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

118 Asymptotic flatness and global quantities
Using this property, as well as expression (7.75) of dVµ with the components nµ = (−N,0,0,0)
given by Eq. (4.38), we get
Vt
∇νA µν
dVµ = −
Vt
∂
∂xν √ µν
−gA nµ
√
γ
√ d
−g 3
x =
Vt
∂
∂xν √ 0ν
γNA d 3 x, (7.78)
where we have also invoked the relation (4.55) between the determinants of g and γ: √ −g =
N √ γ. Now, since A αβ is antisymmetric, A 00 = 0 and we can write ∂/∂x ν √ γNA 0ν =
∂/∂x i √ γ V i where V i = NA 0i are the components of the vector V ∈ T (Σt) defined by
V := −γ(n · A). The above integral then becomes
Vt
∇νA µν
dVµ =
Vt
1 ∂
√
γ ∂xi √ i
γV
√ 3
γ d x = DiV
Vt
i√ γ d 3 x. (7.79)
We can now use the Gauss-Ostrogradsky theorem to get
∇νA µν
dVµ = V i √ 2
si q d y. (7.80)
Vt
Noticing that ∂Vt = Ht ∪ St (cf. Fig. 7.2) and (from the antisymmetry of A µν )
we get the identity (7.74).
∂Vt
V i si = V ν sν = −nµA µν sν = 1
2 Aµν (sµnν − nµsν), (7.81)
Remark : Equation (7.74) can also be derived by applying Stokes’ theorem to the 2-form
4 ǫαβµνA µν , where 4 ǫαβµν is the Levi-Civita alternating tensor (volume element) associated
with the spacetime metric g (see e.g. derivation of Eq. (11.2.10) in Wald’s book [265]).
where
Applying formula (7.74) to A µν = ∇ µ k ν we get, in view of the definition (7.71),
MK = − 1
∇ν∇
4π Vt
µ k ν dVµ + M H K , (7.82)
M H K := 1
∇
8π Ht
µ k ν dS H µν
will be called the Komar mass of the hole. Now, from the Ricci identity
(7.83)
∇ν∇ µ k ν − ∇ µ ∇ ν k ν
=
=0
4 R µ νk ν , (7.84)
where the “= 0” is a consequence of Killing’s equation (7.73). Equation (7.82) becomes then
MK = − 1
4 µ
R
4π
νk
Vt
ν dVµ + M H K = 1
4
Rµνk
4π Vt
ν n µ √ γ d 3 x + M H K . (7.85)