The Lie Derivative on Forms
We’ve defined the Lie derivative of one vector field
by another,
. This worked by using the flow of
to compare nearby points, and used the derivative of the flow to translate vectors.
Well now we know how to translate -forms by pulling back, and thus we can define another Lie derivative:
What happens if is a
-form — a function
? We check
That is, the Lie derivative by acts on
exactly the same as the vector field
does itself.
I also say that the Lie derivative by is a degree-zero derivation of the algebra
. That is, it’s a real-linear transformation, and it satisfies the Leibniz rule:
for any -form
and
-form
. Linearity is straightforward, and given linearity the Leibniz rule follows if we can show
for -forms
. Indeed, we can write
and
as linear combinations of such
– and
-fold wedges, and then the Leibniz rule is obvious.
So, let us calculate:
So we see how we can peel off one of the -forms. A simple induction gives us the general case.
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