assumption". What I'd ultimately be looking to do would be something like using the product rule:

[tex]

\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - \Gamma(\gamma)^0_0V_{\gamma(0)}}{h} [/tex]

[tex]

\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - \Gamma(\gamma)_0^0V_{\gamma(h)}+\Gamma(\gamma)_0^0V_{\gamma(h)}-\Gamma(\gamma)_0^0V_{\gamma(0)}}{h} [/tex]

[tex]

\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - \Gamma(\gamma)_0^0V_{\gamma(h)}}{h}+\lim_{h\to 0}\frac{\Gamma(\gamma)_0^0V_{\gamma(h)}-\Gamma(\gamma)_0^0V_{\gamma(0)}}{h} [/tex]

[tex]

\nabla_X V =V_{\gamma(h)}\frac{d}{dt}\left.\Gamma(\gamma)^t_0\right|_{t=0} +\Gamma(\gamma)^0_0\frac{d}{dt}\left.V_{\gamma(t)}\right|_{h=0}

[/tex]

So, that somehow

[tex]

\nabla_b V^a =V^c_{\gamma(0)}{\Gamma^a}_{cb} +\partial_b V^a_{\gamma(0)}

[/tex]

However, this makes no sense, as it means you are subtracting vectors from different vector spaces (the whole reason the parallel propagator was introduced in the first equation I gave in this post). I've also jumped straight to coordinate components...

Any bright ideas would be much appreciated.

Cheers,

Ianhoolihan.

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# Parallel propagator and covariant derivative of vector

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