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

88 Conformal decomposition
6.2.4 Conformal connection
˜γ being a well defined metric on Σt, let ˜D be the Levi-Civita connection associated to it:
˜D˜γ = 0. (6.27)
Let us denote by ˜ Γ k ij the Christoffel symbols of ˜D with respect to the coordinates (x i ):
Given a tensor field T of type
by the formula
where 2
Γ k ij
DkT i1...ip
j1...jq = ˜ DkT i1...ip
j1...jq +
˜Γ k ij = 1
2 ˜γkl
∂˜γlj
∂x
p
q
∂˜γil ∂˜γij
+ − i ∂xj ∂xl
. (6.28)
on Σt, the covariant derivatives ˜DT and DT are related
p
r=1
C ir i1...l...ip
kl T j1...jq −
q
r=1
C l kjr
i1...ip
T , (6.29)
j1...l...jq
C k ij := Γ k ij − ˜ Γ k ij, (6.30)
being the Christoffel symbols of the connection D. The formula (6.29) follows immediately
from the expressions of DT and ˜DT in terms of respectively the Christoffel symbols Γk ij and
˜Γ k i1...ip
ij . Since DkT j1...jq − ˜ DkT i1...ip
j1...jq are the components of a tensor field, namely DT − ˜DT,
it follows from Eq. (6.29) that the Ck ij are also the components of a tensor field. Hence we recover
a well known property: although the Christoffel symbols are not the components of any tensor
field, the difference between two sets of them represents the components of a tensor field. We
may express the tensor Ck ij in terms of the ˜D-derivatives of the metric γ, by the same formula
than the one for the Christoffel symbols Γk ij , except that the partial derivatives are replaced by
˜D-derivatives:
C k ij = 1
2 γkl Diγlj
˜ + ˜ Djγil − ˜
Dlγij . (6.31)
It is easy to establish this relation by evaluating the right-hand side, expressing the ˜D-derivatives
of γ in terms of the Christoffel symbols ˜ Γ k ij :
1
2 γkl Diγlj
˜ + ˜ Djγil − ˜
Dlγij
= 1
2 γkl
∂γlj
= Γ k ij
∂xi − ˜ Γ m ilγmj − ˜ Γ m ijγlm + ∂γil
∂xj − ˜ Γ m jiγml − ˜ Γ m jlγim − ∂γij
∂x l + ˜ Γ m li γmj + ˜ Γ m lj γim
+ 1
2 γkl (−2) ˜ Γ m ij γlm
= Γ k ij − δ k m ˜ Γ m ij
= C k ij, (6.32)
2 The C k ij are not to be confused with the components of the Cotton tensor discussed in Sec. 6.1. Since we
shall no longer make use of the latter, no confusion may arise.