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

116 Asymptotic flatness and global quantities
For the IWM approximation of general relativity considered in Sec. 6.6, the coordinates
belong to the quasi-isotropic gauge (since ˜γ = f), so we may apply (7.69). Moreover, as a
consequence of ˜γ = f, ˜ R = 0 and in the IWM approximation, K = 0. Therefore Eq. (7.69)
simplifies to
MADM =
Σt
Ψ 5 E + 1
16π Âij Âij Ψ −7
˜γ
3
d x + MH. (7.70)
Within the framework of exact general relativity, the above formula is valid for any maximal
slice Σt with a conformally flat metric.
7.6 Komar mass and angular momentum
In the case where the spacetime (M,g) has some symmetries, one may define global quantities
in a coordinate-independent way by means of a general technique introduced by Komar (1959)
[172]. It consists in taking flux integrals of the derivative of the Killing vector associated with the
symmetry over closed 2-surfaces surrounding the matter sources. The quantities thus obtained
are conserved in the sense that they do not depend upon the choice of the integration 2-surface,
as long as the latter stays outside the matter. We discuss here two important cases: the Komar
mass resulting from time symmetry (stationarity) and the Komar angular momentum resulting
from axisymmetry.
7.6.1 Komar mass
Let us assume that the spacetime (M,g) is stationary. This means that the metric tensor g is
invariant by Lie transport along the field lines of a timelike vector field k. The latter is called a
Killing vector. Provided that it is normalized so that k · k = −1 at spatial infinity, it is then
unique. Given a 3+1 foliation (Σt)t∈R of M, and a closed 2-surface St in Σt, with the topology
of a sphere, the Komar mass is defined by
with the 2-surface element
MK := − 1
∇
8π St
µ k ν dSµν , (7.71)
dSµν = (sµnν − nµsν) √ q d 2 y, (7.72)
where n is the unit timelike normal to Σt, s is the unit normal to St within Σt oriented towards
the exterior of St, (y a ) = (y 1 ,y 2 ) are coordinates spanning St, and q := det(qab), the qab’s being
the components with respect to (y a ) of the metric q induced by γ (or equivalently by g) on St.
Actually the Komar mass can be defined over any closed 2-surface, but in the present context
it is quite natural to consider only 2-surfaces lying in the hypersurfaces of the 3+1 foliation.
A priori the quantity MK as defined by (7.71) should depend on the choice of the 2-surface St.
However, thanks to the fact that k is a Killing vector, this is not the case, as long as St is located
outside any matter content of spacetime. In order to show this, let us transform the surface
integral (7.71) into a volume integral. As in Sec. 7.5.3, we suppose that Σt is diffeomorphic to
either R 3 or R 3 minus one hole, the results being easily generalized to an arbitrary number of