- #1
Blazejr
- 23
- 2
Hello. I'm learning about Lie derivatives and one of the exercises in the book I use (Isham) is to prove that given vector fields X,Y and one-form ω identity [itex]L_X\langle \omega , Y \rangle=\langle L_X \omega, Y \rangle + \langle \omega, L_X Y \rangle[/itex] holds, where LX means Lie derivative with respect to field X.
My idea was to just expand everything in terms of local components. This is where I got (skipping some intermediate steps, because I'm pretty sure I got them right):
[tex]L_X \langle \omega , Y \rangle = \langle \omega , L_X Y \rangle + Y^{\mu}(\omega_{\nu}X^{\nu}_{\; \; ,\mu}+\omega_{\mu , \nu}X^{\nu})[/tex]
where Einstein convention is assumed.
Now if I were able to prove that expression in the brackets is actually equal to [itex](L_X \omega)_{\mu}[/itex] (which is listed as the next exercise in book, by the way) the task would be finished - I have no idea how to do that, though. Alternatively if I knew how to prove the original formula without expanding it in terms of components I could use my results so far in solving the next task. Any tips will be much appreciated.
Edit: I noticed that I wrote in wrong subforum (for which I'm sorry) after wrting this post and I'm not able to delete it now. If whoever is in charge could move it instead of deleting I would be grateful.
My idea was to just expand everything in terms of local components. This is where I got (skipping some intermediate steps, because I'm pretty sure I got them right):
[tex]L_X \langle \omega , Y \rangle = \langle \omega , L_X Y \rangle + Y^{\mu}(\omega_{\nu}X^{\nu}_{\; \; ,\mu}+\omega_{\mu , \nu}X^{\nu})[/tex]
where Einstein convention is assumed.
Now if I were able to prove that expression in the brackets is actually equal to [itex](L_X \omega)_{\mu}[/itex] (which is listed as the next exercise in book, by the way) the task would be finished - I have no idea how to do that, though. Alternatively if I knew how to prove the original formula without expanding it in terms of components I could use my results so far in solving the next task. Any tips will be much appreciated.
Edit: I noticed that I wrote in wrong subforum (for which I'm sorry) after wrting this post and I'm not able to delete it now. If whoever is in charge could move it instead of deleting I would be grateful.
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