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

3.3.5 Evolution of the 3-metric
3.3 Foliation kinematics 45
The evolution of Σt’s metric γ is naturally given by the Lie derivative of γ along the normal
evolution vector m (see Appendix A). By means of Eqs. (A.8) and (3.22), we get
Lmγαβ = m µ ∇µγαβ + γµβ∇αm µ + γαµ∇βm µ
= Nn µ ∇µ(nαnβ) − γµβ (NK µ α + D µ N nα − n µ ∇αN)
−γαµ
= N( n µ ∇µnα
aα
Hence the simple result:
NK µ
β + Dµ N nβ − n µ
∇βN
=N −1 DαN
nβ + nα n µ ∇µnβ
aβ
=N−1Dβ N
) − NKβα − DβN nα − NKαβ − DαN nβ
= −2NKαβ. (3.23)
Lmγ = −2NK . (3.24)
One can deduce easily from this relation the value of the Lie derivative of the 3-metric along
the unit normal n. Indeed, since m = Nn,
Hence
Lmγαβ = LNnγαβ
= Nn µ ∇µγαβ + γµβ∇α(Nn µ ) + γαµ∇β(Nn µ )
= Nn µ ∇µγαβ + γµβn µ
∇αN + Nγµβ∇αn µ + γαµn µ
=0
=0
∇βN + Nγαµ∇βn µ
= NLn γαβ. (3.25)
Consequently, Eq. (3.24) leads to
Ln γ = 1
N Lmγ. (3.26)
K = − 1
2 Ln γ . (3.27)
This equation sheds some new light on the extrinsic curvature tensor K. In addition to being
the projection on Σt of the gradient of the unit normal to Σt [cf. Eq. (2.76)],
K = −γ ∗ ∇n, (3.28)
as well as the measure of the difference between D-derivatives and ∇-derivatives for vectors
tangent to Σt [cf. Eq. (2.83)],
∀(u,v) ∈ T (Σ) 2 , K(u,v)n = Duv − ∇uv, (3.29)
K is also minus one half the Lie derivative of Σt’s metric along the unit timelike normal.