- #1

- 131

- 20

## Main Question or Discussion Point

[Ref. 'Core Concepts in Special and General Relativity' by Luscombe]

Let ##M,M'## be manifolds and ##\psi:M\to M'## a diffeomorphism. Even if ##\psi## weren't a diffeomorphism, and instead just a smooth map, the coordinates of the pushback of ##\mathbf{t}\in T_p(M)##, would be related to the coordinates of ##\mathbf{t}## by ##(\psi_*\mathbf{t})^a=t^i\frac{\partial y^a}{\partial x^i}## (more precisely it's ##y^a\circ\psi## but we ignore the composition in the usual notation.

Note that ##x^i## are the coordinates for the coordinate system covering ##p##, while ##y^a## are those for the coordinate system covering ##\psi(p)##. We thus get the relation $$\psi_*\mathbf{t}=t^i\frac{\partial y^a}{\partial x^i}\mathbf{e}_a$$

I'm dividing this into 3 parts for your convenience.

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Now let a vector field ##\mathbf{v}## be given on ##M##. Let ##\phi_t:M\to M## be a diffeomorphism such that ##\phi_t(\gamma(r))=\gamma(r+t)##, where ##\gamma## is the integral curve associated with ##\mathbf{v}## that passes through ##\gamma(r)##. Then, quoting:

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So the above gives the coordinates of ##p'=\phi_t(p)##. Then it's written:

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Let ##\phi_{-t}## be the inverse of ##\phi_t##. Now we come to evaluating the pullback (under ##\phi_t##) of a tangent vector at ##p'##. This is the same as the pushforward under ##\phi_{-t}##. Quoting:

Let ##M,M'## be manifolds and ##\psi:M\to M'## a diffeomorphism. Even if ##\psi## weren't a diffeomorphism, and instead just a smooth map, the coordinates of the pushback of ##\mathbf{t}\in T_p(M)##, would be related to the coordinates of ##\mathbf{t}## by ##(\psi_*\mathbf{t})^a=t^i\frac{\partial y^a}{\partial x^i}## (more precisely it's ##y^a\circ\psi## but we ignore the composition in the usual notation.

Note that ##x^i## are the coordinates for the coordinate system covering ##p##, while ##y^a## are those for the coordinate system covering ##\psi(p)##. We thus get the relation $$\psi_*\mathbf{t}=t^i\frac{\partial y^a}{\partial x^i}\mathbf{e}_a$$

I'm dividing this into 3 parts for your convenience.

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**Part 1**Now let a vector field ##\mathbf{v}## be given on ##M##. Let ##\phi_t:M\to M## be a diffeomorphism such that ##\phi_t(\gamma(r))=\gamma(r+t)##, where ##\gamma## is the integral curve associated with ##\mathbf{v}## that passes through ##\gamma(r)##. Then, quoting:

It hasn't been specified how the above came about, but I'm guessing it's because the manifold is locally flat in the infinitesimal region, and so we can related coordinates of nearby points as we usually do in vector calculus via directional derivatives in flat space.The Lie derivative of various objects is found by working out their pullbacks. Let ##p\in M## have coordinates ##x^{\mu}##. The coordinates of ##\phi_t(p)## are, forinfinitesimal##t##, generated by the vector ##\mathbf{v}## at ##p##, $$\phi_t(x^{\mu})\equiv \bar x^{\mu}=x^{\mu}+tv^{\mu}+O(t^2)$$

**Is that correct?**---------------

**Part 2**So the above gives the coordinates of ##p'=\phi_t(p)##. Then it's written:

My interpretation of the above is that ##(\phi^*_tf)(p)=f(\phi_t(p))=f(p')## and we want to find the coordinates of ##f(p')## in terms of those of ##p## and those of the vector ##\mathbf{v}|_p##.The pullback of a function at ##\bar x^{\mu}## (which I'm interpreting to mean that ##f## is defined as a smooth function in the neighborhood of ##p'##) is $$(\phi^*_t f)(x^{\mu})=f(x^{\mu}+tv^{\mu})\approx f(x^{\mu})+tv^{\mu}\partial_{\mu}f$$

**How exactly did we conclude that the components of ##f(p')## are given by the RHS**? (Again my guess is that we're using the fact that the patch containing ##p## and ##p'## are locally flat, and some general multivariable form of Taylor expansion, but I'd still appreciate details this)---------------

**Part 3**Let ##\phi_{-t}## be the inverse of ##\phi_t##. Now we come to evaluating the pullback (under ##\phi_t##) of a tangent vector at ##p'##. This is the same as the pushforward under ##\phi_{-t}##. Quoting:

Do I interpret this to mean that ##v^a## represents the coordinates of the vector field ##\mathbf{v}## andThe pushforward of a contravariant vector field ##\mathbf{u}## under ##\phi_{-t}## is, using Eq. (13.7) (the equation given at the start of this question), $$\phi_{-t\ *}(\mathbf{u}|_{\phi_t(p)})=u^a(x^{\lambda}+tv^{\lambda})\frac{\partial x^{\mu}}{\partial\bar x^a}\mathbf{e}_{\mu}$$

**not**the vector ##\mathbf{v}|_{\phi_t(p)}##? Are the components of ##\mathbf{v}## to be considered functions of the coordinates of the point at which we're calculating the vector field?