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

4.4 The Cauchy problem 63
hightest degree (two in the present case). We get, denoting by “· · ·” everything but a second
order derivative of γij:
Rij = ∂Γkij ∂xk − ∂Γk ik + · · ·
∂xj = 1 ∂
2 ∂xk
γ kl
∂γlj ∂γil ∂γij
+ −
∂xi ∂xj ∂xl
− 1 ∂
2 ∂xj
γ kl
∂γlk
∂x ∂x
= 1
2 γkl
∂2γlj Rij = − 1
2 γkl
∂2γij ∂xk∂xl + ∂2γkl ∂xi∂xj − ∂2γlj ∂xi∂xk − ∂2γil ∂xj∂xk
+ Qij
∂γil ∂γik
+ − i k
∂xk∂xi + ∂2γil ∂xk∂xj − ∂2γij ∂xk∂xl − ∂2γlk ∂xj∂xi − ∂2γil ∂xj∂xk + ∂2γik ∂xj∂xl γkl, ∂γkl
∂x m
∂xl
+ · · ·
+ · · ·
, (4.86)
where Qij(γkl,∂γkl/∂x m ) is a (non-linear) expression containing the components γkl and their
first spatial derivatives only. Taking the trace of (4.86) (i.e. contracting with γij ), we get
. (4.87)
Besides
R = γ ik γ jl ∂2γij ∂xk∂xl − γijγ kl ∂2γij ∂xk + Q
∂xl Dj(γ jk ˙γki) = γ jk Dj ˙γki = γ jk
∂ ˙γki
= γ jk ∂2γki ∂xj
+ Qi
∂t
γkl, ∂γkl
∂x m
∂x j − Γl jk ˙γli − Γ l ji ˙γkl
γkl, ∂γkl ∂γkl
∂xm, ∂t
, (4.88)
where Qi(γkl,∂γkl/∂x m ,∂γkl/∂t) is some expression that does not contain any second order
derivative of γkl. Substituting Eqs. (4.86), (4.87) and (4.88) in Eqs. (4.83)-(4.85) gives
− ∂2
γij ∂2γij + γkl
∂t2 ∂xk∂xl + ∂2γkl ∂xi∂xj − ∂2γlj ∂xi∂xk − ∂2γil ∂xj∂xk
= 8π [(S − E)γij − 2Sij]
+Qij γkl, ∂γkl
∂γkl
∂xm, (4.89)
∂t
γ ik γ jl ∂2γij ∂xk∂xl − γijγ kl ∂2γij ∂xk
= 16πE + Q γkl,
∂xl ∂γkl
∂γkl
∂xm, (4.90)
∂t
γ jk ∂2γki ∂xj∂t − γkl ∂2γkl ∂xi∂t = −16πpi
+ Qi γkl, ∂γkl
∂γkl
∂xm, . (4.91)
∂t
Notice that we have incorporated the first order time derivatives into the Q terms.
Equations (4.89)-(4.91) constitute a system of PDEs for the unknowns γij. This system is of
second order and non linear, but quasi-linear, i.e. linear with respect to all the second order
derivatives. Let us recall that, in this system, the γ ij ’s are to be considered as functions of the
γij’s, these functions being given by expressing the matrix (γij) as the inverse of the matrix (γij)
(e.g. via Cramer’s rule).
A key feature of the system (4.89)-(4.91) is that it contains 6 + 1 + 3 = 10 equations for
the 6 unknowns γij. Hence it is an over-determined system. Among the three sub-systems