46
2 Bivector Formalism
with ∂/∂z here referring to partial differentiation with respect to z keeping r fixed.
The general solution of (2.201) is
H (r, z) =
1
2 e 2 r 2 − r 3 m(w)
r 4 + a 2 z 2
, with w =
z
r
,
(2.203)
and m is an arbitrary function of its argument. For a spherically symmetric field
asymptotically we must have m = constant and thus we have arrived at the Kerr–
Newman solution of the Einstein–Maxwell field equations:
ds
2
= −dx
2
− dy
2
− dz
2
+ dt
2
−
2 m r 3 − e 2 r 2
r 4 + a 2 z 2
(k i dX
i )
2 .
(2.204)
2.6
Using the Bianchi Identities
When the electromagnetic field (2.190) with (2.197) (and e = constant) is present
the Weyl conformal curvature tensor has the form
C abcd + i
∗ C abcd = G(r, z) (g abcd + i η abcd + 3 N ab N cd ) ,
(2.205)
with N ab given by the 2-form (2.183), dropping the hat for convenience from
now on. We shall now use the Bianchi identities, and the Einstein–Maxwell field
equations, to determine the complex valued function G in a way analogous to our
use of Maxwell’s equations to determine the complex valued function f (r, z) in
(2.190). In addition to (2.194) and (2.195) we shall require the formula
η abrs k
a k
r;s
= −
2 a r z
r 4 + a 2 z 2 k b .
(2.206)
We now find from (2.205) that
(C abcd + i
∗ C abcd )
;d k
a k
c
=
2 G ,d k
d
+
6 r
r 2 + i a z
G
k b .
(2.207)
With the Einstein–Maxwell field equations
R ab = −2 E ab ,
(2.208)