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

7.6 Komar mass and angular momentum 121
Hence at the Newtonian limit, the Komar mass reduces to the standard total mass. This, along
with the result (7.94) for Schwarzschild spacetime, justifies the name Komar mass.
A natural question which arises then is how does the Komar mass relate to the ADM mass
of Σt ? The answer is not obvious if one compares the defining formulæ (7.13) and (7.71). It
is even not obvious if one compares the 3+1 expressions (7.45) and (7.91): Eq. (7.45) involves
the flux of the gradient of the conformal factor Ψ of the 3-metric, whereas Eq. (7.91) involves
the flux of the gradient of the lapse function N. Moreover, in Eq. (7.45) the integral must be
evaluated at spatial infinity, whereas in Eq. (7.45) it can be evaluated at any finite distance
(outside the matter sources). The answer has been obtained in 1978 by Beig [47], as well as by
Ashtekar and Magnon-Ashtekar the year after [26]: for any foliation (Σt)t∈R whose unit normal
vector n coincides with the timelike Killing vector k at spatial infinity [i.e. N → 1 and β → 0
in Eq. (7.89)],
MK = MADM . (7.99)
Remark : In the quasi-isotropic gauge, we have obtained a volume expression of the ADM
mass, Eq. (7.69), that we may compare to the volume expression (7.96) of the Komar
mass. Even when there is no hole, the two expressions are pretty different. In particular,
the Komar mass integral has a compact support (the matter domain), whereas the ADM
mass integral has not.
7.6.3 Komar angular momentum
If the spacetime (M,g) is axisymmetric, its Komar angular momentum is defined by a
surface integral similar that of the Komar mass, Eq. (7.71), but with the Killing vector k
replaced by the Killing vector φ associated with the axisymmetry:
JK := 1
∇
16π St
µ φ ν dSµν . (7.100)
Notice a factor −2 of difference with respect to formula (7.71) (the so-called Komar’s anomalous
factor [165]).
For the same reason as for MK, JK is actually independent of the surface St as long as the
latter is outside all the possible matter sources and JK can be expressed by a volume integral
over the matter by a formula similar to (7.87) (except for the factor −2):
with
JK = −
Σt
T(n,φ) − 1
√γ 3 H
T n · φ d x + JK , (7.101)
2
J H K := − 1
∇
16π Ht
µ φ ν dS H µν. (7.102)
Let us now establish the 3+1 expression of the Komar angular momentum. It is natural to
choose a foliation adapted to the axisymmetry in the sense that the Killing vector φ is tangent