4.5 Bateman Gravitational Waves
91
+( ˆ
q
3
+ i ˆ
w
3 )(dz ∧ dt − i dx ∧ dz)
,
(4.168)
+ 13 = i|q |
2 ( ˆ
q
2
+ i ˆ
w
2 )
( ˆ
q
1
+ i ˆ
w
2 )(dx ∧ dt − i dy ∧ dz)
+( ˆ
q
2
+ i ˆ
w
2 )(dy ∧ dt + i dx ∧ dz)
+( ˆ
q
3
+ i ˆ
w
3 )(dz ∧ dt − i dx ∧ dz)
,
(4.169)
+ 23 = −i|q |
2 ( ˆ
q
1
+ i ˆ
w
1 )
( ˆ
q
1
+ i ˆ
w
2 )(dx ∧ dt − i dy ∧ dz)
+( ˆ
q
2
+ i ˆ
w
2 )(dy ∧ dt + i dx ∧ dz)
+( ˆ
q
3
+ i ˆ
w
3 )(dz ∧ dt − i dx ∧ dz)
.
(4.170)
Next we make the coordinate transformations used already in the electromagnetic
case:
ζ = x − i y , ¯
ζ = x + i y , u = −t − z , v = t − z .
(4.171)
At the same time it is useful to define
ˆ
k 1 − i ˆ
k 2
1 − ˆ
k 3
= Y ,
(4.172)
from which it follows that
ˆ
q 3 + i ˆ
w 3
ˆ
q 1 + i ˆ
w 1 + i( ˆ
q 2 + i ˆ
w 2 )
= Y and
ˆ
q 1 + i ˆ
w 1 − i ( ˆ
q 2 + i ˆ
w 2 )
ˆ
q 1 + i ˆ
w 1 + i ( ˆ
q 2 + i ˆ
w 2 )
= −Y
2 .
(4.173)
Finally putting
|q |
2 ( ˆ
q
3
+ i ˆ
w
3 )
2
= Y g(ζ, ¯
ζ , u, v) ,
(4.174)
for some complex valued function g, we can write + ij as
+ 12 = −
i
2
g (du + Y d ¯
ζ ) ∧ (dζ − Y dv) ,
(4.175)
+ 13 =
1
4
(Y
−1
+ Y) g (du + Y d ¯
ζ ) ∧ (dζ − Y dv) ,
(4.176)