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## Main Question or Discussion Point

I was reading my lecturer's notes on GR where I came across the geodesic equation for four-velocity. There is a line which read:

[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij} [/tex]

I'm trying to understand how LHS = RHS, surely the indices ##a## and ##i## are different, how can you simply combine them?

I tried writing them out:

[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j[/tex]

[tex] = g_{aj} \partial_i u^i u^j + \left( u^i u^j \partial_i g_{aj} - \frac{1}{2} u^i u^j \partial_a g_{ij} \right) - \frac{1}{2} g_{ij} \partial_a u^i u^j [/tex]

Source: http://physicspages.com/2013/04/02/geodesic-equation-and-four-velocity/

*Summing them up,*[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij} [/tex]

I'm trying to understand how LHS = RHS, surely the indices ##a## and ##i## are different, how can you simply combine them?

I tried writing them out:

[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j[/tex]

[tex] = g_{aj} \partial_i u^i u^j + \left( u^i u^j \partial_i g_{aj} - \frac{1}{2} u^i u^j \partial_a g_{ij} \right) - \frac{1}{2} g_{ij} \partial_a u^i u^j [/tex]

Source: http://physicspages.com/2013/04/02/geodesic-equation-and-four-velocity/