2.3 Bivectors and Gravitational Fields
37
Writing the Riemann tensor in terms of the Weyl tensor via (2.79) this equation
takes the form
C abcd
;d
=
1
2
(R ac;b − R bc;a ) +
1
12
(g bc R ,a − g ac R ,b ) .
(2.133)
In a vacuum space-time R ab = 0 and C abcd = R abcd and by (2.130) and (2.133) we
have
R abcd
;d
= 0 with R abcd = R abcd + i R
∗
abcd .
(2.134)
We note that in a vacuum space-time the left and right duals of the Riemann tensor
are equal. Now if R abcd is algebraically special with k a as degenerate principal null
direction then 0 = 1 = 0 and (2.112) reduces to
1
2
R abpq = − 2 N ab N pq + A ab L pq + L ab A pq ,
(2.135)
with
A ab = 2 M ab + 3 N ab +
1
2
4 L ab .
(2.136)
Substituting (2.135) into (2.134) and multiplying the result by L ps yields
− 2
,q N ab L
s
q − 2 N ab
;q L
s
q − 2 N ab L
ps N pq
;q
+ A ab L
ps L pq
;q
+ L ab
;q A pq L
ps
+ L ab A pq
;q L
ps
= 0 .
(2.137)
Multiplying this successively by L ab , N ab and M ab results in the equations:
2 L
ab N ab
;q L
s
q + 2 2 L
ps L pq
;q
= 0 ,
(2.138)
4 2
,q L
s
q + 4 2 L
ps N pq
;q
− 4 3 L
ps L pq
;q
+ N
ab L ab
;q A pq L
ps
= 0 ,
(2.139)
and
− 2 M
ab N ab
;q L
s
q − 4 L
ps L pq
;q
+ M
ab L ab
;q A pq L
ps
− 2 A pq
;q L
ps
= 0 .
(2.140)
The reader can check that
L
ab N ab
;q L
s
q = 4 L
ps L pq
;q ,
(2.141)