- #1

gnieddu

- 24

- 1

## Homework Statement

Hi,

it's the first time I post here, so apologies if this is not the right place.

I'm trying to self-study GR, but I'm stuck with Lie Derivatives. The book I'm using (Ludvigsen - General Relativity. A geometric approach) starts with the usual definitions and then gives the formula for a vector field (in abstract index notation) as

## Homework Equations

[tex]L_{v}w^a = v^{b}\nabla_{b}w^a - w^{b}\nabla_{b}v^a[/tex]

It then moves on to find the Lie derivative for a covector, and states:

[tex]L_{v}(u_{a}w^a) = (L_{v}u_a)w^a + u_{a}L_{v}w^a[/tex] (1)

which is fair. Then after pointing out that [tex]u_{a}w^a[/tex] is a scalar, gives the (almost) final formula:

[tex](L_{v}u_a)w^a = (v^{c}\nabla_{c}u_a + u_{c}\nabla_{a}v^c)w^a[/tex]

which, simplified by w^a gives the final rule

## The Attempt at a Solution

I tried to work out the missing passages following this line of thought.

[tex]u_{a}w^a[/tex] is a scalar, so:

[tex]L_{v}(u_{a}w^a) = v^{c}\nabla_{c}(u_{a}w^a) = v^{c}w^{a}\nabla_{c}u_a + v^{c}u_{a}\nabla_{c}w^a[/tex] (2)

On the other hand:

[tex]u_{a}L_{v}w^a = u_{a}v^{c}\nabla_{c}w^a - u_{a}w^{c}\nabla_{c}v^a[/tex] (3)

Combining 1, 2 and 3, I get:

[tex](L_{v}u_a)w^a = v^{c}w^{a}\nabla_{c}u_a + u_{a}w^{c}\nabla_{c}v^a[/tex]

which is almost like the final formula, but a couple of indexes don't match. Where have I gone wrong?

Thanks