174
11 Linearized General Relativity and Gravitational Waves
The 0,0 component of the energy-momentum tensor is simply the energy density ρ,
so the integral in (11.61) has a clear physical meaning and is often easy to calculate.
It is generally called the quadrupole integral.
Our final manipulation is to consider the metric field (11.61) at large distances
from the source, where it is asymptotically a plane wave, and to transform it to the
traceless transverse gauge we used previously to calculate the motion of particles.
This has already been done in general in Sect. 11.3 on plane waves. There we found
that the components of the field with an index equal to 0 or 3 could be transformed
away; we have also just shown that these components are constant in time and are not
relevant to a wave analysis, so no more need be said about them. It only remains to
transform the 1,2 block into traceless form which we already did before in (11.40).
Using that equation we may write the metric field in the new system,
h 11 = −
G
c 4 r
∂
2
∂t 2
T
00
(x
2
− y
2
)d
3 x
= −h 22 ,
h 12 = −
2G
c 4 r
∂
2
∂t 2
T
00
(x
, y
)d
3 x
,
(11.62)
in which we no longer need to label the metric field with a prime. Since it is traceless
we also do not need to include the “hat” notation.
Equation (11.62) is also called the quadrupole Formula. It is in convenient form for
calculation; if we know the mass density ρ = T
00 as a function of retarded time and
position it gives the distant metric field in traceless transverse form. It may be applied
to many real-world sources that are small and slowly moving on the astronomical
scale, as we will discuss in the following examples.
Example 11.1 Some examples of the use of the quadrupole formula are in
order. First consider a linear oscillator, that is a system in which all the mass
is concentrated in a point oscillating along the x
axis, which is perpendicular
to the z
axis. The density function is then a Dirac delta function,
T
00
= ρ = Mδ
x
− R cos ωt
δ
y
δ
z
.
(11.63)
The metric field is then purely + polarized, and easily calculated from (11.62)
to be
h 11 = −
G M
c 4 r
∂
2
∂t 2 (R cos ωt)
2
=
2G M
c 2 r
R
2
ω
2
c 2
cos 2ωt. (11.64)
Notice that the quantities in the last two parentheses for h 11 are dimensionless,
and the second parenthesis is the square of a characteristic velocity over c. This
11 Linearized General Relativity and Gravitational Waves
The 0,0 component of the energy-momentum tensor is simply the energy density ρ,
so the integral in (11.61) has a clear physical meaning and is often easy to calculate.
It is generally called the quadrupole integral.
Our final manipulation is to consider the metric field (11.61) at large distances
from the source, where it is asymptotically a plane wave, and to transform it to the
traceless transverse gauge we used previously to calculate the motion of particles.
This has already been done in general in Sect. 11.3 on plane waves. There we found
that the components of the field with an index equal to 0 or 3 could be transformed
away; we have also just shown that these components are constant in time and are not
relevant to a wave analysis, so no more need be said about them. It only remains to
transform the 1,2 block into traceless form which we already did before in (11.40).
Using that equation we may write the metric field in the new system,
h 11 = −
G
c 4 r
∂
2
∂t 2
T
00
(x
2
− y
2
)d
3 x
= −h 22 ,
h 12 = −
2G
c 4 r
∂
2
∂t 2
T
00
(x
, y
)d
3 x
,
(11.62)
in which we no longer need to label the metric field with a prime. Since it is traceless
we also do not need to include the “hat” notation.
Equation (11.62) is also called the quadrupole Formula. It is in convenient form for
calculation; if we know the mass density ρ = T
00 as a function of retarded time and
position it gives the distant metric field in traceless transverse form. It may be applied
to many real-world sources that are small and slowly moving on the astronomical
scale, as we will discuss in the following examples.
Example 11.1 Some examples of the use of the quadrupole formula are in
order. First consider a linear oscillator, that is a system in which all the mass
is concentrated in a point oscillating along the x
axis, which is perpendicular
to the z
axis. The density function is then a Dirac delta function,
T
00
= ρ = Mδ
x
− R cos ωt
δ
y
δ
z
.
(11.63)
The metric field is then purely + polarized, and easily calculated from (11.62)
to be
h 11 = −
G M
c 4 r
∂
2
∂t 2 (R cos ωt)
2
=
2G M
c 2 r
R
2
ω
2
c 2
cos 2ωt. (11.64)
Notice that the quantities in the last two parentheses for h 11 are dimensionless,
and the second parenthesis is the square of a characteristic velocity over c. This