2.4 The Kerr Space-Time
43
with
A = z x + i
{r a x(r 2 − z 2 ) − y (r 4 + a 2 z 2 )}
r (r + a 2 )
,
(2.180)
B = z y + i
{r a y (r 2 − z 2 ) + x (r 4 + a 2 z 2 )}
r (r 2 + a 2 )
,
(2.181)
C =
(r 2 − z 2 )(a z − r 2 )
r 2
.
(2.182)
Using (2.177)–(2.182) in (2.176) results in the following simplified expression for
ˆ
N:
ˆ
N = −
r
r 2 + i a z
x (dx ∧ dt − i dy ∧ dz) + y (dy ∧ dt − i dz ∧ dx)
+(z + i a) (dz ∧ dt − i dx ∧ dy)
=
1
2
ˆ
N ij dX
i
∧ dX
j .
(2.183)
Hence we can write (2.173) in coordinates X i = (x, y, z, t) as
R ij kl + i
∗ R ij kl =
m r 3
(r 2 + i a z) 3 (g ij kl + i η ij kl + 3 ˆ
N ij ˆ
N kl ) ,
(2.184)
with ˆ
N ij given via the 2-form (2.183). We will refer to ˆ
N ij as the components of
the Kerr complex bivector field. We note that with k i given by (2.151), and since
g ij = η ij − 2 H k i k j with k i = η ij k j we have k i = g ij k j and
ˆ
N ij k
j
= k i .
(2.185)
It thus follows that
ˆ
N
ij
= g
ik g
jl ˆ
N kl = η
ik η
jl ˆ
N kl ,
(2.186)
and therefore
ˆ
N ij ˆ
N
ij
= −4 .
(2.187)
Hence from (2.184) we see that
(R ij kl + i
∗ R ij kl ) ˆ
N
kl
= −
4 m r 3
(r 2 + i a z) 3 g ij kl ˆ
N
kl .
(2.188)