Quick Question about the Christoffel Symbol of the Second Kind

[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})])

?

 

[PeterDonis]

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.

 

[GarageDweller]

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

 

[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...