# Active/Passive Diffeomorphisms – clarification on Rovelli’s

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##(\mathcal L_XY)_p =\lim_{t\to 0}\frac{((\phi_{-t})_*Y)_p-Y_p}{t}##...
I stand with you with the demonstration but I think you should use the definition of the Lie derivative by including the flow map instead of p in the first term of the numerator (and similarly in the following lines):

##(\mathcal L_XY)_p =\lim_{t\to 0}\frac{((\phi_{-t})_*Y)_ {\phi_t(p)}-Y_p}{t}\\##

and make use of the definition of the flow map:

##\phi_{t}^{-1}=\phi_{-t}## (together with ##\phi_{t}\circ\phi_{s}=\phi_{t+s}## and ##\phi_{0}=id##)

Fredrik
Staff Emeritus
$$((\phi_{-t})_*Y)_p = (\phi_{-t})_* Y_{(\phi_{-t})^{-1}(p)} =(\phi_{-t})_* Y_{\phi_t(p)}.$$ ##Y_{\phi_t(p)}## is pushed forward to the tangent space at
$$\phi_{-t}(\phi_t(p))=(\phi_{-t}\circ\phi_t)(p)=\phi_{-t+t}(p)=\phi_0(p)=p.$$