4.6 Some ‘Spherical’ Gravitational Waves
95
H
23
=
A [y 2 z − r x 2 ] − C [x y z + r x y]
2 r (r − z)(r 2 − z 2 )
,
(4.211)
H
33
=
−A y + C x
2 r (r − z)
.
(4.212)
Next with w = r − t we can write A(w) = da/dw and C(w) = dc/dw and then
−∂a/∂t = ∂a/∂r = A and −∂c/∂t = ∂c/∂r = C. Consequently we can write
E
αβ
= −
∂σ αβ
∂t
and H
αβ
= −
∂λ αβ
∂t
,
(4.213)
with σ αβ , λ αβ , satisfying σ αβ = σ βα , σ αα = 0 and λ αβ = λ βα , λ αα = 0, given by
σ 11 + i λ 11 = −
i (a + i c)(r y + i xz)
2 (r − z) 2 (r + z)
x (x − i y)
r (r − z)
− 1
,
(4.214)
σ 22 + i λ 22 =
(a + i c) (r x − i y z)
2 (r − z) 2 (r + z)
i y(x − i y)
r (r − z)
− 1
,
(4.215)
σ 33 + i λ 33 =
(a + i c)
2 r
x − i y
r − z
,
(4.216)
σ 12 + i λ 12 = −
(a + i c) (r y + i x z)
2 (r − z) 2 (r + z)
i y(x − i y)
r (r − z)
− 1
,
(4.217)
σ 13 + i λ 13 =
i (a + i c) (r y + i x z)
2 r (r 2 − z 2 )
x − i y
r − z
,
(4.218)
σ 23 + i λ 23 = −
i (a + i c) (r x − i y z)
2 r (r 2 − z 2 )
x − i y
r − z
.
(4.219)
We emphasise that σ αβ has the geometrical interpretation of the perturbed shear
of the t-lines in the perturbed Minkowskian space-time due to the presence of the
gravitational waves. In addition σ αβ , λ αβ satisfy the differential equations
σ αβ,β = 0 and λ αβ,β = 0 .
(4.220)
To verify these it is useful to first express (4.214)–(4.219) in terms of the function
Y in (4.191). This results in
σ 11 + i λ 11 =
(a + i c)
8 r
Y
Y −
1
Y
2
,
(4.221)
σ 22 + i λ 22 = −
(a + i c)
8 r
Y
Y +
1
Y
2
,
(4.222)