Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Parallel propagator and covariant derivative of vector

**Physics Forums | Science Articles, Homework Help, Discussion**