Domain issues in transformation of the coordinate representation of a function

Question

Start with a Manifold $M$ and define a function $f:M\rightarrow\mathbb{R}$. As usual, pick two charts $(U,x)$ and $(V,y)$ with $p \in U\cap V$ and $x:M \supset U \rightarrow x(U) \subset\mathbb{R}^n$. Define $(V,y)$ similarly. If we instead want to work with the coordinate description of $\phi : x(U) \rightarrow \mathbb{R}$ where $\phi(x):= f\circ x^{-1}$. We can also define $\phi'(y)$ on the other chart by $f\circ y^{-1}$.

Now a coordinate transform is defined (in physics, at least) to be $\phi'(x) =(f\circ y^{-1} \circ x )(p)$. Now the usual course of action is to insert $x^{-1} \circ x$ after $f$ to get the result $\phi'(x) = (\phi\circ \Lambda^{-1})(x)$ where $\Lambda = y\circ x^{-1}$ is the chart transition map. This intuitively makes sense, but what is actually happening when we apply $y^{-1}$ to an element in $x(U)$? The function $y^{-1}$ is strictly defined from $y(U) \rightarrow M$, it doesn't have anything to do with the space $x(U)$ except that they are both subsets of $\mathbb{R}^n$.

What exactly is happening here? Am I just confusing myself? Is there some structure that I didn't include here that's assumed? For full context, I'm working in flat minkowski space where each point contains transition functions within the Poincare Lie group (Lorentz group + translations). Thanks.

Answer 1

Let me modify your notation slightly. Given a manifold $M$, a smooth function $f:M \to \Bbb{R}$, and two charts $(U,x), (V,y)$ with non-empty overlap, we can consider the following four functions:

• $f_{(x)}:= f\circ x^{-1} : x[U] \to \Bbb{R}$
• $f_{(y)} := f\circ y^{-1} : y[V] \to \Bbb{R}$
• $\Lambda := y \circ x^{-1}: x[U\cap V] \to y[U\cap V]$
• $\Lambda^{-1} = x\circ y^{-1}: y[U\cap V] \to x[U\cap V]$.

Intuitively, the function $f_{(x)}$ is "the function $f$ represented in $x$-coordinates", similarly for $f_{(y)}$, and the map $\Lambda$ is what physicists call "the transformation from $x$-coordinates to $y$-coordinates" (or something along those lines). With these definitions, it is easily seen that \begin{align} f_{(x)} &= f_{(y)} \circ \Lambda \quad \text{and} \quad f_{(y)} = f_{(x)} \circ \Lambda^{-1}\tag{$*$} \end{align}

That's all you really need to know (I believe physicists refer to these equations as the "transformation law for scalar fields" or something like that). If you write this out in full, it simply says \begin{align} \begin{cases} f\circ x^{-1} &= (f\circ y^{-1}) \circ (y\circ x^{-1}) \\\\ f\circ y^{-1} &= (f\circ x^{-1}) \circ (x\circ y^{-1}) \end{cases} \end{align}

So, the only way to get from $x[U\cap V]$ to $y[U\cap V]$ is through the transition map $\Lambda := y \circ x^{-1}$.