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

20 Geometry of hypersurfaces
Figure 2.1: Embedding Φ of the 3-dimensional manifold ˆ Σ into the 4-dimensional manifold M, defining the
hypersurface Σ = Φ( ˆ Σ). The push-forward Φ∗v of a vector v tangent to some curve C in ˆ Σ is a vector tangent
to Φ(C) in M.
Tp( ˆ Σ) and Tp(M). This mapping is denoted by Φ∗ and is called the push-forward mapping;
thanks to the adapted coordinate systems x α = (t,x,y,z), it can be explicited as follows
Φ∗ : Tp( ˆ Σ) −→ Tp(M)
v = (v x ,v y ,v z ) ↦−→ Φ∗v = (0,v x ,v y ,v z ),
(2.23)
where v i = (v x ,v y ,v z ) denotes the components of the vector v with respect to the natural basis
∂/∂x i of Tp(Σ) associated with the coordinates (x i ).
Conversely, the embedding Φ induces a mapping, called the pull-back mapping and denoted
Φ ∗ , between the linear forms on Tp(M) and those on Tp( ˆ Σ) as follows
Φ ∗ : T ∗
p (M) −→ T ∗
p ( ˆ Σ)
ω ↦−→ Φ ∗ ω : Tp( ˆ Σ) → R
v ↦→ 〈ω,Φ∗v〉.
Taking into account (2.23), the pull-back mapping can be explicited:
Φ ∗ : T ∗
p (M) −→ T ∗
p ( ˆ Σ)
ω = (ωt,ωx,ωy,ωz) ↦−→ Φ ∗ ω = (ωx,ωy,ωz),
(2.24)
(2.25)
where ωα denotes the components of the 1-form ω with respect to the basis dx α associated with
the coordinates (x α ).
In what follows, we identify ˆ Σ and Σ = Φ( ˆ Σ). In particular, we identify any vector on ˆ Σ
with its push-forward image in M, writing simply v instead of Φ∗v.