3.6 Energy-Momentum-Stress Tensor
65
and the asymptotic flux of angular momentum
S
ij
= lim
r→+∞
r
2
u 1
u 0
du
(T
ki X
j
− T
kj X
i ) r ,k sin θ dθ dφ
(3.91)
crossing r = constant → +∞ outwards in the direction of increasing r between
the future null cones u = u 0 and u = u 1 > u 0 (say). The polar angles θ, φ arise
from the parametrisation of (3.76):
X =
r 2 + a 2 sin θ cos φ , Y =
r 2 + a 2 sin θ sin φ , Z = r cos θ ,
(3.92)
with 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2 π. Detailed derivations of the 3-volume elements in
(3.90) and (3.91) can be found in [19]. We deduce from (3.90) and (3.91) that P i (u)
and S ij (u) satisfy
dP i
du
= lim
r→+∞
r
2
T
ij r ,j sin θ dθ dφ ,
(3.93)
and
dS ij
du
= lim
r→+∞
r
2
(T
ki X
j
− T
kj X
i ) r ,k sin θ dθ dφ ,
(3.94)
respectively. Denoting by ˆ
T ij the leading term in the expansion of T ij in inverse
powers of r which contributes to (3.93) and (3.94) we find that
8 π ˆ
T
ij
= −
2 ˙
m
r 2 −
3
r 3
d
du
(m a) ( ˆ
k
i ˆ
λ
j
+ ˆ
k
j ˆ
λ
i ) ,
(3.95)
with
ˆ
k
i
=
1,
X
r
,
Y
r
,
Z
r
and ˆ
λ
i
=
0,
Y
r
, −
X
r
, 0
.
(3.96)
The region of space-time in which (3.95) holds corresponds to large positive values
of r and in this region of space-time, in which we are evaluating the integrals (3.95)
and (3.96), we can take X = r sin θ cos φ , Y = r sin θ sin φ , Z = r cos θ and
thus r 2 = X 2 + Y 2 + Z 2 . We see from (3.78) that if ˆ
v i = δ i
0 we can write, for large
values of r,
r ,i = ˆ
v i − ˆ
k i .
(3.97)
We note the useful formulas:
ˆ
v
i
ˆ
v i = 1 , ˆ
k
i
ˆ
v i = 1 , ˆ
λ
i
ˆ
v i = 0 , ˆ
k
i ˆ
λ i = 0 .
(3.98)
65
and the asymptotic flux of angular momentum
S
ij
= lim
r→+∞
r
2
u 1
u 0
du
(T
ki X
j
− T
kj X
i ) r ,k sin θ dθ dφ
(3.91)
crossing r = constant → +∞ outwards in the direction of increasing r between
the future null cones u = u 0 and u = u 1 > u 0 (say). The polar angles θ, φ arise
from the parametrisation of (3.76):
X =
r 2 + a 2 sin θ cos φ , Y =
r 2 + a 2 sin θ sin φ , Z = r cos θ ,
(3.92)
with 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2 π. Detailed derivations of the 3-volume elements in
(3.90) and (3.91) can be found in [19]. We deduce from (3.90) and (3.91) that P i (u)
and S ij (u) satisfy
dP i
du
= lim
r→+∞
r
2
T
ij r ,j sin θ dθ dφ ,
(3.93)
and
dS ij
du
= lim
r→+∞
r
2
(T
ki X
j
− T
kj X
i ) r ,k sin θ dθ dφ ,
(3.94)
respectively. Denoting by ˆ
T ij the leading term in the expansion of T ij in inverse
powers of r which contributes to (3.93) and (3.94) we find that
8 π ˆ
T
ij
= −
2 ˙
m
r 2 −
3
r 3
d
du
(m a) ( ˆ
k
i ˆ
λ
j
+ ˆ
k
j ˆ
λ
i ) ,
(3.95)
with
ˆ
k
i
=
1,
X
r
,
Y
r
,
Z
r
and ˆ
λ
i
=
0,
Y
r
, −
X
r
, 0
.
(3.96)
The region of space-time in which (3.95) holds corresponds to large positive values
of r and in this region of space-time, in which we are evaluating the integrals (3.95)
and (3.96), we can take X = r sin θ cos φ , Y = r sin θ sin φ , Z = r cos θ and
thus r 2 = X 2 + Y 2 + Z 2 . We see from (3.78) that if ˆ
v i = δ i
0 we can write, for large
values of r,
r ,i = ˆ
v i − ˆ
k i .
(3.97)
We note the useful formulas:
ˆ
v
i
ˆ
v i = 1 , ˆ
k
i
ˆ
v i = 1 , ˆ
λ
i
ˆ
v i = 0 , ˆ
k
i ˆ
λ i = 0 .
(3.98)