3+1 formalism and bases of numerical relativity - LUTh ...

A.2 Generalization to any tensor field 191
where use has been made of the standard notation Lu v α := (Lu v) α . The above relation shows
that the Lie derivative of a vector with respect to another one is nothing but the commutator
of these two vectors:
Lu v = [u,v] . (A.5)
A.2 Generalization to any tensor field
The Lie derivative is extended to any tensor field by (i) demanding that for a scalar field f,
Lu f = 〈df,u〉 and (ii) using the Leibniz rule. As a result, the Lie derivative Lu
T of a tensor
k
field T of type ℓ is a tensor field of the same type, the components of which with respect to
a given coordinate system (x α ) are
Lu T α1...αk
β1...βℓ
∂ α1...αk
= uµ
∂x µT β1...βℓ −
k
T α1...
i=1
i
↓
σ...αk
β1...βℓ
∂uαi +
∂xσ ℓ
T α1...αk
i=1
β1... σ...βℓ ↑
i
∂uσ . βi ∂x
(A.6)
In particular, for a 1-form,
Luωα = u µ∂ωα ∂u
+ ωµ
∂x µ µ
∂xα. (A.7)
Notice that the partial derivatives in Eq. (A.6) can be remplaced by any connection without
torsion, such as the Levi-Civita connection ∇ associated with the metric g, yielding
LuT α1...αk
β1...βℓ = uµ ∇µT α1...αk
β1...βℓ −
k
T α1...
i=1
i
↓
σ...αk
β1...βℓ ∇σu αi +
ℓ
T α1...αk
i=1
β1... σ...βℓ ↑
i
∇βiuσ .
(A.8)