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

30 Geometry of hypersurfaces
Since it constitutes an “extension” of γ to all vectors in Tp(M), we shall denote it by the same
symbol:
γ := γ ∗ Mγ . (2.67)
This extended γ can be expressed in terms of the metric tensor g and the linear form n dual to
the normal vector n according to
In components:
γ = g + n ⊗ n . (2.68)
γαβ = gαβ + nα nβ. (2.69)
Indeed, if v and u are vectors both tangent to Σ, γ(u,v) = g(u,v)+〈n,u〉〈n,v〉 = g(u,v)+0 =
g(u,v), and if u = λn, then, for any v ∈ Tp(M), γ(u,v) = λg(n,v) + λ〈n,n〉〈n,v〉 =
λ[g(n,v) − 〈n,v〉] = 0. This establishes Eq. (2.68). Comparing Eq. (2.69) with Eq. (2.64)
justifies the notation γ employed for the orthogonal projector onto Σ, according to the convention
set in Sec. 2.2.2 [see Eq. (2.11)]: γ is nothing but the ”extended” induced metric γ with the
first index raised by the metric g.
Similarly, we may use the γ ∗ M operation to extend the extrinsic curvature tensor K, defined
a priori as a bilinear form on Σ [Eq. (2.42)], to a bilinear form on M, and we shall use the same
symbol to denote this extension:
K := γ ∗ MK . (2.70)
Remark : In this lecture, we will very often use such a “four-dimensional point of view”, i.e.
we shall treat tensor fields defined on Σ as if they were defined on M. For covariant
tensors (multilinear forms), if not mentioned explicitly, the four-dimensional extension is
performed via the γ ∗ M operator, as above for γ and K. For contravariant tensors, the
identification is provided by the push-forward mapping Φ∗ discussed in Sec. 2.3.1. This
four-dimensional point of view has been advocated by Carter [80, 81, 82] and results in
an easier manipulation of tensors defined in Σ, by treating them as ordinary tensors on
M. In particular this avoids the introduction of special coordinate systems and complicated
notations.
In addition to the extension of three dimensional tensors to four dimensional ones, we use
the orthogonal projector γ to define an “orthogonal projection operation” for all tensors on M
in the following way. Given a tensor T of type on M, we denote by γ ∗T another tensor on
p
q
M, of the same type and such that its components in any basis (eα) of Tp(M) are expressed in
terms of those of T by
(γ ∗ T) α1...αp
= γα1
β1...βq µ1 ...γαp µpγ ν1
µ1...µp
... γνq T β1 βq ν1...νq. (2.71)
Notice that for any multilinear form A on Σ, γ ∗ (γ ∗ MA) = γ∗ MA, for a vector v ∈ Tp(M),
γ ∗v = γ(v), for a linear form ω ∈ T ∗
p (M), γ∗ ω = ω ◦ γ, and for any tensor T, γ ∗T is tangent
to Σ, in the sense that γ ∗T results in zero if one of its arguments is n or n.