4.6 Some ‘Spherical’ Gravitational Waves
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4.6
Some ‘Spherical’ Gravitational Waves
We now set out to construct the gravitational analogue of the spherical waves
described in electromagnetic theory. The propagation direction in R 3 is
ˆ
k =
x
r
,
y
r
,
z
r
with r =
x 2 + y 2 + z 2 .
(4.190)
The function Y, which plays a key role in the construction of the Bateman waves, is
Y =
ˆ
k 1 − i ˆ
k 2
1 − ˆ
k 3
=
x − i y
r − z
,
(4.191)
and the functions X 1 , X 2 are given by
X 1 = r − t , X 2 =
x − i y
r − z
(r − t) ,
(4.192)
resulting in
G(X 1 , X 2 ) dX 1 ∧ dX 2 = ˆ
G
r − t,
x − i y
r − z
(dr − dt) ∧ d
x − i y
r − z
,
(4.193)
for some function ˆ
G of its arguments. For arguably the simplest ‘spherical’ waves
we take
ˆ
G = A(r − t) + i C(r − t) ,
(4.194)
where A, C are arbitrary real-valued functions of r − t. Now the 2-forms (4.184)–
(4.189) read
+ 12 = −
i
2
(A + i C) (dr − dt) ∧ d
x − i y
r − z
,
(4.195)
+ 13 =
1
4
1
Y
+ Y
(A + i C) (dr − dt) ∧ d
x − i y
r − z
,
(4.196)
+ 23 = −
i
4
1
Y
− Y
(A + i C) (dr − dt) ∧ d
x − i y
r − z
, (4.197)
+ 01 = −
1
4
1
Y
− Y
(A + i C) (dr − dt) ∧ d
x − i y
r − z
, (4.198)