- #1
help1please
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The work I have been following has me very confused... and I am almost sure I am making a mistake somewhere!
After working up to this equation:
[tex]\delta V = dX^{\mu}\delta X^{\nu} [\nabla_{\mu} \nabla_{\nu}]V[/tex]
I am asked to calculate the curvature tensor. Now the way I did it, turned out different to the way it is shown at the end of the work... it took a bit of time to understand what method I was using was different but I did work it out nonetheless, and what I want to know is which method is correct (most likely mine is wrong but I need some guidance.)
Ignoring [tex]V[/tex] and just working out the commutator relationship, I expand:
[tex](\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})[/tex]
The first part of the calculation, gives
[tex]\partial_{\nu}\partial_{\mu} - \partial_{\mu} \partial_{\nu}[/tex]
Which is just zero, because the ordinary derivatives commute, so they go to zero. Fine. Now, according to the work I am following, the next set of terms should have been:
[tex]\Gamma_{\nu} \partial_{\mu} - \partial_{\nu}\Gamma_{\mu}[/tex]
But I ended up with
[tex]\partial_{\nu}\Gamma_{\mu} - \partial_{\mu} \Gamma_{\nu}[/tex]
and I only arrived at this because it is well known that once you calculate the first set of terms, for instance, using this guide:
[tex](a+b)(c+d)[/tex]
ignoring that we are taking this part away from another part, the first term arises because you multiply [tex]a[/tex] with [tex]c[/tex]. Then you multiply [tex]a[/tex] with [tex]d[/tex].
In the work I am following, it seems that in this:
[tex](\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})[/tex]
You get
[tex]\partial_{\nu} \partial{\mu}[/tex]
first of all by following that rule (then the extra's of course [tex]- \partial_{\mu} \partial_{\nu}[/tex] but according the second lot, the work has
[tex]\Gamma_{\nu} \partial_{\mu} - \partial_{\mu} \Gamma_{\nu}[/tex]
My brain agrees with the [tex]-\partial_{\mu} \Gamma_{\nu}[/tex] term but I do not understand how it gathers the
[tex]\Gamma_{\nu} \partial_{\mu}[/tex]
Because for that to be true, it would mean using my expression again for simplicity that
[tex](a+b)(c+d) - (a'+b')(c'+d')[/tex]
It seems right to multiply [tex](a' \cdot d')[/tex] but with a steady analysis of the works example, it shows [tex]b \cdot c[/tex] which would give the first term [tex]\Gamma_{\nu}\partial_{\mu}[/tex]... but that isn't right is it? Or am I wrong? Am I doing it wrong?
Thanks