Playing with the Cristoffel symbol

  • Thread starter redstone
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  • #1
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I'm learning about the Christoffel symbol and playing around with it, so I'm curious... Does the math work below, or have I done something wrong?

[tex]\Gamma^{j}_{cd}=\Gamma^{j}_{cd}[/tex]
[tex]g^{cd}\Gamma^{j}_{cd}=g^{cd}\Gamma^{j}_{cd}[/tex]
[tex]\frac{1}{2}g^{cd}g^{jm}\left(g_{md,c}+g_{mc,d}-g_{cd,m}\right)=\frac{1}{2}g^{cd}g^{jm}\left(g_{md,c}+g_{mc,d}-g_{cd,m}\right)[/tex]
[tex]\frac{1}{2}g^{cd}g^{jm}\left(g_{md,c}+g_{mc,d}-g_{cd,m}\right)=\frac{1}{2}g^{ab}g^{jm}\left(g_{mb,a}+g_{ma,b}-g_{ab,m}\right)[/tex]
[tex]\frac{1}{2}g^{jm}\left(g_{md,c}+g_{mc,d}-g_{cd,m}\right)=\frac{1}{2}g_{cd}g^{ab}g^{jm}\left(g_{mb,a}+g_{ma,b}-g_{ab,m}\right)[/tex]
[tex]\Gamma^{j}_{cd}=g_{cd}g^{ab}\Gamma^{j}_{ab}[/tex]

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
fzero
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No, the problem is that you're summing over both [tex]c,d[/tex] in

[tex]
g^{cd}\Gamma^{j}_{cd}=g^{ab}\Gamma^{j}_{ab}
[/tex]

and there's no operation that we can do to move [tex]g^{cd}[/tex] over to the RHS. There's no free index available to use

[tex]g_{ec} g^{cd} = \delta^d_e[/tex]

or similar identities.
 
  • #3
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OK, so I guess there's something wrong with the following too then?

1) [tex]g_{ab}A^{ab}=g_{cd}A^{cd}[/tex]
2) [tex]g_{ab}g^a_eA^{eb}=g_{cd}A^{cd}[/tex]
3) [tex]g_{ab}g^a_eA^{eb}=g^e_ag^a_eg_{cd}A^{cd}[/tex]
4) [tex]g_{ab}A^{eb}=g^e_ag_{cd}A^{cd}[/tex]
5) [tex]g^f_bg_{af}A^{eb}=g^e_ag_{cd}A^{cd}[/tex]
6) [tex]g^f_bg_{af}A^{eb}=g^f_bg^b_fg^e_ag_{cd}A^{cd}[/tex]
7) [tex]g_{af}A^{eb}=g^b_fg^e_ag_{cd}A^{cd}[/tex]
8) [tex]g^{af}g_{af}A^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}[/tex]
9) [tex]g^a_aA^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}[/tex]
10) [tex]A^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}[/tex]
11) [tex]A^{eb}=g^{ab}g^e_ag_{cd}A^{cd}[/tex]
12) [tex]A^{eb}=g^{eb}g_{cd}A^{cd}[/tex]
13) [tex]A^{ab}=g^{ab}g_{cd}A^{cd}[/tex]

Where did I go wrong here? Also, maybe you know of a good reference that gives a good overview of what is and isn't allowed in index gymnastics?
 
  • #4
fzero
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OK, so I guess there's something wrong with the following too then?

1) [tex]g_{ab}A^{ab}=g_{cd}A^{cd}[/tex]

This is just a relabeling of indices and is correct.

2) [tex]g_{ab}g^a_eA^{eb}=g_{cd}A^{cd}[/tex]

Here [tex]g^a_e = g^{af}g_{fe} = \delta^a_e[/tex] so this equation is correct.

3) [tex]g_{ab}g^a_eA^{eb}=g^e_ag^a_eg_{cd}A^{cd}[/tex]

The LHS is [tex]g_{ae} A^{eb}=A^b_a[/tex], while the RHS involves the trace of [tex]A[/tex], so this equation is not generally correct.

4) [tex]g_{ab}A^{eb}=g^e_ag_{cd}A^{cd}[/tex]
5) [tex]g^f_bg_{af}A^{eb}=g^e_ag_{cd}A^{cd}[/tex]
6) [tex]g^f_bg_{af}A^{eb}=g^f_bg^b_fg^e_ag_{cd}A^{cd}[/tex]
7) [tex]g_{af}A^{eb}=g^b_fg^e_ag_{cd}A^{cd}[/tex]

These are all equivalent to 3 and incorrect.

8) [tex]g^{af}g_{af}A^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}[/tex]
9) [tex]g^a_aA^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}[/tex]

For both, the LHS involves the trace of the metric, while the RHS involves the trace of A, this is incorrect in general.

10) [tex]A^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}[/tex]
11) [tex]A^{eb}=g^{ab}g^e_ag_{cd}A^{cd}[/tex]
12) [tex]A^{eb}=g^{eb}g_{cd}A^{cd}[/tex]
13) [tex]A^{ab}=g^{ab}g_{cd}A^{cd}[/tex]

These are all the same equation and are incorrect.


Where did I go wrong here? Also, maybe you know of a good reference that gives a good overview of what is and isn't allowed in index gymnastics?

Any undergrad relativity tex should spend some time explaining index notation. You should know that [tex]g^{ab}[/tex] is the inverse of [tex]g_{bc}[/tex] as a matrix so that [tex] g^{ab}g_{bc}=\delta^a_c[/tex]. You should also know that you generally never repeat an index twice in an expression.

Finally you could have actually thought about what your expressions look like in components as a sanity check. For instance, equation 13 above is telling you that

[tex]A^{11} = g^{11} \left( g_{11}A^{11} + g_{12}( A^{12} + A^{21}) + \cdots \right). [/tex]

The RHS is drastically different from the LHS and would not be satisfied by an arbitrary tensor A.

Do lots of exercises and the formalism will start to sink in.
 
  • #5
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[tex]g_{ab}g^a_eA^{eb}=g^e_ag^a_eg_{cd}A^{cd}[/tex]

The LHS is [tex]g_{ae} A^{eb}=A^b_a[/tex], while the RHS involves the trace of [tex]A[/tex], so this equation is not generally correct.

This confuses me a bit. The only difference between this step and step 2, is that here the RHS includes [tex]g^e_a[/tex] and [tex]g^a_e[/tex]
which I thought were both just identity matrices? so doesn't that mean it should be the same as step 2?
 
  • #6
fzero
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This confuses me a bit. The only difference between this step and step 2, is that here the RHS includes [tex]g^e_a[/tex] and [tex]g^a_e[/tex]
which I thought were both just identity matrices? so doesn't that mean it should be the same as step 2?

OK, I did make a mistake there. The LHS is

[tex]
g_{ab}g^a_eA^{eb}= g_{eb} A^{eb}
[/tex]

but the RHS is

[tex]
g^e_ag^a_eg_{cd}A^{cd} = \delta^a_a g_{cd}A^{cd}.
[/tex]

[tex]\delta^a_a = D,[/tex] the dimension of the space, so this is still incorrect.
 

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