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

28 Geometry of hypersurfaces
The non vanishing of the Riemann tensor is reflected by the well-known property that the
sum of angles of any triangle drawn at the surface of a sphere is larger than π (cf. Fig. 2.4).
The unit vector n normal to Σ (and oriented towards the exterior of the sphere) has the
following components with respect to the coordinates (X α ) = (x,y,z):
n α
x
=
x2 + y2 + z2 ,
y
x 2 + y 2 + z 2 ,
It is then easy to compute ∇βn α = ∂n α /∂X β to get
∇βn α = (x 2 + y 2 + z 2 ) −3/2
⎛
⎝
z
x2 + y2 + z2 . (2.54)
y 2 + z 2 −xy −xz
−xy x 2 + z 2 −yz
−xz −yz x 2 + y 2
The natural basis associated with the coordinates (x i ) = (θ,ϕ) on Σ is
∂θ = (x 2 + y 2 ) −1/2 xz ∂x + yz ∂y − (x 2 + y 2
)∂z
⎞
⎠. (2.55)
(2.56)
∂ϕ = −y ∂x + x∂y. (2.57)
The components of the extrinsic curvature tensor in this basis are obtained from Kij =
K(∂i,∂j) = −∇βnα (∂i) α (∂j) β . We get
Kij =
Kθθ Kθϕ
Kϕθ Kϕϕ
=
The trace of K with respect to γ is then
−R 0
0 −Rsin 2 θ
= − 1
R γij. (2.58)
K = − 2
. (2.59)
R
With these examples, we have encountered hypersurfaces with intrinsic and extrinsic curvature
both vanishing (the plane), the intrinsic curvature vanishing but not the extrinsic one (the
cylinder), and with both curvatures non vanishing (the sphere). As we shall see in Sec. 2.5, the
extrinsic curvature is not fully independent from the intrinsic one: they are related by the Gauss
equation.
2.4 Spacelike hypersurface
From now on, we focus on spacelike hypersurfaces, i.e. hypersurfaces Σ such that the induced
metric γ is definite positive (Riemannian), or equivalently such that the unit normal vector n
is timelike (cf. Secs. 2.3.1 and 2.3.2).