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

86 Conformal decomposition
As an example of background metric, let us consider a coordinate system (x i ) = (x,y,z)
on Σt and define the metric f as the bilinear form whose components with respect to that
coordinate system are fij = diag(1,1,1) (in this example, f is flat).
The inverse metric is denoted by f ij :
f ik fkj = δ i j. (6.9)
In particular note that, except for the very special case γij = fij, one has
We denote by D the Levi-Civita connection associated with f:
and define
f ij = γ ik γ jl fkl. (6.10)
Dkfij = 0, (6.11)
D i = f ij Dj. (6.12)
The Christoffel symbols of the connection D with respect to the coordinates (xi ) are denoted
by ¯ Γk ij ; they are given by the standard expression:
¯Γ k 1
ij =
2 fkl
∂flj ∂fil ∂fij
+ −
∂xi ∂xj ∂xl
. (6.13)
6.2.3 Conformal metric
Thanks to f, we define
where
Ψ :=
˜γ := Ψ −4 γ , (6.14)
1/12 γ
, γ := det(γij), f := det(fij). (6.15)
f
The key point is that, contrary to γ, Ψ is a tensor field on Σt. Indeed a change of coordinates
(xi ) ↦→ (xi′ ) induces the following changes in the determinants:
where J denotes the Jacobian matrix
γ ′ = (det J) 2 γ (6.16)
f ′ = (det J) 2 f, (6.17)
J i ∂xi
i ′ :=
∂xi′ . (6.18)
From Eqs. (6.16)-(6.17) it is obvious that γ ′ /f ′ = γ/f, which shows that γ/f, and hence Ψ, is
a scalar field. Of course, this scalar field depends upon the choice of the background metric f.
Ψ being a scalar field, the quantity ˜γ defined by (6.14) is a tensor field on Σt. Moreover, it is a
Riemannian metric on Σt. We shall call it the conformal metric. By construction, it satisfies
det(˜γij) = f . (6.19)