1 8 D O C . 1 G R A V I T A T I O N A L W A V E S
We note first that with respect to (15) the scheme of the coefficients of an
energy-free wave is
(17)
where are four mutually independent selectable numbers.
,
Next, we look at a field-free space whose line element ds with respect to suitably
chosen coordinates can be expressed in the form
. (18)
We now introduce new coordinates x1, x2, x3, x4 by means of the substitution
(19)
The four are real-valued, infinitesimally small constants, and is a real-valued
function of the argument . If quantities of the second degree in are
neglected, it follows from (18) and (19) that
.
From this follow for the associated the values
0
0
αμν
α β γ iα
αμν=) (
β δ iβ
γ
δ iγ
iα iβ iγ –α
α β γ δ , , ,
x1 ' x2 ' x3 ' x4 ' , , , ( )
–ds
2
dx1 '
2
dx2 '
2
dx3 '
2
dx4 '
2
+ + + =
x'ν xν λνφ( x1 ix4). + – =
λν φ
x1 ix4) + ( λν
ds2
ν
2
–∑dx'ν
2
2φ'( + dx1 idx4) +
ν
∑λνdxν
ν
–∑dxν
= =
[p. 161]
γμν
2λ1 λ2 λ3 iλ1 λ4 +
1
φ'
---γμν -
=⎞
⎝ ⎠
⎛
λ2
0 0
iλ2
λ3
0 0
iλ3
iλ1 λ4 + iλ2 iλ3 2iλ4