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

2.4.1 The orthogonal projector
2.4 Spacelike hypersurface 29
At each point p ∈ Σ, the space of all spacetime vectors can be orthogonally decomposed as
Tp(M) = Tp(Σ) ⊕ Vect(n) , (2.60)
where Vect(n) stands for the 1-dimensional subspace of Tp(M) generated by the vector n.
Remark : The orthogonal decomposition (2.60) holds for spacelike and timelike hypersurfaces,
but not for the null ones. Indeed for any normal n to a null hypersurface Σ, Vect(n) ⊂
Tp(Σ).
The orthogonal projector onto Σ is the operator γ associated with the decomposition (2.60)
according to
γ : Tp(M) −→ Tp(Σ)
v ↦−→ v + (n · v)n.
In particular, as a direct consequence of n · n = −1, γ satisfies
(2.61)
γ(n) = 0. (2.62)
Besides, it reduces to the identity operator for any vector tangent to Σ:
∀v ∈ Tp(Σ), γ(v) = v. (2.63)
According to Eq. (2.61), the components of γ with respect to any basis (eα) of Tp(M) are
γ α β = δα β + nα nβ . (2.64)
We have noticed in Sec. 2.3.1 that the embedding Φ of Σ in M induces a mapping Tp(Σ) →
Tp(M) (push-forward) and a mapping T ∗
p (M) → T ∗
p (Σ) (pull-back), but does not provide any
mapping in the reverse ways, i.e. from Tp(M) to Tp(Σ) and from T ∗
p (Σ) to T ∗
p (M). The
orthogonal projector naturally provides these reverse mappings: from its very definition, it is a
mapping Tp(M) → Tp(Σ) and we can construct from it a mapping γ ∗ M : T ∗
p (Σ) → T ∗
p (M) by
setting, for any linear form ω ∈ T ∗
p (Σ),
γ ∗ Mω : Tp(M) −→ R
v ↦−→ ω(γ(v)).
(2.65)
This clearly defines a linear form belonging to T ∗
p (M). Obviously, we can extend the operation
γ ∗ M to any multilinear form A acting on Tp(Σ), by setting
γ ∗ M A : Tp(M) n −→ R
(v1,... ,vn) ↦−→ A(γ(v1),... ,γ(vn)) .
(2.66)
Let us apply this definition to the bilinear form on Σ constituted by the induced metric γ: γ ∗ M γ
is then a bilinear form on M, which coincides with γ if its two arguments are vectors tangent
to Σ and which gives zero if any of its argument is a vector orthogonal to Σ, i.e. parallel to n.