170
11 Linearized General Relativity and Gravitational Waves
x =
1
√
2
( ˜
x + ˜
y), y =
1
√
2
( ˜
x − ˜
y).
(11.44)
For the line element (11.29) with h 11 = 0 this gives for the tilde system
ds
2
= c
2 dt
2
− dx
2
− dy
2
+ 2h 12 dxdy − dz
2
= c
2 dt
2
− (1 − h 12 )d ˜
x
2
− (1 + h 12 )d ˜
y
2
− dz
2
.
(11.45)
Since this is exactly the same line element we have just analyzed we need do no
more. The motion of test bodies is the same as we obtained above but with everything
rotated by 45 degrees. For this reason we refer to the waves with only nonzero h 11
as + polarized and those with only nonzero h 12 as × polarized, and sometimes write
h 11 = h + and h 12 = h × .
Notice an analogy between gravitational waves and electromagnetic waves. We
see that the two polarizations of gravitational waves are related by a rotation of
45 degrees, whereas the 2 polarizations of electromagnetic waves are related by a
rotation of 90 degrees. In the quantum field theory of these fields this is associated
with the spin of a photon being 1 and the spin of a graviton being 2 (Bjorken 1965).
For people who are not interested in relativity theory, but are interested in the
detection of gravitational waves, it is useful to express the dynamics of bodies in
gravitational waves in terms of equivalent Newtonian tidal forces. From the expression (11.43) for the physical distance from a test body at the origin to a nearby freely
falling test body we can calculate the relative velocity and acceleration to be
v x =
d
dt
x = −
1
2
d
dt
(h + x 0 ) = −
1
2
d
dt
(h + x ),
a x =
dv x
dt
= −
1
2
d
2
dt 2 (h + x ).
(11.46)
According to Newton’s second law this is the same relative acceleration that bodies
would experience under a Newtonian tidal force,
F x /m = −
1
2
d
2
dt 2 (h + x ) = −
1
2
¨
h + x ,
(11.47)
where the dot indicates a derivative with respect to time. This tells us for example
that the tidal force exerted by a monochromatic gravitational wave is proportional to
the square of the frequency.
F x /m =
1
2
h + x ω
2
.
(11.48)
11 Linearized General Relativity and Gravitational Waves
x =
1
√
2
( ˜
x + ˜
y), y =
1
√
2
( ˜
x − ˜
y).
(11.44)
For the line element (11.29) with h 11 = 0 this gives for the tilde system
ds
2
= c
2 dt
2
− dx
2
− dy
2
+ 2h 12 dxdy − dz
2
= c
2 dt
2
− (1 − h 12 )d ˜
x
2
− (1 + h 12 )d ˜
y
2
− dz
2
.
(11.45)
Since this is exactly the same line element we have just analyzed we need do no
more. The motion of test bodies is the same as we obtained above but with everything
rotated by 45 degrees. For this reason we refer to the waves with only nonzero h 11
as + polarized and those with only nonzero h 12 as × polarized, and sometimes write
h 11 = h + and h 12 = h × .
Notice an analogy between gravitational waves and electromagnetic waves. We
see that the two polarizations of gravitational waves are related by a rotation of
45 degrees, whereas the 2 polarizations of electromagnetic waves are related by a
rotation of 90 degrees. In the quantum field theory of these fields this is associated
with the spin of a photon being 1 and the spin of a graviton being 2 (Bjorken 1965).
For people who are not interested in relativity theory, but are interested in the
detection of gravitational waves, it is useful to express the dynamics of bodies in
gravitational waves in terms of equivalent Newtonian tidal forces. From the expression (11.43) for the physical distance from a test body at the origin to a nearby freely
falling test body we can calculate the relative velocity and acceleration to be
v x =
d
dt
x = −
1
2
d
dt
(h + x 0 ) = −
1
2
d
dt
(h + x ),
a x =
dv x
dt
= −
1
2
d
2
dt 2 (h + x ).
(11.46)
According to Newton’s second law this is the same relative acceleration that bodies
would experience under a Newtonian tidal force,
F x /m = −
1
2
d
2
dt 2 (h + x ) = −
1
2
¨
h + x ,
(11.47)
where the dot indicates a derivative with respect to time. This tells us for example
that the tidal force exerted by a monochromatic gravitational wave is proportional to
the square of the frequency.
F x /m =
1
2
h + x ω
2
.
(11.48)