Ok, done :)
Just plug in, replace ab by ml, use symmetry of g_ml.
Thank you both.
Now I know what to prove next - or maybe I should finally start a serious book about this :rolleyes:
I could do that for sure, but I haven't been that much into this, that I think about things like...
It wasn't much work, actually.
But that still boils down to ##\frac{\partial g^{ml}}{\partial x^p} \frac{\partial g_{ml}}{\partial {x^i}} == \frac{\partial g^{ml}}{\partial x^i} \frac{\partial g_{ml}}{\partial {x^p}}##, doesn't it?
Using your definition, I get the first term to be...
Hey,
I have been doing a few proofs and stumbled across this little problem.
Trying to show the symmetry of the Ricci tensor by using the Riemann tensor definition
##R^m_{\ ikp} = \partial_k \Gamma^m_{\ ip} - \partial_p \Gamma^m_{\ ki} + \Gamma^a_{\ ip} \Gamma^m_{\ ak} - \Gamma^a_{\ ik}...
Oh yes, I know.
I just wanted to see what I get.
Taking the trace / contracting with ##\eta## made it easier to compare both equations.
If you have ##h_{ab}## or ##h^{ab}## shouldn't matter, right?
It just has to be on the same levels as in the other equation.
At the moment I am just...
Okay, after 1 hour of 'index porn' I got this result...
##\partial_\mu \partial^\mu h - \partial_\mu \partial_\nu h^{\mu\nu} = 0##
1) I treated ##h^{\mu\nu}## as symmetric
2) I contracted out the ##\eta##'s
3) Divided by the common factors (0.5 in mine, 4 in 3.38)
If I do the same for...
Wow, the obvious again...
I have seen way too many indices today, thanks alot!
Kind of stressed to finish this, not my own book...
I will do the whole thing again tomorrow, and will post my results here.
Any tips/hints to make the calculation easier?
I think my starting point should be...
I was wondering about the whole thing using ##\delta## notation...
Using the Lagrangian
##\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}##
where
##\delta F_{\mu\nu} = \delta (\partial_\mu A_\nu - \partial_\nu A_\mu) = \delta (\partial_\mu A_\nu) - \delta (\partial_\nu A_\mu) =...
Wow, nice explanation. Thanks!
I need to find a book which does this a lot more, the other books didn't spend too much time on anything :(
What does Wald say?
Two pages and 400 indices later, yes.
It was a total mess, especially that little index mistake :D
NOW I will try your way.
Where
##\frac{\partial F_{\rho\sigma}}{\partial (\partial_{\mu}A_{\nu})} = \frac{\partial (\partial_\rho A_\sigma - \partial_\sigma A_\rho )}{\partial...