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

3.4 Last part of the 3+1 decomposition of the Riemann tensor 49
can limit the range of variation of the indices to {1,2,3} since all the involved tensors are spatial
ones [including LmK, thanks to the property (3.32)] Hence
γ µν4 Rµν = − 1
N γij LmKij − 1
N DiD i N + R + K 2 − 2KijK ij . (3.46)
Now γ µν4 Rµν = (g µν + n µ n ν ) 4 Rµν = 4 R + 4 Rµνn µ n ν and
− γ ij LmKij = −Lm(γ ij Kij
with Lmγ ij evaluted from the very definition of the inverse 3-metric:
γikγ kj = δ j
i
⇒ Lmγik γ kj + γik Lmγ kj = 0
⇒ γ il γ kj Lmγlk + γ il γlk
) + KijLmγ
=K
ij , (3.47)
=δ i k
⇒ Lmγ ij = −γ ik γ jl Lmγkl
⇒ Lmγ ij = 2Nγ ik γ kl Kkl
Lmγ lj = 0
⇒ Lmγ ij = 2NK ij , (3.48)
where we have used Eq. (3.24). Pluging Eq. (3.48) into Eq. (3.47) gives
Consequently Eq. (3.46) becomes
− γ ij LmKij = −LmK + 2NKijK ij . (3.49)
4 R + 4 Rµνn µ n ν = R + K 2 − 1
N LmK − 1
N DiD i N . (3.50)
It is worth to combine with equation with the scalar Gauss relation (2.95) to get rid of the
Ricci tensor term 4Rµνn µ nν and obtain an equation which involves only the spacetime scalar
curvature 4R: 4 2
R = R + K + KijK ij − 2
N LmK − 2
N DiD i N . (3.51)