3+1 formalism and bases of numerical relativity - LUTh ...
3+1 formalism and bases of numerical relativity - LUTh ... 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.
2.4.2 Relation between K and ∇n 2.4 Spacelike hypersurface 31 A priori the unit vector n normal to Σ is defined only at points belonging to Σ. Let us consider some extension of n in an open neighbourhood of Σ. If Σ is a level surface of some scalar field t, such a natural extension is provided by the gradient of t, according to Eq. (2.30). Then the tensor fields ∇n and ∇n are well defined quantities. In particular, we can introduce the vector a := ∇nn. (2.72) Since n is a timelike unit vector, it can be regarded as the 4-velocity of some observer, and a is then the corresponding 4-acceleration. a is orthogonal to n and hence tangent to Σ, since n · a = n · ∇nn = 1/2∇n(n · n) = 1/2∇n(−1) = 0. Let us make explicit the definition of the tensor K extend to M by Eq. (2.70). From the definition of the operator γ ∗ M [Eq. (2.66)] and the original definition of K [Eq. (2.43)], we have ∀(u,v) ∈ Tp(M) 2 , K(u,v) = K(γ(u), γ(v)) = −γ(u) · ∇ γ(v)n = −γ(u) · ∇ v+(n·v)n n = −[u + (n · u)n] · [∇vn + (n · v)∇nn] = −u · ∇vn − (n · v)u · ∇nn =a −(n · u)(n · v)n · ∇nn =0 = −u · ∇vn − (a · u)(n · v), −(n · u)n · ∇vn =0 = −∇n(u,v) − 〈a,u〉〈n,v〉, (2.73) where we have used the fact that n·n = −1 to set n·∇xn = 0 for any vector x. Since Eq. (2.73) is valid for any pair of vectors (u,v) in Tp(M), we conclude that In components: ∇n = −K − a ⊗ n . (2.74) ∇β nα = −Kαβ − aα nβ . (2.75) Notice that Eq. (2.74) implies that the (extended) extrinsic curvature tensor is nothing but the gradient of the 1-form n to which the projector operator γ ∗ is applied: K = −γ ∗ ∇n . (2.76) Remark : Whereas the bilinear form ∇n is a priori not symmetric, its projected part −K is a symmetric bilinear form. Taking the trace of Eq. (2.74) with respect to the metric g (i.e. contracting Eq. (2.75) with g αβ ) yields a simple relation between the divergence of the vector n and the trace of the extrinsic curvature tensor: K = −∇ · n . (2.77)
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2.4.2 Relation between K <strong>and</strong> ∇n<br />
2.4 Spacelike hypersurface 31<br />
A priori the unit vector n normal to Σ is defined only at points belonging to Σ. Let us consider<br />
some extension <strong>of</strong> n in an open neighbourhood <strong>of</strong> Σ. If Σ is a level surface <strong>of</strong> some scalar field<br />
t, such a natural extension is provided by the gradient <strong>of</strong> t, according to Eq. (2.30). Then the<br />
tensor fields ∇n <strong>and</strong> ∇n are well defined quantities. In particular, we can introduce the vector<br />
a := ∇nn. (2.72)<br />
Since n is a timelike unit vector, it can be regarded as the 4-velocity <strong>of</strong> some observer, <strong>and</strong> a<br />
is then the corresponding 4-acceleration. a is orthogonal to n <strong>and</strong> hence tangent to Σ, since<br />
n · a = n · ∇nn = 1/2∇n(n · n) = 1/2∇n(−1) = 0.<br />
Let us make explicit the definition <strong>of</strong> the tensor K extend to M by Eq. (2.70). From the<br />
definition <strong>of</strong> the operator γ ∗ M [Eq. (2.66)] <strong>and</strong> the original definition <strong>of</strong> K [Eq. (2.43)], we have<br />
∀(u,v) ∈ Tp(M) 2 , K(u,v) = K(γ(u), γ(v)) = −γ(u) · ∇ γ(v)n<br />
= −γ(u) · ∇ v+(n·v)n n<br />
= −[u + (n · u)n] · [∇vn + (n · v)∇nn]<br />
= −u · ∇vn − (n · v)u · ∇nn<br />
<br />
=a<br />
−(n · u)(n · v)n · ∇nn<br />
<br />
=0<br />
= −u · ∇vn − (a · u)(n · v),<br />
−(n · u)n · ∇vn<br />
<br />
=0<br />
= −∇n(u,v) − 〈a,u〉〈n,v〉, (2.73)<br />
where we have used the fact that n·n = −1 to set n·∇xn = 0 for any vector x. Since Eq. (2.73)<br />
is valid for any pair <strong>of</strong> vectors (u,v) in Tp(M), we conclude that<br />
In components:<br />
∇n = −K − a ⊗ n . (2.74)<br />
∇β nα = −Kαβ − aα nβ . (2.75)<br />
Notice that Eq. (2.74) implies that the (extended) extrinsic curvature tensor is nothing but the<br />
gradient <strong>of</strong> the 1-form n to which the projector operator γ ∗ is applied:<br />
K = −γ ∗ ∇n . (2.76)<br />
Remark : Whereas the bilinear form ∇n is a priori not symmetric, its projected part −K is<br />
a symmetric bilinear form.<br />
Taking the trace <strong>of</strong> Eq. (2.74) with respect to the metric g (i.e. contracting Eq. (2.75) with<br />
g αβ ) yields a simple relation between the divergence <strong>of</strong> the vector n <strong>and</strong> the trace <strong>of</strong> the extrinsic<br />
curvature tensor:<br />
K = −∇ · n . (2.77)