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

32 Geometry of hypersurfaces
2.4.3 Links between the ∇ and D connections
Given a tensor field T on Σ, its covariant derivative DT with respect to the Levi-Civita connection
D of the metric γ (cf. Sec. 2.3.3) is expressible in terms of the covariant derivative ∇T
with respect to the spacetime connection ∇ according to the formula
the component version of which is [cf. Eq. (2.71)]:
DρT α1...αp
= γα1
β1...βq µ1
DT = γ ∗ ∇T , (2.78)
· · · γαp
µp γ ν1
β1
· · · γνq
βq γσ µ1...µp
ρ ∇σT ν1...νq . (2.79)
Various comments are appropriate: first of all, the T in the right-hand side of Eq. (2.78) should
be the four-dimensional extension γ ∗ MT provided by Eq. (2.66). Following the remark made
above, we write T instead of γ ∗ MT. Similarly the right-hand side should write γ∗ MDT, so that
Eq. (2.78) is a equality between tensors on M. Therefore the rigorous version of Eq. (2.78) is
γ ∗ MDT = γ∗ [∇(γ ∗ MT)]. (2.80)
Besides, even if T := γ ∗ MT is a four-dimensional tensor, its suppport (domain of definition)
remains the hypersurface Σ. In order to define the covariant derivative ∇T, the support must
be an open set of M, which Σ is not. Accordingly, one must first construct some extension T ′ of
T in an open neighbourhood of Σ in M and then compute ∇T ′ . The key point is that thanks
to the operator γ ∗ acting on ∇T ′ , the result does not depend of the choice of the extension T ′ ,
provided that T ′ = T at every point in Σ.
The demonstration of the formula (2.78) takes two steps. First, one can show easily that
γ ∗ ∇ (or more precisely the pull-back of γ ∗ ∇γ ∗ M) is a torsion-free connection on Σ, for it satisfies
all the defining properties of a connection (linearity, reduction to the gradient for a scalar
field, commutation with contractions and Leibniz’ rule) and its torsion vanishes. Secondly, this
connection vanishes when applied to the metric tensor γ: indeed, using Eqs. (2.71) and (2.69),
(γ ∗ ∇γ) αβγ = γ µ αγ ν βγργ∇ργµν = γ µ αγ ν βγργ(∇ρ gµν +∇ρnµ nν + nµ∇ρnν)
=0
= γ ρ γ (γµ α γνβnν ∇ρnµ + γ
=0
µ αnµ ∇ρnν)
=0
= 0. (2.81)
Invoking the uniqueness of the torsion-free connection associated with a given non-degenerate
metric (the Levi-Civita connection, cf. Sec. 2.IV.2 of N. Deruelle’s lecture [108]), we conclude
that necessarily γ ∗∇ = D.
One can deduce from Eq. (2.78) an interesting formula about the derivative of a vector field
v along another vector field u, when both vectors are tangent to Σ. Indeed, from Eq. (2.78),
(Duv) α = u σ Dσv α = u σ γ ν σ
=uν γ α µ∇νv µ = u ν δ α µ + n α
nµ ∇νv µ
= u ν ∇νv α + n α u ν nµ∇νv µ
=−v µ = u
∇νnµ
ν ∇νv α − n α u ν v µ ∇µnν, (2.82)