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

26 Geometry of hypersurfaces
Figure 2.3: Cylinder Σ as a hypersurface of the Euclidean space R 3 . Notice that the unit normal vector n stays
constant when z varies at fixed ϕ, whereas its direction changes as ϕ varies at fixed z. Consequently the extrinsic
curvature of Σ vanishes in the z direction, but is non zero in the ϕ direction. Besides, the sum of angles of any
triangle lying in Σ is α + β + γ = π, which shows that the intrinsic curvature of (Σ, γ) is identically zero.
which exhibits the standard Cartesian shape.
To evaluate the extrinsic curvature of Σ, let us consider the unit normal n to Σ. Its
components with respect to the Cartesian coordinates (X α ) = (x,y,z) are
n α
x
=
x2 + y2 ,
It is then easy to compute ∇βn α = ∂n α /∂X β . We get
∇βn α = (x 2 + y 2 ) −3/2
y
, 0
x2 + y2 . (2.47)
⎛
⎝
y 2 −xy 0
−xy x 2 0
0 0 0
⎞
⎠ . (2.48)
From Eq. (2.43), the components of the extrinsic curvature K with respect to the basis
(x i ) = (ϕ,z) are
Kij = K(∂i,∂j) = −∇βnα (∂i) α (∂j) β , (2.49)
where (∂i) = (∂ϕ,∂z) = (∂/∂ϕ,∂/∂z) denotes the natural basis associated with the coordinates
(ϕ,z) and (∂i) α the components of the vector ∂i with respect to the natural basis
(∂α) = (∂x,∂y,∂z) associated with the Cartesian coordinates (X α ) = (x,y,z). Specifically,