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

4.2 Coordinates adapted to the foliation 55
Figure 4.1: Coordinates (x i ) on the hypersurfaces Σt: each line x i = const cuts across the foliation (Σt)t∈R and
defines the time vector ∂t and the shift vector β of the spacetime coordinate system (x α ) = (t, x i ).
spacetime manifold M, which we have not done yet. Coordinates adapted to the foliation
(Σt)t∈R are set in the following way. On each hypersurface Σt one introduces some coordinate
system (x i ) = (x 1 ,x 2 ,x 3 ). If this coordinate system varies smoothly between neighbouring
hypersurfaces, then (x α ) = (t,x 1 ,x 2 ,x 3 ) constitutes a well-behaved coordinate system on M.
We shall call (x i ) = (x 1 ,x 2 ,x 3 ) the spatial coordinates.
Let us denote by (∂α) = (∂t,∂i) the natural basis of Tp(M) associated with the coordinates
(x α ), i.e. the set of vectors
∂t := ∂
∂t
(4.23)
∂i := ∂
∂xi, i ∈ {1,2,3}. (4.24)
Notice that the vector ∂t is tangent to the lines of constant spatial coordinates, i.e. the curves
of M defined by (x 1 = K 1 ,x 2 = K 2 ,x 3 = K 3 ), where K 1 , K 2 and K 3 are three constants (cf.
Fig. 4.1). We shall call ∂t the time vector.
Remark : ∂t is not necessarily a timelike vector. This will be discussed further below [Eqs. (4.33)-
(4.35)].
For any i ∈ {1,2,3}, the vector ∂i is tangent to the lines t = K 0 , x j = K j (j = i), where K 0
and K j (j = i) are three constants. Having t constant, these lines belong to the hypersurfaces
Σt. This implies that ∂i is tangent to Σt:
∂i ∈ Tp(Σt), i ∈ {1,2,3}. (4.25)