190
11 Linearized General Relativity and Gravitational Waves
density thus becomes
ρ =
1
16π
c
2
G
( ˙
h 11 )
2
+ ( ˙
h 12 )
2
where the numerical factor 1/16π must be gotten from a more detailed
analysis such as Schutz (2009).
11.13 Verify the expression (11.72) for the total energy of the orbiting system
according to classical mechanics. Then use (11.70) and (11.71) to calculate
the wave energy in a thin spherical shell of thickness cdt. Balance this energy
with the energy the orbiting system must lose in a time dt to verify (11.73).
11.14 Verify (11.74) which gives the frequency of the waves as a function of time.
11.15 Take the size of the orbiting bodies in Example 11.3 to be nonzero and calculate a more realistic coalescence time than given by (11.74). Also calculate
the maximum frequency due to the finite size.
11.16 We noted in the text that the production and detection of gravitational waves
in a laboratory is not likely in the foreseeable future. Use the order of
magnitude relation in Exercise 11.6 to show this; you need only estimate
the maximum mass and velocity one might hope to achieve in a terrestrial
laboratory setting.
11 Linearized General Relativity and Gravitational Waves
density thus becomes
ρ =
1
16π
c
2
G
( ˙
h 11 )
2
+ ( ˙
h 12 )
2
where the numerical factor 1/16π must be gotten from a more detailed
analysis such as Schutz (2009).
11.13 Verify the expression (11.72) for the total energy of the orbiting system
according to classical mechanics. Then use (11.70) and (11.71) to calculate
the wave energy in a thin spherical shell of thickness cdt. Balance this energy
with the energy the orbiting system must lose in a time dt to verify (11.73).
11.14 Verify (11.74) which gives the frequency of the waves as a function of time.
11.15 Take the size of the orbiting bodies in Example 11.3 to be nonzero and calculate a more realistic coalescence time than given by (11.74). Also calculate
the maximum frequency due to the finite size.
11.16 We noted in the text that the production and detection of gravitational waves
in a laboratory is not likely in the foreseeable future. Use the order of
magnitude relation in Exercise 11.6 to show this; you need only estimate
the maximum mass and velocity one might hope to achieve in a terrestrial
laboratory setting.