2.6 Using the Bianchi Identities
47
where E ab is the electromagnetic energy-momentum tensor
E ab =
1
2
g
cd
F ac ¯
F bd
=
1
2
e 2 r 4
(r 4 + a 2 z 2 ) 2 g
cd N ac ¯
N bd
=
1
2
e 2 r 4
(r 4 + a 2 z 2 ) 2 (η
cd N ac ¯
N bd − 2 H k a k b ) ,
(2.209)
with the bar denoting complex conjugation, we easily recover the second of (2.199).
The Bianchi identities read
C abcd
;d
= E cb;a − E ca;b ,
(2.210)
from which we have, using the second of (2.199),
C abcd
;d k
a k
c
=
2 e 2 r 7
(r 4 + a 2 z 2 ) 3 k b .
(2.211)
Next using (2.210) again we have
∗ C abcd
;d
= −η ab
rs E cr;s ,
(2.212)
from which we deduce, using (2.199) and (2.206), that
∗ C abcd
;d k
a k
c
= −
e 2 a r 5 z
(r 4 + a 2 z 2 ) 3 k b + η ab
rs k
a k
c ;s E cr .
(2.213)
Using (2.209) and the fact that
¯
N ab = −i
∗ ¯
N ab ,
(2.214)
We can write
η ab
rs k
a k
c ;s E cr =
e 2 r 4
2 (r 4 + a 2 z 2 ) 2 η ab
rs k
a k
c ;s (η
pq N pc ¯
N qr − 2 H k c k r )
= −
i e 2 r 4
2 (r 4 + a 2 z 2 ) 2 η ab
rs k
a k
c ;s η
pq N pc
∗ ¯
N qr
= −
i e 2 r 4
4 (r 4 + a 2 z 2 ) 2 η ab
rs η qrlm k
a k
c ;s η
pq N pc ¯
N
lm