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

or equivalently, since v µ = γ µ σv σ ,
2.5 Gauss-Codazzi relations 35
γ µ αγ ν β γγ ργ σ λ 4 R ρ σµνv λ = R γ
λαβ vλ +
K γ αKλβ − K γ
βKαλ
v λ . (2.91)
In this identity, v can be replaced by any vector of T (M) without changing the results, thanks
to the presence of the projector operator γ and to the fact that both K and Riem are tangent
to Σ. Therefore we conclude that
γ µ αγ ν β γγ ργ σ δ 4 R ρ σµν = R γ
δαβ + Kγ αKδβ − K γ
β Kαδ . (2.92)
This is the Gauss relation.
If we contract the Gauss relation on the indices γ and α and use γ µ αγ α ρ = γ µ ρ = δ µ ρ +n µ nρ,
we obtain an expression that lets appear the Ricci tensors 4 R and R associated with g and γ
respectively:
γ µ αγ ν β 4 Rµν + γαµn ν γ ρ
βnσ 4 R µ νρσ = Rαβ + KKαβ − KαµK µ
β . (2.93)
We call this equation the contracted Gauss relation. Let us take its trace with respect to γ,
taking into account that K µ µ = K i i = K, KµνK µν = KijK ij and
γ αβ γαµn ν γ ρ
β nσ 4 R µ νρσ = γ ρ µn ν n σ4 R µ νρσ = 4 R µ νµσ
We obtain
= 4 Rνσ
n ν n σ + 4 R µ νρσn ρ nµn ν n σ
=
=0
4 Rµνn µ n ν . (2.94)
4 R + 2 4 Rµνn µ n ν = R + K 2 − KijK ij . (2.95)
Let us call this equation the scalar Gauss relation. It constitutes a generalization of Gauss’
famous Theorema Egregium (remarkable theorem) [52, 53]. It relates the intrinsic curvature
of Σ, represented by the Ricci scalar R, to its extrinsic curvature, represented by K 2 − KijK ij .
Actually, the original version of Gauss’ theorem was for two-dimensional surfaces embedded in
the Euclidean space R 3 . Since the curvature of the latter is zero, the left-hand side of Eq. (2.95)
vanishes identically in this case. Moreover, the metric g of the Euclidean space R 3 is Riemannian,
not Lorentzian. Consequently the term K 2 − KijK ij has the opposite sign, so that Eq. (2.95)
becomes
R − K 2 + KijK ij = 0 (g Euclidean). (2.96)
This change of sign stems from the fact that for a Riemannian ambient metric, the unit normal
vector n is spacelike and the orthogonal projector is γ α β = δα β −nα nβ instead of γ α β = δα β +nα nβ
[the latter form has been used explicitly in the calculation leading to Eq. (2.87)]. Moreover, in
dimension 2, formula (2.96) can be simplified by letting appear the principal curvatures κ1 and
κ2 of Σ (cf. Sec. 2.3.4). Indeed, K can be diagonalized in an orthonormal basis (with respect
to γ) so that Kij = diag(κ1,κ2) and Kij = diag(κ1,κ2). Consequently, K = κ1 + κ2 and
KijK ij = κ2 1 + κ22 and Eq. (2.96) becomes
R = 2κ1κ2 (g Euclidean, Σ dimension 2). (2.97)