Appendix 3: Electromagnetic Wave Sources
189
show that the + polarized wave picks up a factor of
1 + cos
2
θ
/2 and the
× polarized wave picks up a factor of cos θ .
Thus verify (11.67).
11.6 The amplitude solution (11.64) has a general order of magnitude form
involving a characteristic velocity v ch and we may write it as
h ∼
G M
c 2 r
v
2
ch
c 2
=
m
r
v ch
c
2 .
Can you show heuristically that this should be roughly true for a fairly general
source? See also Exercise 11.16.
11.7 In the text we discussed as sources of gravitational waves orbiting black
holes and neutron stars. Can you think of any other possibly interesting
astronomical sources?
11.8 In Sect. 11.2 on Newtonian limits we neglected some velocity dependent
effects in the geodesic motion of particles and also velocity dependent effects
on the sources of gravity. Such effects are interesting, although usually very
small in the real world. They are called gravitomagnetic effects or sometimes
“frame dragging” effects; they have been observed in the orbits of satellites
and on the precession of an orbiting gyroscope. Work out the effects for the
field produced by a spinning body and on the motion of bodies; see Adler
(2000).
11.9 Verify Kepler’s law expressed in (11.69) for the orbiting system of Example
11.3.
11.10 Verify that the wave metric may be written in terms of the chirp time as in
(11.70). What is the approximate chirp time in seconds for a pair of orbiting
solar mass black holes?
11.11. It is fairly straight-forward to derive the energy density in a gravitational
wave using linearized general relativity, but a bit tedious and lengthy. We
may take a shortcut and obtain the result heuristically using the analogy
with electromagnetic waves and dimensional analysis. The energy density
in an electric field is well-known by physics students to be proportional
to
E
2 and for an electromagnetic wave it is thus proportional to
˙
A
2 .
Use the analogy between electromagnetism and linearized general relativity
discussed in Appendix 2 and (11.96) to see that the analogous expression
for the gravitational wave should have the form
ρ ∝
( ˙
h 11 )
2
+ ( ˙
h 12 )
2
.
The angle brackets in this expression indicate that the quantity is to be averaged over a wavelength or so. Think a bit about why the averaging should
occur and see Schutz (2009).
11.12 Next use dimensional analysis to see that a factor of c
2
/G should be included
in the expression for the energy density in the above exercise. The energy
189
show that the + polarized wave picks up a factor of
1 + cos
2
θ
/2 and the
× polarized wave picks up a factor of cos θ .
Thus verify (11.67).
11.6 The amplitude solution (11.64) has a general order of magnitude form
involving a characteristic velocity v ch and we may write it as
h ∼
G M
c 2 r
v
2
ch
c 2
=
m
r
v ch
c
2 .
Can you show heuristically that this should be roughly true for a fairly general
source? See also Exercise 11.16.
11.7 In the text we discussed as sources of gravitational waves orbiting black
holes and neutron stars. Can you think of any other possibly interesting
astronomical sources?
11.8 In Sect. 11.2 on Newtonian limits we neglected some velocity dependent
effects in the geodesic motion of particles and also velocity dependent effects
on the sources of gravity. Such effects are interesting, although usually very
small in the real world. They are called gravitomagnetic effects or sometimes
“frame dragging” effects; they have been observed in the orbits of satellites
and on the precession of an orbiting gyroscope. Work out the effects for the
field produced by a spinning body and on the motion of bodies; see Adler
(2000).
11.9 Verify Kepler’s law expressed in (11.69) for the orbiting system of Example
11.3.
11.10 Verify that the wave metric may be written in terms of the chirp time as in
(11.70). What is the approximate chirp time in seconds for a pair of orbiting
solar mass black holes?
11.11. It is fairly straight-forward to derive the energy density in a gravitational
wave using linearized general relativity, but a bit tedious and lengthy. We
may take a shortcut and obtain the result heuristically using the analogy
with electromagnetic waves and dimensional analysis. The energy density
in an electric field is well-known by physics students to be proportional
to
E
2 and for an electromagnetic wave it is thus proportional to
˙
A
2 .
Use the analogy between electromagnetism and linearized general relativity
discussed in Appendix 2 and (11.96) to see that the analogous expression
for the gravitational wave should have the form
ρ ∝
( ˙
h 11 )
2
+ ( ˙
h 12 )
2
.
The angle brackets in this expression indicate that the quantity is to be averaged over a wavelength or so. Think a bit about why the averaging should
occur and see Schutz (2009).
11.12 Next use dimensional analysis to see that a factor of c
2
/G should be included
in the expression for the energy density in the above exercise. The energy