# Playing with the Cristoffel symbol

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?

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

## The Attempt at a Solution

fzero
Homework Helper
Gold Member
No, the problem is that you're summing over both $$c,d$$ in

$$g^{cd}\Gamma^{j}_{cd}=g^{ab}\Gamma^{j}_{ab}$$

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

$$g_{ec} g^{cd} = \delta^d_e$$

or similar identities.

OK, so I guess there's something wrong with the following too then?

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

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?

fzero
Homework Helper
Gold Member
OK, so I guess there's something wrong with the following too then?

1) $$g_{ab}A^{ab}=g_{cd}A^{cd}$$

This is just a relabeling of indices and is correct.

2) $$g_{ab}g^a_eA^{eb}=g_{cd}A^{cd}$$

Here $$g^a_e = g^{af}g_{fe} = \delta^a_e$$ so this equation is correct.

3) $$g_{ab}g^a_eA^{eb}=g^e_ag^a_eg_{cd}A^{cd}$$

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

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

These are all equivalent to 3 and incorrect.

8) $$g^{af}g_{af}A^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}$$
9) $$g^a_aA^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}$$

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

10) $$A^{eb}=g^{af}g^b_fg^e_ag_{cd}A^{cd}$$
11) $$A^{eb}=g^{ab}g^e_ag_{cd}A^{cd}$$
12) $$A^{eb}=g^{eb}g_{cd}A^{cd}$$
13) $$A^{ab}=g^{ab}g_{cd}A^{cd}$$

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 $$g^{ab}$$ is the inverse of $$g_{bc}$$ as a matrix so that $$g^{ab}g_{bc}=\delta^a_c$$. 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

$$A^{11} = g^{11} \left( g_{11}A^{11} + g_{12}( A^{12} + A^{21}) + \cdots \right).$$

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.

$$g_{ab}g^a_eA^{eb}=g^e_ag^a_eg_{cd}A^{cd}$$

The LHS is $$g_{ae} A^{eb}=A^b_a$$, while the RHS involves the trace of $$A$$, 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 $$g^e_a$$ and $$g^a_e$$
which I thought were both just identity matrices? so doesn't that mean it should be the same as step 2?

fzero
Homework Helper
Gold Member
This confuses me a bit. The only difference between this step and step 2, is that here the RHS includes $$g^e_a$$ and $$g^a_e$$
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

$$g_{ab}g^a_eA^{eb}= g_{eb} A^{eb}$$

but the RHS is

$$g^e_ag^a_eg_{cd}A^{cd} = \delta^a_a g_{cd}A^{cd}.$$

$$\delta^a_a = D,$$ the dimension of the space, so this is still incorrect.