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

126 The initial data problem
Notice that Eqs. (8.1)-(8.2) involve a single hypersurface Σ0, not a foliation (Σt) t∈R . In particular,
neither the lapse function nor the shift vector appear in these equations. Facing them, a
naive way to proceed would be to choose freely the metric γ, thereby fixing the connection D
and the scalar curvature R, and to solve Eqs. (8.1)-(8.2) for K. Indeed, for fixed γ, E, and p,
Eqs. (8.1)-(8.2) form a quasi-linear system of first order for the components Kij. However, as
discussed by Choquet-Bruhat [128], this approach is not satisfactory because we have only four
equations for six unknowns Kij and there is no natural prescription for choosing arbitrarily two
among the six components Kij.
Lichnerowicz (1944) [177] has shown that a much more satisfactory split of the initial data
(γ,K) between freely choosable parts and parts obtained by solving Eqs. (8.1)-(8.2) is provided
by the conformal decomposition introduced in Chap. 6. Lichnerowicz method has been extended
by Choquet-Bruhat (1956, 1971) [128, 86], by York and Ó Murchadha (1972, 1974, 1979) [273,
274, 196, 276] and more recently by York and Pfeiffer (1999, 2003) [278, 202]. Actually, conformal
decompositions are by far the most widely spread techniques to get initial data for the 3+1
Cauchy problem. Alternative methods exist, such as the quasi-spherical ansatz introduced by
Bartnik in 1993 [37] or a procedure developed by Corvino (2000) [98] and by Isenberg, Mazzeo
and Pollack (2002) [162] for gluing together known solutions of the constraints, thereby producing
new ones. Here we shall limit ourselves to the conformal methods. Standard reviews on this
subject are the articles by York (1979) [276] and Choquet-Bruhat and York (1980) [88]. Recent
reviews are the articles by Cook (2000) [94], Pfeiffer (2004) [201] and Bartnik and Isenberg
(2004) [39].
8.1.2 Conformal decomposition of the constraints
The conformal form of the constraint equations has been derived in Chap. 6. We have introduced
there the conformal metric ˜γ and the conformal factor Ψ such that the metric γ induced by the
spacetime metric on some hypersurface Σ0 is [cf. Eq. (6.22)]
γij = Ψ 4 ˜γij, (8.3)
and have decomposed the traceless part A ij of the extrinsic curvature K ij according to [cf.
Eq. (6.82)]
A ij = Ψ −10 Â ij . (8.4)
We consider here the decomposition involving Âij [α = −10 in Eq. (6.58)] and not the alternative
one, which uses Ãij (α = −4), because we have seen in Sec. 6.4.2 that the former is
well adapted to the momentum constraint. Using the decompositions (8.3) and (8.4), we have
rewritten the Hamiltonian constraint (8.1) and the momentum constraint (8.2) as respectively
the Lichnerowicz equation [Eq. (6.111)] and an equation involving the divergence of Âij with
respect to the conformal metric [Eq. (6.112)] :
˜Di ˜ D i Ψ − 1
8 ˜ RΨ + 1
8 Âij Âij Ψ −7 + 2π ˜ EΨ −3 − 1
12 K2 Ψ 5 = 0 , (8.5)
˜Dj Âij − 2
3 Ψ6 ˜ D i K = 8π˜p i , (8.6)