2.3 Bivectors and Gravitational Fields
35
3 = −
1
2
β 3 = C abcd k
a l
b
¯
m
c l
d ,
(2.116)
4 = −
1
2
β 1 = C abcd ¯
m
a l
b
¯
m
c l
d .
(2.117)
The (in general) five complex components of the Weyl tensor on the null tetrad
k a , l a , m a , ¯
m a are listed in (2.113)–(2.117). These are generally referred to as the
Newman-Penrose [7] components of the Weyl tensor. If we transform the tetrad
by the null rotation given in (1.50) of the previous chapter then these components
transform to ˆ
0 , ˆ
1 , ˆ
2 , ˆ
3 and ˆ
4 related to (2.113)–(2.117) by
ˆ
0 = 0 − 4 1 β − 6 2 β
2
+ 4 3 β
3
+ 4 β
4 ,
(2.118)
ˆ
1 = 1 + 3 2 β − 3 3 β
2
− 4 β
3 ,
(2.119)
ˆ
2 = 2 − 2 3 β − 4 β
2 ,
(2.120)
ˆ
3 = 3 + 4 β ,
(2.121)
ˆ
4 = 4 .
(2.122)
This sequence of equations has the interesting property that the derivative of ˆ
0 with
respect to β is a constant multiple of ˆ
1 , the derivative of ˆ
1 with respect to β is a
constant multiple of ˆ
2 , the derivative of ˆ
2 with respect to β is a constant multiple
of ˆ
3 and the derivative of ˆ
3 with respect to β is ˆ
4 . We can make immediate use
of this fact. We first assume that 4 = 0. If this is not the case then a null rotation
(2.51) of the previous chapter will make it so. Now ˆ
0 = 0 is a quartic equation for
β over the field of complex numbers and therefore, by the fundamental theorem of
algebra, has at most four (complex) roots. Thus there are in general four null vectors
ˆ
k a , given by (2.50) of the previous chapter, for which ˆ
0 = 0. These are principal
null directions of the Weyl tensor. On account of the properties just mentioned of
the sequence (2.118)–(2.122), it follows that if two roots of ˆ
0 = 0 coincide then
the coincident root is a solution of the equation ˆ
1 = 0. If three roots of ˆ
0 = 0
coincide then the coincident root is a solution of the equations ˆ
1 = ˆ
2 = 0 and if
the four roots of ˆ
0 = 0 coincide then this root is a solution of ˆ
1 = ˆ
2 = ˆ
3 = 0.
Since L ab k b = 0, M ab k b = ¯
m a , N ab k b = k a we have from (2.112) the equations
1
2
C abpq k
b k
p
= − 0 ¯
m a ¯
m q − 1 ( ¯
m a k q + k a ¯
m q ) + 2 k a k q ,
(2.123)
and
1
2
C abpq k
b
= 0 ¯
m a M pq + 1 ( ¯
m a N pq + k a M pq ) + 2 (−k a N pq + ¯
m a L pq )
+ 3 k a L pq .
(2.124)
35
3 = −
1
2
β 3 = C abcd k
a l
b
¯
m
c l
d ,
(2.116)
4 = −
1
2
β 1 = C abcd ¯
m
a l
b
¯
m
c l
d .
(2.117)
The (in general) five complex components of the Weyl tensor on the null tetrad
k a , l a , m a , ¯
m a are listed in (2.113)–(2.117). These are generally referred to as the
Newman-Penrose [7] components of the Weyl tensor. If we transform the tetrad
by the null rotation given in (1.50) of the previous chapter then these components
transform to ˆ
0 , ˆ
1 , ˆ
2 , ˆ
3 and ˆ
4 related to (2.113)–(2.117) by
ˆ
0 = 0 − 4 1 β − 6 2 β
2
+ 4 3 β
3
+ 4 β
4 ,
(2.118)
ˆ
1 = 1 + 3 2 β − 3 3 β
2
− 4 β
3 ,
(2.119)
ˆ
2 = 2 − 2 3 β − 4 β
2 ,
(2.120)
ˆ
3 = 3 + 4 β ,
(2.121)
ˆ
4 = 4 .
(2.122)
This sequence of equations has the interesting property that the derivative of ˆ
0 with
respect to β is a constant multiple of ˆ
1 , the derivative of ˆ
1 with respect to β is a
constant multiple of ˆ
2 , the derivative of ˆ
2 with respect to β is a constant multiple
of ˆ
3 and the derivative of ˆ
3 with respect to β is ˆ
4 . We can make immediate use
of this fact. We first assume that 4 = 0. If this is not the case then a null rotation
(2.51) of the previous chapter will make it so. Now ˆ
0 = 0 is a quartic equation for
β over the field of complex numbers and therefore, by the fundamental theorem of
algebra, has at most four (complex) roots. Thus there are in general four null vectors
ˆ
k a , given by (2.50) of the previous chapter, for which ˆ
0 = 0. These are principal
null directions of the Weyl tensor. On account of the properties just mentioned of
the sequence (2.118)–(2.122), it follows that if two roots of ˆ
0 = 0 coincide then
the coincident root is a solution of the equation ˆ
1 = 0. If three roots of ˆ
0 = 0
coincide then the coincident root is a solution of the equations ˆ
1 = ˆ
2 = 0 and if
the four roots of ˆ
0 = 0 coincide then this root is a solution of ˆ
1 = ˆ
2 = ˆ
3 = 0.
Since L ab k b = 0, M ab k b = ¯
m a , N ab k b = k a we have from (2.112) the equations
1
2
C abpq k
b k
p
= − 0 ¯
m a ¯
m q − 1 ( ¯
m a k q + k a ¯
m q ) + 2 k a k q ,
(2.123)
and
1
2
C abpq k
b
= 0 ¯
m a M pq + 1 ( ¯
m a N pq + k a M pq ) + 2 (−k a N pq + ¯
m a L pq )
+ 3 k a L pq .
(2.124)