Quick question - Christoffel Symbol Transformation Law

[osnarf]

Hey everyone,

This formula was just provided in a book and I was trying to prove it but I'm having a hard time understanding what it's saying. The formula is attached, along with the definitions given for the Christoffel symbols. In the definitions the i's are the standard basis vectors and the x's are the coordinates associated with them.

I've been playing with it for a while and can't figure out what each symbol is supposed to mean in the context of the formula.

Just to make it easier to type, in this context does a symbol of the first kind [i j, k] with a bar over it mean

$\frac{\partial^{2}\vartheta^{m}}{\partial\overline{\vartheta}^{i}\partial\overline{\vartheta}^{j}}$*$\frac{\partial\vartheta^{m}}{\partial\overline{\vartheta}^{k}}$,

or does it mean the same but with the thetas on top replaced with the standard basis coordinates?

The same question goes for the first order symbol with no bar, would this be the same as defined in the picture (w/r/t the standard basis), or is it the same as the Tex above but with the bars moved from the thetas on bottom to the thetas on top?

Thanks, your help is very appreciated.

edit - no idea why that one theta won't show up, sorry

 

[Fredrik]

edit - no idea why that one theta won't show up, sorry

I don't have time to think about the actual question right now, so I'll just quickly answer this. When you type 50 characters without a space, vBulletin will insert a space, and this will often break your code.

 

[osnarf]

Ah awesome, thanks. I never use the Tex because it always did that and I couldn't figure out why. Good to know.

Still haven't figured it out by the way, if anybody can explain it.

Thanks again

 

[osnarf]

Alright, I think I got it. Along the way I had this:

$\overline{\Gamma}^{m}_{r s} = \frac{\partial^{2}\vartheta^{n}}{ \partial\overline{\vartheta}^{r}\partial\overline{\vartheta}^{s}} \frac{\partial\overline{\vartheta}^{m} }{\partial\vartheta^{n}} + \Gamma^{n}_{n q} \frac{\partial\vartheta^{q} }{\partial\overline{\vartheta}^{s} } \delta^{m}_{r} = \frac{\partial^{2}x^{z} }{\partial\overline{\vartheta}^{r }\partial\overline{\vartheta}^{s }} \frac{\partial\overline{\vartheta}^{m }}{\partial x^{z}}$

So my questions are:

-Are the symbols always defined in some arbitrary variables with respect to the standard basis coordinates (like on the far right), or can they be derived completely in arbitrary coordinates?

-Is how I have it correct: you can define the symbol just as you would on the right, but with respect to some other arbitrary coordinates instead of the standard ones, if one of the lower indices is the same as the raised one?

-Is there more significance to the symbols, or is it just a convenient way to write something you see over and over? I feel like I am missing the point..

Thanks again

--

Edit - Also, thanks again Fredrik. That was definitely what was happening.

 

[Fredrik]

I haven't really tried to understand what your book is doing. I will just describe how I'm used to thinking about these things. (I'm leaving out some details about domains and stuff, so this is a bit non-rigorous). Let V be the set of smooth vector fields on a manifold M. A connection is map $(X,Y)\mapsto \nabla_XY$ from V×V into V that satisfies a few conditions that you can look up if you're interested. Let x be an arbitrary coordinate system, and let $\partial_\mu$ be the $\mu$th basis vector associated with x. Then we define $\Gamma^\mu_{\nu\rho}$ in this coordinate system by $$\nabla_{\partial_\nu}\partial_\rho =\Gamma^\mu_{\nu\rho}\partial_\mu.$$