1. Aug 14, 2014

space-time

This expression:

$\Gamma$avc$\Gamma$cab

Can someone please show me how to multiply the two Christoffel symbol formulas for these Christoffel symbols without overloading any indices?

2. Aug 14, 2014

HallsofIvy

Staff Emeritus
I would do this as two separate sums. If you sum over "a" first, you get $$\Gamma^a_{vc}\Gamma^d_{ab}= \Gamma^1_{vc}\Gamma^c_{1b}+ \Gamma^2_{vc}\Gamma^c_{2b}+ \Gamma^3_{vc}\Gamma^c_{ab}+ \Gamma^4_{vc}\Gamma^c_{4b}$$

Now, in each of those replace c with 1, 2, 3, and 4. There will be a sum of 16 such products. The first four are $$\Gamma^1_{vc}\Gamma^c_{1b}= \Gamma^1_{v1}\Gamma^2_{1b}+ \Gamma^1_{v2}\Gamma^2_{1b}+ \Gamma^1_{v3}\Gamma^3_{1b}+ \Gamma^1_{v4}\Gamma^4_{1b}$$.