Spacetime and Geometry: Vanishing commutators#2

[George Keeling]

This is a refinement of a previous thread (here). I hope I am following correct protocol.

Homework Statement

I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can be thought of as differential operator which transforms smooth functions to smooth functions on the manifold. For a vector field X and a function f(xⁱ) we write

X(f) = g, where g is another function. We then define the commutator of two fields X and Y as

[X,Y](f) = X(Y(f)) - Y(X(f)

In the exercise I am working on, we are asked to find two vector fields whose commutator does not vanish. An important step is to show that if the commutator vanishes for one non-trivial function f, it vanishes for all non-trivial functions. This is implied by the question but not proven. (f = a constant everywhere would be trivial)

Homework Equations

See above

The Attempt at a Solution

I proved it this way using 'Reductio ad absurdum'.

Our starting point is f is non-trivial and [X,Y](f) = 0. We have another function g is non-trivial and [X,Y](g) ≠ 0.

We already know that commutators are linear (from the previous exercise), so

[X,Y](f + g) = [X,Y](f) + [X,Y](g)
or
[X,Y](f + g) = [X,Y](g)

Therefore f is trivial, which breaks our starting assumption, with which there must be some error. The only non trivial possibility is that [X,Y](g) = 0. QED?

Is my solution correct?
Is there a more obvious solution? (I.e. Am I missing something? See below)
I would like a good definition of 'non-trivial'.

In another thread on Commutator of two vector fields (here), I read "[X,Y] describes how far the endpoints of a rectangle vary if you go along X followed by Y or the other way around. Commuting vector fields mean the two path end at the same point". This does not mention a particular function. I think it supports my conclusion.

 

[stevendaryl]

There was no reason to create a new thread just to make a second post.

Our starting point is f is non-trivial and [X,Y](f) = 0. We have another function g is non-trivial and [X,Y](g) ≠ 0.

We already know that commutators are linear (from the previous exercise), so

[X,Y](f + g) = [X,Y](f) + [X,Y](g)
or
[X,Y](f + g) = [X,Y](g)

Therefore f is trivial, which breaks our starting assumption, with which there must be some error. The only non trivial possibility is that [X,Y](g) = 0. QED?

I think you're going down a wrong path. $f$ being a constant is not the only possibility for $[X,Y](f) = 0$. But in any case, how does $[X,Y](f+g) = [X,Y](g)$ imply that $[X,Y](g) = 0$?

I think that you need to go back to what a vector field is, and what it means to apply it to a function. If you pick a particular coordinate system $x^1, x^2, ...$, then you can write a vector $X$ in terms of components $X^1, X^2, ...$ and then $X(f) = X^1 \frac{\partial f}{\partial x^1} + X^2 \frac{\partial f}{\partial x^2} + ...$. Applying $X$ to $f$ means applying the "directional derivative". So what happens when you compute $X(Y(f))$? (In general, the components $X^1, X^2, ...$ are not constants).

 

[George Keeling]

Thanks. You are right. As you have noticed I am still struggling with what X(f) really means. And thanks fort the advice on threads.

 

[stevendaryl]

As I said $X(f)$ means take the directional derivative of $f$ in the direction $X$. In the first course in vector calculus, they tend to write this as $X \cdot \nabla f$. In terms of components, it's $X^x \frac{\partial f}{\partial x} + X^y \frac{\partial f}{\partial y} + ...$

 

[George Keeling]

I would rewrite that expression for $X(f)$ as
$$X(f) = X^i \frac{\partial f}{\partial x^i}$$ from there I can get pretty quickly to the components of $[X,Y]$
$$[X,Y]^i = X^j \frac {\partial Y^i}{\partial x^j} - Y^j \frac {\partial X^i}{\partial x^j}$$ That also gives $[X,Y]$
The exercise was very useful. I had done it in a previous exercise from Carroll's book, but in a coordinate independent fashion. (A field is equivalent to the derivative along a line in the direction of the field. There are no partial derivatives on coordinates.)
I have also realized that I have committed the heinous error of not reading the question in the current exercise. It said that the "commutator does not vanish", not that "the effect of the commutator on a function does not vanish". My initial question vanishes! Thanks again. The exercise was useful and luckily I have plenty of time. (I even learned a bit of LaTeX. Ugh.)