36
2 Bivector Formalism
From (2.123) we can readily deduce that if k a is a simple principal null direction of
the Weyl tensor then 0 = 0 and it satisfies
k [f C a]bp[q k g] k
b k
p
= 0 ,
(2.125)
with the square brackets, as always, denoting skewsymmetrisation. If k a is a doubly
degenerate principal null direction then 0 = 1 = 0 and so k a satisfies
k [f C a]bpq k
b k
p
= 0 .
(2.126)
If k a is a triply degenerate principal null direction then 0 = 1 = 2 = 0 and,
from (2.124), we have
k [f C a]bpq k
b
= 0 .
(2.127)
If k a is a quadruply degenerate principal null direction then 0 = 1 = 2 =
3 = 0 and by (2.124) the vector k a satisfies
C abpq k
b
= 0 .
(2.128)
If any of (2.126)–(2.128) hold we say that the Weyl tensor is algebraically special.
This Weyl tensor algebra constitutes the so-called Petrov-Pirani [8, 9] classification
while the principal null directions are referred to as Debever-Penrose [10, 11]
directions. Finally the complex components (2.113)–(2.117) of the Weyl tensor on
the null tetrad are the Newman-Penrose [7] components of the Weyl tensor.
The Bianchi identities satisfied by the Riemann tensor read
R abcd;e + R abec;d + R abde;c = 0 .
(2.129)
An equivalent way of writing this is
η
f cde R abcd;e = 0 ⇔ R
∗
ab
f e ;e = 0 .
(2.130)
On multiplication by η rsab this reads
∗ R
∗rsf e ;e = 0 .
(2.131)
With ∗ R ∗ abcd given by (2.85) we can write (2.131) as
R abcd
;d
= R ac;b − R bc;a .
(2.132)