Affine connection transformation

[Mr-R]

Dear All,


I am teaching myself tensors for the first time. I am using D'Inverno's book and got stuck at page 73. Basically, he says: demand that the first term on the left of the equation to be a type (1,1) tensor. Then he gets the affine connection transformation.

I basically wrote the first term as a second rank mixed tensor transformation. Then I got stuck. I am not sure on how to isolate (?) the affine connection and show how it transforms. I tried many times but failed due to my lack of knowledge of tensors. Could someone help me understand this please?

\nabla_{c}X^{a}= \partial_{c}X^{a}+\Gamma_{bc}^{a}X^{b}

Thanks in advance! (Sorry if my post isn't very informative as I have to go for 6 hours. When I come back I will be more than happy to upload pictures of my attempts)

 

[Matterwave]

You can make a coordinate transformation on the term:

$$\partial_c X^a +\Gamma^a_{bc}X^b$$

You know how $\partial_c X^a$ transform (equation 6.1), and it does not transform as a tensor, so now you need that the transformation of $\Gamma^a_{bc}X^b$ to cancel out the "wrong terms".

Are you having troubles with the details?

 

[haushofer]

Carroll does this in detail in his gr notes, if i remember correctly.

 

[Mr-R]

You can make a coordinate transformation on the term:

$$\partial_c X^a +\Gamma^a_{bc}X^b$$

You know how $\partial_c X^a$ transform (equation 6.1), and it does not transform as a tensor, so now you need that the transformation of $\Gamma^a_{bc}X^b$ to cancel out the "wrong terms".

Are you having troubles with the details?

Oh I didn't think of it in that way. Yes my trouble was in the details. I keep messing up the dummy variables and didn't know that I can leave the connection alone while transforming the tensors I know. Like the way Carroll did.

Edit: I managed to derive it finally but I have a question. In equation 6.1, why do we make \frac{\partial}{\partial x^{'c}}= \frac{\partial x^{d}}{\partial x^{'c}}\frac{\partial}{\partial x^{d}} ? at first I didnt do this and it didnt work

 

[Mr-R]

Carroll does this in detail in his gr notes, if i remember correctly.

This indeed helped me a lot. Made me discover where I need improvements in. Thanks

 

[Matterwave]

Oh I didn't think of it in that way. Yes my trouble was in the details. I keep messing up the dummy variables and didn't know that I can leave the connection alone while transforming the tensors I know. Like the way Carroll did.

Edit: I managed to derive it finally but I have a question. In equation 6.1, why do we make \frac{\partial}{\partial x^{'c}}= \frac{\partial x^{d}}{\partial x^{'c}}\frac{\partial}{\partial x^{d}} ? at first I didnt do this and it didnt work

This is just the chain rule from multi-variable calculus.