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

18 Geometry of hypersurfaces
2.2.3 Curvature tensor
We follow the MTW convention [189] and define the Riemann curvature tensor of the
spacetime connection ∇ by 2
4 Riem : T ∗ (M) × T (M) 3 −→ C ∞ (M, R)
(ω,w,u,v) ↦−→
ω, ∇u∇vw − ∇v∇uw
−∇ [u,v]w ,
(2.12)
where C∞ (M, R) denotes the space of smooth scalar fields on M. As it is well known, the
above formula does define a tensor field on M, i.e. the value of 4Riem(ω,w,u,v) at a given
point p ∈ M depends only upon the values of the fields ω, w, u and v at p and not upon their
behaviors away from p, as the gradients in Eq. (2.12) might suggest. We denote the components
of this tensor in a given basis (eα), not by 4Riem γ
δαβ , but by 4R γ
δαβ . The definition (2.12) leads
then to the following writing (called Ricci identity):
∀w ∈ T (M), (∇α∇β − ∇β∇α) w γ = 4 R γ
µαβ wµ , (2.13)
From the definition (2.12), the Riemann tensor is clearly antisymmetric with respect to its last
two arguments (u,v). The fact that the connection ∇ is associated with a metric (i.e. g) implies
the additional well-known antisymmetry:
∀(ω,w) ∈ T ∗ (M) × T (M), 4 Riem(ω,w, ·, ·) = − 4 Riem(w, ω, ·, ·). (2.14)
In addition, the Riemann tensor satisfies the cyclic property
∀(u,v,w) ∈ T (M) 3 ,
4 Riem(·,u,v,w) + 4 Riem(·,w,u,v) + 4 Riem(·,v,w,u) = 0 . (2.15)
The Ricci tensor of the spacetime connection ∇ is the bilinear form 4 R defined by
4 R : T (M) × T (M) −→ C ∞ (M, R)
(u,v) ↦−→ 4 Riem(e µ ,u,eµ,v).
(2.16)
This definition is independent of the choice of the basis (eα) and its dual counterpart (e α ).
Moreover the bilinear form 4 R is symmetric. In terms of components:
4
Rαβ = 4 R µ
αµβ . (2.17)
Note that, following the standard usage, we are denoting the components of both the Riemann
and Ricci tensors by the same letter R, the number of indices allowing to distinguish between
the two tensors. On the contrary we are using different symbols, 4 Riem and 4 R, when dealing
with the ‘intrinsic’ notation.
2 the superscript ‘4’ stands for the four dimensions of M and is used to distinguish from Riemann tensors that
will be defined on submanifolds of M