11.5 Gravitational Wave Sources
175
Fig. 11.3 Two bodies orbit in a plane perpendicular to the line to earth. The wave metric they
produce at a distance r is given in (11.66)
is a typical form for gravitational waves and can be useful in making rough
estimates. See Exercise 11.6.
Example 11.2 A more realistic example is a pair of equal mass points in
circular orbit about a common center, which does occur in nature. We assume
the orbit plane is fortuitously perpendicular to the earth direction in Fig. 11.3
so the geometry is simple, with θ = 0. From the figure the density function is
T
00
= ρ =
1
2
Mδ
x
− R cos ωt
δ
y
− R sin ωt
δ
z
+
1
2
Mδ
x
+ R cos ωt
δ
y
+ R sin ωt
δ
z
(11.65)
From the expressions in (11.62) the metric field is then
h 11 = −
G M
c 4 r
∂
2
∂t 2
(R cos ωt)
2
− (R sin ωt)
2
=
4G M
c 2 r
R
2
ω
2
c 2
cos 2ωt,
h 12 = −
2G M
c 4 r
∂
2
∂t 2
R
2 cos ωt sin ωt
=
4G M
c 2 r
R
2
ω
2
c 2
sin 2ωt.
(11.66)
Thus the metric field has both + and × polarizations; it is the gravitational
analog of circularly polarized light.
In Exercise 11.5 you are asked to work out the wave metric for a general
angle θ > 0, rather than θ = 0. The result is