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

8.2 Conformal transverse-traceless method 129
8.2.2 Conformal transverse-traceless form of the constraints
Inserting the longitudinal/transverse decomposition (8.9) into the constraint equations (8.5) and
(8.6) and making use of Eq. (8.15) yields to the system
where
˜Di ˜ D i Ψ − 1
8 ˜ RΨ + 1
(
8
˜ LX)ij + ÂTT ij
( ˜ LX) ij + Âij
TT
Ψ −7 + 2π ˜ EΨ −3 − 1
12 K2 Ψ 5 = 0 ,
(8.20)
˜∆L X i − 2
3 Ψ6 ˜ D i K = 8π˜p i , (8.21)
( ˜ LX)ij := ˜γik˜γjl( ˜ LX) kl
(8.22)
 TT
ij := ˜γik˜γjl Âkl TT . (8.23)
With the constraint equations written as (8.20) and (8.21), we see clearly which part of the
initial data on Σ0 can be freely chosen and which part is “constrained”:
• free data:
– conformal metric ˜γ;
– symmetric traceless and transverse tensor Âij
TT
with respect to ˜γ: ˜γij Âij
TT = 0 and ˜ DjÂij TT
– scalar field K;
– conformal matter variables: ( ˜ E, ˜p i );
• constrained data (or “determined data”):
(traceless and transverse are meant
= 0);
– conformal factor Ψ, obeying the non-linear elliptic equation (8.20) (Lichnerowicz
equation)
– vector X, obeying the linear elliptic equation (8.21) .
Accordingly the general strategy to get valid initial data for the Cauchy problem is to choose
(˜γij, Âij
TT ,K, ˜ E, ˜p i ) on Σ0 and solve the system (8.20)-(8.21) to get Ψ and X i . Then one constructs
γij = Ψ 4 ˜γij (8.24)
K ij
−10
= Ψ ( ˜ LX) ij + Âij
TT + 1
3 Ψ−4 K˜γ ij
(8.25)
E = Ψ −8 ˜ E (8.26)
p i = Ψ −10 ˜p i
(8.27)
and obtains a set (γ,K,E,p) which satisfies the constraint equations (8.1)-(8.2). This method
has been proposed by York (1979) [276] and is naturally called the conformal transverse
traceless (CTT) method.