42
2 Bivector Formalism
to rewrite (2.171) in the form
R abcd + i
∗ R abcd =
m
(r + i a cos θ) 3 (g abcd + i η abcd + 3 ˆ
N ab ˆ
N cd ) .
(2.173)
Finally we want to express this in terms of the coordinates X i = (x, y, z, t) in
which the metric tensor components, given via the line element (2.154), have the
Kerr–Schild form
g ij = η ij + 2 H k i k j with H = −
m r 3
r 4 + a 2 z 2 ,
(2.174)
with k i given by (2.151) and η ij = diag(1, −1, −1, −1). We note that for this form
of metric det(g ij ) = det(η ij ) = −1 (see Appendix A) and so η ij kl = ij kl . Under
the tetrad transformation (2.163) the bivector N ab transforms as
ˆ
N ab = N ab − 2 ¯
α L ab ,
(2.175)
following from the definition of the bivector basis. Hence we can write the 2-form
ˆ
N =
1
2
ˆ
N ab dx
a
∧ dx
b ,
= i (r
2
+ a
2 cos
2 θ) sin θ dθ ∧ dφ + (du + a sin
2 θ dφ) ∧ (dr − a sin
2 θ dφ)
+
√
2 ¯
α (r + i a cos θ) (du + a sin
2 θ dφ) ∧ (dθ + i sin θ dφ) ,
(2.176)
and, with α given by (2.170),
√
2 ¯
α (r + i a cos θ) (dθ + i sin θ dφ) = −i a sin θ (dθ + i sin θ dφ) .
(2.177)
Making the transformation to coordinates X i = (x, y, z, t) given by (2.148), and
using (2.150) and (2.151), we find the intermediate results:
(r
2
+ a
2 cos
2 θ) sin θ dθ ∧ dφ =
a x − r y
r 2 + a 2
dx ∧ dz +
a y + r x
r 2 + a 2
dy ∧ dz
+
z
r
dx ∧ dy ,
(2.178)
and
sin θ (dθ + i sin θ dφ) =
r
r 4 + a 2 z 2 (A dx + B dy + C dz) ,
(2.179)