3+1 formalism and bases of numerical relativity - LUTh ...
3+1 formalism and bases of numerical relativity - LUTh ... 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)
8.2 Conformal transverse-traceless method 127 where we have introduce the rescaled matter quantities and ˜E := Ψ 8 E (8.7) ˜p i := Ψ 10 p i . (8.8) The definition of ˜p i is clearly motivated by Eq. (6.112). On the contrary the power 8 in the definition of ˜ E is not the only possible choice. As we shall see in § 8.2.4, it is chosen (i) to guarantee a negative power of Ψ in the ˜ E term in Eq. (8.5), resulting in some uniqueness property of the solution and (ii) to allow for an easy implementation of the dominant energy condition. 8.2 Conformal transverse-traceless method 8.2.1 Longitudinal/transverse decomposition of Âij In order to solve the system (8.5)-(8.6), York (1973,1979) [274, 275, 276] has decomposed Âij into a longitudinal part and a transverse one, by setting  ij = ( ˜ LX) ij + Âij TT , (8.9) where Âij TT is both traceless and transverse (i.e. divergence-free) with respect to the metric ˜γ: ˜γij Âij TT = 0 and Dj ˜ Âij TT = 0, (8.10) and ( ˜ LX) ij is the conformal Killing operator associated with the metric ˜γ and acting on the vector field X: ( ˜ LX) ij := ˜ D i X j + ˜ D j X i − 2 3 ˜ DkX k ˜γ ij . (8.11) The properties of this linear differential operator are detailed in Appendix B. Let us retain here that ( ˜ LX) ij is by construction traceless: ˜γij( ˜ LX) ij = 0 (8.12) (it must be so because in Eq. (8.9) both Âij and Âij TT are traceless) and the kernel of ˜L is made of the conformal Killing vectors of the metric ˜γ, i.e. the generators of the conformal isometries (cf. Sec. B.1.3). The symmetric tensor ( ˜ LX) ij is called the longitudinal part of Âij , whereas Âij TT is called the transverse part. Given Âij , the vector X is determined by taking the divergence of Eq. (8.9): taking into account property (8.10), we get ˜Dj( ˜ LX) ij = ˜ Dj Âij . (8.13) The second order operator ˜ Dj( ˜ LX) ij acting on the vector X is the conformal vector Laplacian ˜∆L: ˜∆L X i := ˜ Dj( ˜ LX) ij = ˜ Dj ˜ D j X i + 1 3 ˜ D i ˜ DjX j + ˜ R i j Xj , (8.14)
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8.2 Conformal transverse-traceless method 127<br />
where we have introduce the rescaled matter quantities<br />
<strong>and</strong><br />
˜E := Ψ 8 E (8.7)<br />
˜p i := Ψ 10 p i . (8.8)<br />
The definition <strong>of</strong> ˜p i is clearly motivated by Eq. (6.112). On the contrary the power 8 in the<br />
definition <strong>of</strong> ˜ E is not the only possible choice. As we shall see in § 8.2.4, it is chosen (i)<br />
to guarantee a negative power <strong>of</strong> Ψ in the ˜ E term in Eq. (8.5), resulting in some uniqueness<br />
property <strong>of</strong> the solution <strong>and</strong> (ii) to allow for an easy implementation <strong>of</strong> the dominant energy<br />
condition.<br />
8.2 Conformal transverse-traceless method<br />
8.2.1 Longitudinal/transverse decomposition <strong>of</strong> Âij<br />
In order to solve the system (8.5)-(8.6), York (1973,1979) [274, 275, 276] has decomposed Âij<br />
into a longitudinal part <strong>and</strong> a transverse one, by setting<br />
 ij = ( ˜ LX) ij + Âij<br />
TT , (8.9)<br />
where Âij<br />
TT is both traceless <strong>and</strong> transverse (i.e. divergence-free) with respect to the metric ˜γ:<br />
˜γij Âij<br />
TT = 0 <strong>and</strong> Dj<br />
˜ Âij<br />
TT = 0, (8.10)<br />
<strong>and</strong> ( ˜ LX) ij is the conformal Killing operator associated with the metric ˜γ <strong>and</strong> acting on<br />
the vector field X:<br />
( ˜ LX) ij := ˜ D i X j + ˜ D j X i − 2<br />
3 ˜ DkX k ˜γ ij . (8.11)<br />
The properties <strong>of</strong> this linear differential operator are detailed in Appendix B. Let us retain here<br />
that ( ˜ LX) ij is by construction traceless:<br />
˜γij( ˜ LX) ij = 0 (8.12)<br />
(it must be so because in Eq. (8.9) both Âij <strong>and</strong> Âij<br />
TT are traceless) <strong>and</strong> the kernel <strong>of</strong> ˜L is<br />
made <strong>of</strong> the conformal Killing vectors <strong>of</strong> the metric ˜γ, i.e. the generators <strong>of</strong> the conformal<br />
isometries (cf. Sec. B.1.3). The symmetric tensor ( ˜ LX) ij is called the longitudinal part <strong>of</strong><br />
Âij , whereas Âij<br />
TT is called the transverse part.<br />
Given Âij , the vector X is determined by taking the divergence <strong>of</strong> Eq. (8.9): taking into<br />
account property (8.10), we get<br />
˜Dj( ˜ LX) ij = ˜ Dj Âij . (8.13)<br />
The second order operator ˜ Dj( ˜ LX) ij acting on the vector X is the conformal vector Laplacian<br />
˜∆L:<br />
˜∆L X i := ˜ Dj( ˜ LX) ij = ˜ Dj ˜ D j X i + 1<br />
3 ˜ D i ˜ DjX j + ˜ R i j Xj , (8.14)
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