# What is meant by “The Lie derivative commutes with contraction”?

This was stated recently in a GR course I am taking, and I found it also stated on Wikipedia (second paragraph). I simply don’t know what is meant by this. For a vector $X$ and 1-form $\eta$, I would define the contraction as

$$\eta(X)$$

Whilst the Lie derivative of this quantity is

$$\mathcal{L}_Y(\eta(X)) = (\mathcal{L}_Y\eta)(X) + \eta(\mathcal{L}_Y X)$$

By the Leibniz rule. So I can’t see what “commutes” could mean in this context.

#### Solutions Collecting From Web of "What is meant by “The Lie derivative commutes with contraction”?"

Write $C$ as the contraction, so $\eta (X) = C(\eta \otimes X)$.

Then

\begin{split}
\mathcal{L}_Y(\eta(X)) &= \mathcal{L}_Y (C (\eta \otimes X) \\
&= C \mathcal{L}_Y(\eta \otimes X) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{Contraction commutes with Lie derivative}) \\
&= C (\mathcal{L}_Y \eta \otimes X + \eta \otimes \mathcal{L}_Y X) \ \ \ \ \ \ (\text{Leibniz rule}) \\
&= (\mathcal{L}_Y\eta)(X) + \eta(\mathcal{L}_Y X)
\end{split}

So what you wrote down is true because contraction commutes with Lie derivative. (The rule is used so naturally that you don’t even realize)