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

24 Geometry of hypersurfaces
The eigenvalues of the Weingarten map, which are all real numbers since χ is self-adjoint,
are called the principal curvatures of the hypersurface Σ and the corresponding eigenvectors
define the so-called principal directions of Σ. The mean curvature of the hypersurface Σ
is the arithmetic mean of the principal curvature:
where the κi are the three eigenvalues of χ.
H := 1
3 (κ1 + κ2 + κ3) (2.41)
Remark : The curvatures defined above are not to be confused with the Gaussian curvature
introduced in Sec. 2.3.3. The latter is an intrinsic quantity, independent of the way the
manifold (Σ,γ) is embedded in (M,g). On the contrary the principal curvatures and mean
curvature depend on the embedding. For this reason, they are qualified of extrinsic.
The self-adjointness of χ implies that the bilinear form defined on Σ’s tangent space by
K : Tp(Σ) × Tp(Σ) −→ R
(u,v) ↦−→ −u · χ(v)
(2.42)
is symmetric. It is called the second fundamental form of the hypersurface Σ. It is also called
the extrinsic curvature tensor of Σ (cf. the remark above regarding the qualifier ’extrinsic’).
K contains the same information as the Weingarten map.
Remark : The minus sign in the definition (2.42) is chosen so that K agrees with the convention
used in the numerical relativity community, as well as in the MTW book [189].
Some other authors (e.g. Carroll [79], Poisson [205], Wald [265]) choose the opposite
convention.
If we make explicit the value of χ in the definition (2.42), we get [see Eq. (2.7)]
∀(u,v) ∈ Tp(Σ) × Tp(Σ), K(u,v) = −u · ∇vn . (2.43)
We shall denote by K the trace of the bilinear form K with respect to the metric γ; it is the
opposite of the trace of the endomorphism χ and is equal to −3 times the mean curvature of Σ:
K := γ ij Kij = −3H. (2.44)
2.3.5 Examples: surfaces embedded in the Euclidean space R 3
Let us illustrate the previous definitions with some hypersurfaces of a space which we are very
familiar with, namely R 3 endowed with the standard Euclidean metric. In this case, the dimension
is reduced by one unit with respect to the spacetime M and the ambient metric g is
Riemannian (signature (+,+,+)) instead of Lorentzian. The hypersurfaces are 2-dimensional
submanifolds of R 3 , namely they are surfaces by the ordinary meaning of this word.
In this section, and in this section only, we change our index convention to take into account
that the base manifold is of dimension 3 and not 4: until the next section, the Greek indices run
in {1,2,3} and the Latin indices run in {1,2}.