44
2 Bivector Formalism
In this sense ˆ
N ij is an eigenbivector of the Kerr Riemann tensor. We note in passing
that, in coordinates X i ,
∗ ˆ
N ij =
1
2
ij kl ˆ
N
kl
= −i ˆ
N ij .
(2.189)
2.5
Passage to Charged Kerr Space-Time
We can make use of the Kerr eigenbivector ˆ
N ij to obtain the charged Kerr
solution of the vacuum Einstein–Maxwell field equations [15, 16] assuming that
the metric tensor in the charged case has the Kerr–Schild form (2.174) with H
to be determined. This can be achieved without using all of the field equations
but with the addition of a boundary condition which states that asymptotically
(for r → +∞) the electromagnetic field and the gravitational field should both
be spherically symmetric. Starting with the electromagnetic case, in coordinates
X i = (x, y, z, t), we look for a Maxwell field in the form
F
ij
= F
ij
+ i
∗ F
ij
= f (r, z) ˆ
N
ij ,
(2.190)
from which, using (2.185), we have
F
ij k j = f k
i .
(2.191)
Using
F
ij ;i k j = 0 ,
(2.192)
which follows from Maxwell’s equations (F ij ;i = 0) but is weaker than them, we
arrive at
f ,i k
i
+ f k
i ;i = f ˆ
N
ij k j ;i =
1
2
ˆ
N
ij (k j,i − k i,j ) .
(2.193)
With k i given via (2.151) and ˆ
N ij given via (2.183) we find that
k
i ;i = k
i
,i =
2 r 3
r 4 + a 2 z 2 ,
(2.194)
and
ˆ
N
ij k j ;i =
2 i a r z
r 4 + a 2 z 2 .
(2.195)