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

56 3+1 decomposition of Einstein equation
4.2.2 Shift vector
The dual basis associated with (∂α) is the gradient 1-form basis (dxα ), which is a basis of the
space of linear forms T ∗
p (M):
〈dx α ,∂β〉 = δ α β . (4.26)
In particular, the 1-form dt is dual to the vector ∂t:
〈dt,∂t〉 = 1. (4.27)
Hence the time vector ∂t obeys to the same property as the normal evolution vector m, since
〈dt,m〉 = 1 [Eq. (3.11)]. In particular, ∂t Lie drags the hypersurfaces Σt, as m does (cf.
Sec. 3.3.2). In general the two vectors ∂t and m differ. They coincide only if the coordinates
(x i ) are such that the lines x i = const are orthogonal to the hypersurfaces Σt (cf. Fig. 4.1). The
difference between ∂t and m is called the shift vector and is denoted β:
∂t =: m + β . (4.28)
As for the lapse, the name shift vector has been coined by Wheeler (1964) [267]. By combining
Eqs. (4.27) and (3.11), we get
or equivalently, since dt = −N −1 n [Eq. (3.7)],
〈dt,β〉 = 〈dt,∂t〉 − 〈dt,m〉 = 1 − 1 = 0, (4.29)
n · β = 0 . (4.30)
Hence the vector β is tangent to the hypersurfaces Σt.
The lapse function and the shift vector have been introduced for the first time explicitly,
although without their present names, by Y. Choquet-Bruhat in 1956 [128].
It usefull to rewrite Eq. (4.28) by means of the relation m = Nn [Eq. (3.8)]:
∂t = Nn + β . (4.31)
Since the vector n is normal to Σt and β tangent to Σt, Eq. (4.31) can be seen as a 3+1
decomposition of the time vector ∂t.
The scalar square of ∂t is deduced immediately from Eq. (4.31), taking into account n·n = −1
and Eq. (4.30):
∂t · ∂t = −N 2 + β · β. (4.32)
Hence we have the following:
∂t is timelike ⇐⇒ β · β < N 2 , (4.33)
∂t is null ⇐⇒ β · β = N 2 , (4.34)
∂t is spacelike ⇐⇒ β · β > N 2 . (4.35)