Quick Question about the Christoffel Symbol of the Second Kind

1. Aug 11, 2012

TheEtherWind

The Christoffel symbol is as followed:

$${\Gamma ^{m}}_{ab}=\frac{1}{2}g^{mk}(g_{ak,b}+g_{bk,a}-g_{ab,k})$$

where $k$ is a dummy index. What values is it summed over? If I had to guess I'd say 0 to 3, but it seems somewhat counter intuitive.

Does the Christoffel symbol become

$${\Gamma ^{m}}_{ab}=\frac{1}{2}([g^{m0}(g_{a0,b}+g_{b0,a}-g_{ab,0})]+[g^{m1}(g_{a1,b}+g_{b1,a}-g_{ab,1})]+[g^{m2}(g_{a2,b}+g_{b2,a}-g_{ab,2})]+[g^{m3}(g_{a3,b}+g_{b3,a}-g_{ab,3})])$$

?

2. Aug 11, 2012

Staff: Mentor

You've got it right, at least for the "standard" case in relativity of a 4-D spacetime. The general rule is that summed indexes range over the full range of indexes in whatever manifold you are working with.

3. Aug 11, 2012

GarageDweller

Why is this counter intuitive? It simply follows from the einstein summnation notation.

4. Aug 11, 2012

TheEtherWind

I understand the Einstein summation notation. I found it somewhat counter intuitive seeing as how it was a dummy index that didn't seem to "be" the 4 coordinates of space-time in any equation or tensor, etc. but it makes more sense the way PeterDonis said it... it's on a 4-dimensional manifold...