The Affine Connection components

  • #1
Hello!

I have a few questions about how the Affine connection works.

I know the geodesic equation;


[tex]\Gamma^{\lambda}_{\mu \nu} = \frac{\partial x^{\lambda}}{\partial \xi^{\alpha}} \frac{\partial^{2} \xi^{\alpha}}{\partial x^{\nu} \partial x^{\mu}}[/tex]

So, for example, if I had the expression

[tex]\ddot{t} + \frac{A'}{A} \frac{dr}{d \tau} \frac{dt}{d \tau} = 0[/tex]

I can read off

[tex]\Gamma^{t}_{rt} = \frac{A'}{2A}[/tex]

Where the expression is divided by 2 because they are off diagonal terms, since the subscript indices are not the same.

I am not sure how this then applies to the actual


[tex]\Gamma_{t}[/tex]


matrix. Should the


[tex]\Gamma^{t}_{rt} = \Gamma^{t}_{tr} [/tex] terms be exactly the same? Because in an example the

[tex]\Gamma^{t}_{tr} = \frac{A'}{2A} [/tex]

Whereas the

[tex]\Gamma^{t}_{rt} = \frac{A'}{2B} [/tex]

And I don't understand why there should be a different letter on the denominator, I had always thought that if they are off-diagonal terms they should be identical?? Is that not the case?

And finally, as for all of the other components in the matrix, would these all just be 0? I think that because, I know that the Affine connection is not a tensor, and so will not have diagonal "1" terms like the metric tensor, so all other terms must be 0? Is that true?
ie
[tex]\Gamma^{t}_{tt} = 0[/tex] ???

Many thanks in advance!

Hannah
 

Answers and Replies

  • #2
649
3
The Christoffell symbols are symmetric in the lower two indicies:

[tex]\Gamma^{a}{}_{bc}=\Gamma^{a}{}_{cb}[/tex]

So they should be the same. Check the reference.
 
  • #3
5,428
291
The Christoffell symbols are symmetric in the lower two indicies:
It follows from the first equation by the commutativity of partial differentiation.
 

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