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

3.3 Foliation kinematics 41
Figure 3.2: The point p ′ deduced from p ∈ Σt by the displacement δtm belongs to Σt+δt, i.e. the hypersurface
Σt is transformed to Σt+δt by the vector field δtm (Lie dragging).
3.3 Foliation kinematics
3.3.1 Lapse function
As already noticed in Sec. 2.3.2, the timelike and future-directed unit vector n normal to the
slice Σt is necessarily collinear to the vector ∇t associated with the gradient 1-form dt. Hence
we may write
with
N :=
n := −N ∇t (3.4)
− ∇t · −1/2
∇t = −〈dt, −1/2 ∇t〉 . (3.5)
The minus sign in (3.4) is chosen so that the vector n is future-oriented if the scalar field t
is increasing towards the future. Notice that the value of N ensures that n is a unit vector:
n · n = −1. The scalar field N hence defined is called the lapse function. The name lapse has
been coined by Wheeler in 1964 [267].
Remark : In most of the numerical relativity literature, the lapse function is denoted α instead
of N. We follow here the ADM [23] and MTW [189] notation.
Notice that by construction [Eq. (3.5)],
N > 0. (3.6)
In particular, the lapse function never vanishes for a regular foliation. Equation (3.4) also
says that −N is the proportionality factor between the gradient 1-form dt and the 1-form n
associated to the vector n by the metric duality:
n = −N dt . (3.7)