- #1
dingo_d
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Homework Statement
I've searched everywhere, and I cannot find an example of calculation of Lie derivation of a metric.
If I have some vector field [itex]\alpha[/itex], and a metric [itex]g[/itex], a lie derivative is (by definition, if I understood it):
[itex]\mathcal{L}_\alpha g=\nabla_\mu \alpha_\nu+\nabla_\nu \alpha_\mu[/itex]
So if my vector field is given in this form (polar coordinates for instance):
[itex]\alpha=r\sin^2\theta \partial_t+r\partial_\varphi[/itex]
(this is something I made up btw), so if I got this right (and I'm not sure, that's why I'm asking) I should find the Christoffel symbols from my metric, and use the definition of covariant derivative, and just calculate term by term (for [itex]\mu,\ \nu=t,\ r,\ \theta,\ \varphi[/itex])?
In that case, is [itex]\alpha_t=r\sin^2\theta[/itex] ? And so on? Or did I missed the point entirely, because I'm at loss :\