Well, first off, I did specifically ask to not be given spoilers, that in mind, I read about the first line of your post then immediately closed my app, since you said you were going to provide a solution. That in mind, I started over from scratch, and feel as though I am closer, but I am not sure what to do from here:
##D_m T^k \to D_m g^{\mu k} T_{\mu}##
##D_m g^{\mu k} T_{\mu} = T_{\mu}D_m g^{\mu k} + g^{\mu k} D_m T_{\mu}##
##= 0 + g^{\mu k} [ \partial_m T_{\mu} - \Gamma_{m \mu}^z T_z ]##
##= g^{\mu k} [ \partial_m T_{\mu} - g^{zr}(\partial_m g_{\mu r} + \partial_{\mu} g_{mr} - \partial_r g_{m \mu})T_z]##
##= "..." + (-g^{zr}g^{\mu k} \partial_m g_{\mu r} - g^{zr}g^{\mu k} \partial_{\mu} g_{mr} + g^{zr}g^{\mu k}\partial_r g_{m \mu}) T_z##
##= "..." + (-\delta_r^k \partial_m g^{zr} - \delta_m^z \partial_{\mu} g^{\mu k} + \delta_m^k \partial_r g^{zr})T_z##
##= "..." + (-\delta_r^k \partial_m g^{zr} - g^{zk}g_{mk} \partial_{\mu} g^{\mu k} + \delta_m^k \partial_r g^{zr})T_z##
##= "..." + (-\delta_r^k \partial_m g^{zr} - \delta_m^{\mu} \partial_{\mu} g^{zk} + \delta_m^k \partial_r g^{zr})T_z##
The middle term obviously stands out, you can see it in the partial_mu g^mu k. I'm just not sure what to do about it. The last couple steps might even be redundant. My thinking was I shouldn't have a partial of g^mu k w.r.t. the muth component. That just seems weird, and it doesn't happen in the christoffel symbols. However, I now have two upper and 1 lower components...
I'm still of the idea that I should have partial_something g^zr for each term, as I need to transform T_z to T^r. This warrants a lot of these steps useless, and leaves me at the 4th from last line:
##= "..." + (-g^{zr}g^{\mu k} \partial_m g_{\mu r} - g^{zr}g^{\mu k} \partial_{\mu} g_{mr} + g^{zr}g^{\mu k}\partial_r g_{m \mu}) T_z##
Which can be simplified to:
##= "..." + (-\delta_r^k \partial_m g^{zr} - g_{mr}g^{\mu k} \partial_{\mu} g^{zr} + \delta_m^k\partial_r g^{zr}) T_z##
and the g_zr can be factored out and contracted with T_z to give
##= "..." + (-\delta_r^k \partial_m - g_{mr}g^{\mu k} \partial_{\mu} + \delta_m^k\partial_r ) T^r##
My thinking is that I need to use the "..." and rearrange, but I'm having trouble seeing how I can get rid of the T_mu in the ...
So, I am, again, at another unique impasse. Fun.
I honestly feel like I'm doing something completely idiotic, I'm just not sure what, when, or where. I expanded the deltas as sums over mu to get a g^mu k attached to every term, then I can factor that out, but that seem to get me anywhere useful, either. The closest that gets me to anything (from what I can see) is:
##g^{\mu k}\partial_m T_{\mu} + g^{\mu k}(\partial_r g_{\mu m} - \partial_{\mu} g_{mr} - \partial_m g_{\mu r}) T^r##
Which almost looks like ##\Gamma_{m r}^k## but I have a floating negative sign somewhere. If I could get ##\partial_m g_{\mu r}## to be positive, then I would have the christoffel, but it's not. However, the other partial term (the one not nested in the christoffel) is a partial w.r.t. the same variable, the only problem is the term is the partial of T_mu, and not T^r. What's even more peculiar is that if you distribute the T^r to the -partial_m, and group it with the other term, you can collapse T^r to T_mu and then you have ##g^{\mu k} \partial_m(T_{\mu} - T_{\mu})##
I'm either making mistakes, doing stupid things, or both. Can someone point out where I've went astray?
