11.4 Motion of Test Bodies in Gravitational Waves
169
Here the prime denotes a derivative with respect to the argument of h 11 , or U = ct −z.
The solution to these equations is surprisingly easy for bodies that are at rest initially,
before the wave arrives. Both constants in the x and y equations are then equal to
zero so ˙
x = ˙
y = 0 and bodies remains at the same x and y positions. This implies
furthermore that from the z equation ¨
z = 0, so bodies initially at rest at z = 0 do not
move in the z direction. Finally the t equation tells us that c¨ t = 0 so we may choose
the coordinate time along the geodesic to be equal to the proper time, or ct = s.
In summary, the motion is very simple: bodies initially at coordinate rest in the
traceless transverse gauge remain at coordinate rest as the wave passes. For this reason
the coordinates are called co-moving. However coordinate distances and physical
distances are different, so bodies at rest in the coordinate system are not physically
at rest.
Let’s first consider two test bodies at coordinate rest, both at y = z = 0 and
separated by a small coordinate distance 0 . Their physical separation is, from
(11.29),
x =
1 − h 11 0 = (1 − h 11 /2) 0 .
(11.43)
(see Exercise 11.4). Thus the physical separation changes in time according to the
time dependence of the function h 11 . A useful example is to take the function to be an
oscillation like a sine or cosine, so the separation of the bodies oscillates about 0
by a small amount h 11 0 /2, corresponding to a fractional distance change h 11 /2.
Exactly the same considerations for two test bodies separated in the y disrection
gives us a change of −h 11 0 /2 and a fractional change of −h 11 /2. In Fig. 11.1 we
show the effect on a circular ring of test bodies in the x, y plane for a wave moving
in the z direction. Because of the motion pattern a wave of this sort is referred to as
polarized in the + direction and the metric function is often written h 11 = h + . From
Fig. 11.1 it is already clear how one might try to detect a gravitational wave using a
distance measuring device such as an interferometer.
In the above we considered a wave with only an h 11 component and h 12 = 0. Next
we will consider a wave with nonzero h 12 and h 11 = 0. There is a very easy way to
do this and also display the nature of polarization for the waves. The coordinates in
the x, y plane may be rotated by 45 degrees to a tilde system using the transformation
Fig. 11.1 Qualitative nature of motion produced by an oscillatory plane gravitational wave on a
circle of test bodies for the + polarization. The pictures are a half cycle apart. For the × polarization
the pictures are rotated by 45°
169
Here the prime denotes a derivative with respect to the argument of h 11 , or U = ct −z.
The solution to these equations is surprisingly easy for bodies that are at rest initially,
before the wave arrives. Both constants in the x and y equations are then equal to
zero so ˙
x = ˙
y = 0 and bodies remains at the same x and y positions. This implies
furthermore that from the z equation ¨
z = 0, so bodies initially at rest at z = 0 do not
move in the z direction. Finally the t equation tells us that c¨ t = 0 so we may choose
the coordinate time along the geodesic to be equal to the proper time, or ct = s.
In summary, the motion is very simple: bodies initially at coordinate rest in the
traceless transverse gauge remain at coordinate rest as the wave passes. For this reason
the coordinates are called co-moving. However coordinate distances and physical
distances are different, so bodies at rest in the coordinate system are not physically
at rest.
Let’s first consider two test bodies at coordinate rest, both at y = z = 0 and
separated by a small coordinate distance 0 . Their physical separation is, from
(11.29),
x =
1 − h 11 0 = (1 − h 11 /2) 0 .
(11.43)
(see Exercise 11.4). Thus the physical separation changes in time according to the
time dependence of the function h 11 . A useful example is to take the function to be an
oscillation like a sine or cosine, so the separation of the bodies oscillates about 0
by a small amount h 11 0 /2, corresponding to a fractional distance change h 11 /2.
Exactly the same considerations for two test bodies separated in the y disrection
gives us a change of −h 11 0 /2 and a fractional change of −h 11 /2. In Fig. 11.1 we
show the effect on a circular ring of test bodies in the x, y plane for a wave moving
in the z direction. Because of the motion pattern a wave of this sort is referred to as
polarized in the + direction and the metric function is often written h 11 = h + . From
Fig. 11.1 it is already clear how one might try to detect a gravitational wave using a
distance measuring device such as an interferometer.
In the above we considered a wave with only an h 11 component and h 12 = 0. Next
we will consider a wave with nonzero h 12 and h 11 = 0. There is a very easy way to
do this and also display the nature of polarization for the waves. The coordinates in
the x, y plane may be rotated by 45 degrees to a tilde system using the transformation
Fig. 11.1 Qualitative nature of motion produced by an oscillatory plane gravitational wave on a
circle of test bodies for the + polarization. The pictures are a half cycle apart. For the × polarization
the pictures are rotated by 45°