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I'm having a small issue with the notion of Lie-derivatives after rereading Carroll's notes

https://arxiv.org/abs/gr-qc/9712019

page 135 onward. The Lie derivative of a tensor T w.r.t. a vector field V is defined in eqn.(5.18) via a diffeomorphism ##\phi##. In this definition, both terms are "tensors at the point p", as he remarks after eqn.(5.17). My issue is with the term

[tex]\phi_{* t}\Bigl[ T(\phi_t (p))\Bigr] [/tex]

in eqn.(5.17). As I read this, you first evaluate the tensor T at a shifted point ##\phi_t (p)##, and after that you pull this back via ##\phi_{* t}## at the point p. After eqn.(5.21) however, I get confused. In this part, Carroll tries to show that in a particular coordinate system the Lie derivative becomes an ordinary partial derivative, so he can introduce Lie brackets. He takes as a vector field ##V=\partial_1##, and states:

"The magic of this coordinate system is that a diffeomorphism by t amounts to a coordinate transformation from ##x^{\mu} = (x^1, x^2, . . . , x^n)## to ##y^{\mu} = (x^1 + t, x^2, . . . , x^n)##."

All right, you go along the flow in the ##x^1##-direction.

My confusion is with eqn.(5.23): he evaluates the tensor components in ##y^{\mu} = (x^1 + t, x^2, . . . , x^n)##. But why? Aren't we supposed to evaluate the terms in the original point p, that is, with coordinates ##x^{\mu} = (x^1, x^2, . . . , x^n)##? The confusion also arises, because I thought the point of Lie derivaties was that you compare tensor components in the very same point (and hence, evaluated at the same coordinate values!). But in eqn.(5.24), the Lie derivative becomes an ordinary partial derivative because in Carroll's magic coordinate system it falls down to

[tex]

\lim_{t \rightarrow 0 }\frac{T^{\mu \ldots}_{\nu \ldots}(x^1 + t, x^2, . . . , x^n) - T^{\mu \ldots}_{\nu \ldots}(x^1, x^2, . . . , x^n)}{t} = \partial_{x^1} T^{\mu \ldots}_{\nu \ldots}(x^1, x^2, . . . , x^n)

[/tex]

But here we are comparing the same tensor components in two different points! I understand that this is the whole idea of a partial derivative, but I'm confused in the context of Lie derivatives and Carroll's remark after eqn.(5.17).

So I guess my question really is: if a diffeomorphism brings us from a point with coordinates ##x^{\mu}## to a point with coordinates ##y^{\mu}##, how do the components of

[tex]\phi_{* t}\Bigl[ T(\phi_t (p))\Bigr] [/tex]

look like? I thought it would be ##T^{'\mu \ldots}_{'\nu \ldots}(x)##, but because of Carroll's discussion above I'm confused.