188
11 Linearized General Relativity and Gravitational Waves
Thus the integral reduces to the time derivative of an integral involving the charge
density J
0 that we will call a dipole integral.
A( x, t)
k = −
1
4π
1
r
∂
∂t
J
0 x
k d
3 x
.
(11.102)
This result lets us calculate some simple and interesting cases of radiation. For
example consider a point charge q oscillating along the x axis with amplitude L at
frequency ω. Then the charge density function is
J
0
= qδ
x
− L cos ωt
δ
y
δ
z
,
(11.103)
and the field from (11.102) is
A( x, t)
1 =
q
4π
Lω
r
sin ωt.
(11.104)
This is a reasonable model for a short dipole antenna. Note that the corresponding
electric field is the derivative of this vector potential and thus is proportional to the
square of the frequency.
Exercises
11.1 Write out the approximate metric in (11.22) for a source which has monopole
and quadrupole moments. What of a dipole moment? What of higher
moments? Where might this equation be useful?
11.2 Do the two functions in the traceless transverse gauge solution (11.28) need
to be related to each other? Can you imagine a source in which the elements
of the metric have different time dependence?
11.3. Design a simple gravitational wave detector using springs and masses.
Design one consisting of elastic rods. Equations (11.47) and (11.48) should
be a help.
11.4 The current official definition of physical distance is that of light travel
time. Consider then the line element (11.29). Light moves on a null line,
ds = 0, at constant physical velocity c, so in the x direction it obeys cdt =
(1 − h 11 /2)dx. The definition of distance thus means the relation between
coordinate distance and physical distance is d = cdt = (1 − h 11 /2)dx.
This gives justification for the relation (11.43) for test bodies in a gravitational wave. Now use this to analyze the operation of an interferometer
wave detector that is not of negligible size compared to the wavelength of
the gravitational wave. This analysis is relevant for very large machines such
as LISA.
11.5 We studied in Example 11.2 an orbiting pair of equal mass bodies in a plane
perpendicular to the line toward earth; see Fig. 11.3. Work out the metric
field if the line to earth is at an angle θ from the perpendicular, and thereby
11 Linearized General Relativity and Gravitational Waves
Thus the integral reduces to the time derivative of an integral involving the charge
density J
0 that we will call a dipole integral.
A( x, t)
k = −
1
4π
1
r
∂
∂t
J
0 x
k d
3 x
.
(11.102)
This result lets us calculate some simple and interesting cases of radiation. For
example consider a point charge q oscillating along the x axis with amplitude L at
frequency ω. Then the charge density function is
J
0
= qδ
x
− L cos ωt
δ
y
δ
z
,
(11.103)
and the field from (11.102) is
A( x, t)
1 =
q
4π
Lω
r
sin ωt.
(11.104)
This is a reasonable model for a short dipole antenna. Note that the corresponding
electric field is the derivative of this vector potential and thus is proportional to the
square of the frequency.
Exercises
11.1 Write out the approximate metric in (11.22) for a source which has monopole
and quadrupole moments. What of a dipole moment? What of higher
moments? Where might this equation be useful?
11.2 Do the two functions in the traceless transverse gauge solution (11.28) need
to be related to each other? Can you imagine a source in which the elements
of the metric have different time dependence?
11.3. Design a simple gravitational wave detector using springs and masses.
Design one consisting of elastic rods. Equations (11.47) and (11.48) should
be a help.
11.4 The current official definition of physical distance is that of light travel
time. Consider then the line element (11.29). Light moves on a null line,
ds = 0, at constant physical velocity c, so in the x direction it obeys cdt =
(1 − h 11 /2)dx. The definition of distance thus means the relation between
coordinate distance and physical distance is d = cdt = (1 − h 11 /2)dx.
This gives justification for the relation (11.43) for test bodies in a gravitational wave. Now use this to analyze the operation of an interferometer
wave detector that is not of negligible size compared to the wavelength of
the gravitational wave. This analysis is relevant for very large machines such
as LISA.
11.5 We studied in Example 11.2 an orbiting pair of equal mass bodies in a plane
perpendicular to the line toward earth; see Fig. 11.3. Work out the metric
field if the line to earth is at an angle θ from the perpendicular, and thereby