Need help MTW's Gravitation exercise 16.1

[qinglong.1397]

Need help! MTW's Gravitation exercise 16.1!

I am working on MTW's Gravitation and I came across a problem. In the attachment https://www.physicsforums.com/attachments/50292, I show my calculation and, finally, I show the contradiction that is,

you can get a vanishing $\Gamma^k_{\phantom{k}kk}$ with Cartan's equation

but, at the same time,

you can also get a non-vanishing $\Gamma^k_{\phantom{k}kk}$ by simply inserting metric into the definition of connection coefficients.

What's wrong with my calculation? I need your help. Thank you!

 

[Ben Niehoff]

You should number all your equations, because the ones I need to refer to are not.

I have not checked your calculation of $\omega$. However, when you change from the orthonormal basis to the coordinate basis, are you sure you used the correct change of basis? A connection is not a tensor; a change of basis includes an extra derivative term.

 

[qinglong.1397]

You should number all your equations, because the ones I need to refer to are not.

I have not checked your calculation of $\omega$. However, when you change from the orthonormal basis to the coordinate basis, are you sure you used the correct change of basis? A connection is not a tensor; a change of basis includes an extra derivative term.

Thanks for your reply, and I've already uploaded a new document where I numbered all the equations. Sorry for this inconvenience.

You have a good point. A connection isn't a tensor. However, I'm concerning how to add that extra derivative term. You know, you get that extra derivative term when you transfer from one coordinate basis to another coordinate basis, but here, you go from one non-coordinate basis to a coordinate one. Would you please give a hint? Thank you!

 

[Ben Niehoff]

In all the places you see something like

$$\frac{\partial x^\mu}{\partial y^\nu}$$
you can put a more general change-of-basis matrix.

 

[qinglong.1397]

In all the places you see something like

$$\frac{\partial x^\mu}{\partial y^\nu}$$
you can put a more general change-of-basis matrix.

Thanks for your reply! But I think my way of changing basis is correct. You can see this example on Page 19 :http://physicssusan.mono.net/upl/9111/Lotsofcalculationsp.1326.pdf [Broken]

 

[Ben Niehoff]

Equation 5.14 in your reference is wrong, and could only have worked for most of the Christoffel symbols by coincidence. See "Change of frame" here:

http://en.wikipedia.org/wiki/Connection_form

 

[qinglong.1397]

Equation 5.14 in your reference is wrong, and could only have worked for most of the Christoffel symbols by coincidence. See "Change of frame" here:

http://en.wikipedia.org/wiki/Connection_form

Thank you very much! This exactly solves my puzzle.

 

[qinglong.1397]

Asking for help with MTW's problem 16.1

I am working on MTW's Gravitation and I came across an annoying problem. I describe the problem and show my calculation in the attachment. Please download it and help me out.

Thank you very much!

 

[qinglong.1397]

I am working on MTW's Gravitation and I came across an annoying problem. I describe the problem and show my calculation in the attachment. Please download it and help me out.

Thank you very much!

This is a new post. I need your help! Thank you!