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6.2 Conformal decomposition of the 3-metric 85
6.2 Conformal decomposition of the 3-metric
6.2.1 Unit-determinant conformal “metric”
A somewhat natural representative of a conformal equivalence class is the unit-determinant
conformal “metric”
ˆγ := γ −1/3 γ, (6.5)
where γ := det(γij). This would correspond to the choice Ψ = γ 1/12 in Eq. (6.1). All the metrics
γ in the same conformal equivalence class lead to the same value of ˆγ. However, since the
determinant γ depends upon the choice of coordinates to express the components γij, Ψ = γ 1/12
would not be a scalar field. Actually, the quantity ˆγ is not a tensor field, but a tensor density,
of weight −2/3.
Let us recall that a tensor density of weight n ∈ Q is a quantity τ such that
where T is a tensor field.
τ = γ n/2 T, (6.6)
Remark : The conformal “metric” (6.5) has been used notably in the BSSN formulation [233,
43] for the time evolution of 3+1 Einstein system, to be discussed in Chap. 9. An “associated”
connection ˆD has been introduced, such that ˆDˆγ = 0. However, since ˆγ is a tensor
density and not a tensor field, there is not a unique connection associated with it (Levi-
Civita connection). In particular one has Dˆγ = 0, so that the connection D associated
with the metric γ is “associated” with ˆγ, in addition to ˆD. As a consequence, some of
the formulæ presented in the original references [233, 43] for the BSSN formalism have a
meaning only for Cartesian coordinates.
6.2.2 Background metric
To clarify the meaning of ˆD (i.e. to avoid to work with tensor densities) and to allow for the
use of spherical coordinates, we introduce an extra structure on the hypersurfaces Σt, namely
a background metric f [63]. It is asked that the signature of f is (+,+,+), i.e. that f is a
Riemannian metric, as γ. Moreover, we tight f to the coordinates (x i ) by demanding that the
components fij of f with respect to (x i ) obey to
An equivalent writing of this is
∂fij
∂t
= 0. (6.7)
L∂tf = 0, (6.8)
i.e. the metric f is Lie-dragged along the coordinate time evolution vector ∂t.
If the topology of Σt enables it, it is quite natural to choose f to be flat, i.e. such that its
Riemann tensor vanishes. However, in this chapter, we shall not make such hypothesis, except
in Sec. 6.6.