11.4 Motion of Test Bodies in Gravitational Waves
171
If one wants to design a mechanical wave detector consisting of springs and masses or
solid bars this effective force can be quite useful, and it is not necessary to understand
general relativity. See Exercises 11.3 and 11.4.
11.5 Gravitational Wave Sources
After solving the equations for plane gravitational waves in vacuum and seeing how
they affect test bodies we now turn to understanding some possible sources of such
waves. For this we use the linearized equations, which we repeat from (11.18),
¯
h
,λ
μν ,λ =
2 ¯
h μν = −
16π G
c 2
T μν , ¯
h
μν
,ν = 0.
(11.49)
The layout of the source region and the distant detection region that we assume is
shown in Fig. 11.2.
This field equation (11.49) occurs often in physics, in particular in electrodynamics; we discuss it in Appendix 1 for the reader who is not familiar with it or
desires a brief review. The retarded solution is, from (11.83) in Appendix 1,
¯
h( x, t) μν = −
4G
c 2
1
r
T
x
, t ret
μν
d
3 x
,
r =
x − −
x
, t ret = t − r/c.
(11.50)
Here t ret is referred to as the retarded time for obvious reasons. For the small source
approximation, in which the source size and characteristic frequency obey Lω ch c,
the radiation from all parts of the source is in phase, and the solution far from the
source reduces to an integral over the source at a single retarded time,
¯
h( x, t) μν = −
4G
c 2
1
r
T
x
, t ret
μν
d
3 x
, r = | x|, t ret = t − r/c. (11.51)
Fig. 11.2 The small source on the right emits gravitational waves that are to be detected at a large
distance r on the left side, where they are approximately plane waves moving in the local z direction
171
If one wants to design a mechanical wave detector consisting of springs and masses or
solid bars this effective force can be quite useful, and it is not necessary to understand
general relativity. See Exercises 11.3 and 11.4.
11.5 Gravitational Wave Sources
After solving the equations for plane gravitational waves in vacuum and seeing how
they affect test bodies we now turn to understanding some possible sources of such
waves. For this we use the linearized equations, which we repeat from (11.18),
¯
h
,λ
μν ,λ =
2 ¯
h μν = −
16π G
c 2
T μν , ¯
h
μν
,ν = 0.
(11.49)
The layout of the source region and the distant detection region that we assume is
shown in Fig. 11.2.
This field equation (11.49) occurs often in physics, in particular in electrodynamics; we discuss it in Appendix 1 for the reader who is not familiar with it or
desires a brief review. The retarded solution is, from (11.83) in Appendix 1,
¯
h( x, t) μν = −
4G
c 2
1
r
T
x
, t ret
μν
d
3 x
,
r =
x − −
x
, t ret = t − r/c.
(11.50)
Here t ret is referred to as the retarded time for obvious reasons. For the small source
approximation, in which the source size and characteristic frequency obey Lω ch c,
the radiation from all parts of the source is in phase, and the solution far from the
source reduces to an integral over the source at a single retarded time,
¯
h( x, t) μν = −
4G
c 2
1
r
T
x
, t ret
μν
d
3 x
, r = | x|, t ret = t − r/c. (11.51)
Fig. 11.2 The small source on the right emits gravitational waves that are to be detected at a large
distance r on the left side, where they are approximately plane waves moving in the local z direction