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

84 Conformal decomposition
equivalence class is defined as the set of all metrics that can be related to a given metric γ
by a transform like (6.1). The argument of York is based on the Cotton tensor [99], which is
a rank-3 covariant tensor defined from the covariant derivative of the Ricci tensor R of γ by
Cijk := Dk
Rij − 1
4 Rγij
− Dj Rik − 1
4 Rγik
. (6.2)
The Cotton tensor is conformally invariant and shows the same property with respect to 3dimensional
metric manifolds than the Weyl tensor [cf. Eq. (2.18)] for metric manifolds of
dimension strictly greater than 3, namely its vanishing is a necessary and sufficient condition for
the metric to be conformally flat, i.e. to be expressible as γ = Ψ 4 f, where Ψ is some scalar
field and f a flat metric. Let us recall that in dimension 3, the Weyl tensor vanishes identically.
More precisely, York [271] constructed from the Cotton tensor the following rank-2 tensor
C ij := − 1
2 ǫiklCmklγ mj = ǫ ikl
Dk R j 1
l −
4 Rδj
l , (6.3)
where ǫ is the Levi-Civita alternating tensor associated with the metric γ. This tensor is called
the Cotton-York tensor and exhibits the following properties:
• symmetric: C ji = C ij
• traceless: γijC ij = 0
• divergence-free (one says also transverse): DjC ij = 0
Moreover, if one consider, instead of C, the following tensor density of weight 5/3,
C ij
∗ := γ5/6 C ij , (6.4)
where γ := det(γij), then one gets a conformally invariant quantity. Indeed, under a conformal
transformation of the type (6.1), ǫikl = Ψ−6˜ǫ ikl , Cmkl = ˜ Cmkl (conformal invariance of the Cotton
tensor), γml = Ψ−4˜γ ml and γ5/6 = Ψ10˜γ 5/6 , so that C ij
∗ = ˜ C ij
∗ . The traceless and transverse
(TT) properties being characteristic of the pure spin 2 representations of the gravitational field
(cf. T. Damour’s lectures [103]), the conformal invariance of C ij
∗ shows that the true degrees of
freedom of the gravitational field are carried by the conformal equivalence class.
Remark : The remarkable feature of the Cotton-York tensor is to be a TT object constructed
from the physical metric γ alone, without the need of some extra-structure on the manifold
Σt. Usually, TT objects are defined with respect to some extra-structure, such as privileged
Cartesian coordinates or a flat background metric, as in the post-Newtonian approach to
general relativity (see L. Blanchet’s lectures [58]).
Remark : The Cotton and Cotton-York tensors involve third derivatives of the metric tensor.