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

Chapter 6
Conformal decomposition
Contents
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Conformal decomposition of the 3-metric . . . . . . . . . . . . . . . . 85
6.3 Expression of the Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Conformal decomposition of the extrinsic curvature . . . . . . . . . 91
6.5 Conformal form of the 3+1 Einstein system . . . . . . . . . . . . . . 95
6.6 Isenberg-Wilson-Mathews approximation to General Relativity . . 99
6.1 Introduction
Historically, conformal decompositions in 3+1 general relativity have been introduced in two
contexts. First of all, Lichnerowicz [177] 1 has introduced in 1944 a decomposition of the
induced metric γ of the hypersurfaces Σt of the type
γ = Ψ 4 ˜γ, (6.1)
where Ψ is some strictly positive scalar field and ˜γ an auxiliary metric on Σt, which is necessarily
Riemannian (i.e. positive definite), as γ is. The relation (6.1) is called a conformal transformation
and ˜γ will be called hereafter the conformal metric. Lichnerowicz has shown
that the conformal decomposition of γ, along with some specific conformal decomposition of the
extrinsic curvature provides a fruitful tool for the resolution of the constraint equations to get
valid initial data for the Cauchy problem. This will be discussed in Chap. 8.
Then, in 1971-72, York [271, 272] has shown that conformal decompositions are also important
for the time evolution problem, by demonstrating that the two degrees of freedom of the
gravitational field are carried by the conformal equivalence classes of 3-metrics. A conformal
1 see also Ref. [178] which is freely accessible on the web