ñòð. 87 
We have seen now the electric eld of a single charge, this is called the monopole, see
gure 17.4. The eld of this charge decays as 1=r2 . If we put two opposite charges close
together we have seen we can create a dipole, see gure 17.4, as derived in expression
(17.45) this eld decays as 1=r3 . We can also put two opposite dipoles together as shown
in gure 17.4. The resulting charge distribution is called a quadrupole. To leading order
the electric elds of the dipoles that constitute the quadrupole cancel, and we will see in
the next section that the electric potential for quadrupole decays as 1=r3 so that the eld
decays with distance as 1=r4 .
You may wonder whether the concept of a dipole or quadrupole can be used as well
for the gravity eld because these concept are for the electric eld based on the presence
of both positive and negative charges whereas we know that only positive mass occurs in
nature. However, there is nothing that should keep us from computing the gravitational
eld for a negative mass, and this is actually quite useful. As an example, let us consider
a double star that consists of two heavy stars that rotate around their joint center of
gravity. The rst order eld is the monopole eld that is generated by the joint mass of
the stars. However, as shown in gure 17.5 the mass of the two stars can approximately

= +
+
+ + +
+

Figure 17.5: The decomposition of a double star in a gravitational monopole and a gravi
tational quadrupole.
be seen as the sum of a monopole and a quadrupole consisting of two positive and two
negative charges. Since the stars rotate, the gravitational quadrupole rotates as well, and
this is the reason why rotating double stars are seen as a prime source for the generation
of gravitational waves 42]. However, gravitational waves that spread in space with time
cannot be described by the classic expression (17.1) for the gravitational potential.
Problem g: Can you explain why (17.1) cannot account for propagating gravitational
waves?
CHAPTER 17. POTENTIAL THEORY
254
A proper description of gravitational waves depends on the general theory of relativity
42]. This is the reason why huge detectors for gravitational waves are being developed,
because these waves can be used to investigate the theory of general relativity.
17.6 The multipole expansion
Now that we have learned that the concepts of monopole, dipole and quadrupole are
relevant for both the electric eld and the gravity eld we will continue the analysis with
the gravitational eld. In this section we will derive the multipole expansion where the
total eld is written as a superposition of a monopole eld, a dipole eld, a quadrupole
eld, an octupole eld, etc.
r
râ€™
Figure 17.6: De nition of the integration variable r0 within the mass and the observation
point r outside the mass.
Consider the situation shown in gure 17.6 where a nite body has a mass density
(r0 ). The gravitational potential generated by this mass is given by:
Z (r0)
0
V (r) = 0 dV (17.9) again:
r
;G
jr ; j
We will consider the potential at a distance that is much larger than the size of the body.
Since the integration variable r0 is limited by the size of the body a \large distance"
means in this context that r r0 . We will therefore make a Taylor expansion of the term
r0 in the small parameter (r0=r) which is much smaller than unity.
1=jr ; j
Problem a: Show that
0 = qr2 2 (r r0 ) + r02
rr (17.46)
; ;
Problem b: Use a Taylor expansion in the small parameter r0=r to show that:
( )
;^ r0 + 1 3 ;^ r0 2 r02 + O r0 3 :
1 = 1 1+ 1 r
2r2 r (17.47)
r0 r r r
;
jr ; j
Be careful that you properly account for all the terms of order r0 correctly. Also be
aware of the distinction between the di erence between the position vector r and
the unit vector ^.
r
17.6. THE MULTIPOLE EXPANSION 255
From this point on we will ignore the terms of order (r0 =r)3 .
Problem c: Insert the expansion (17.47) in (17.9) and show that the gravitational po
tential can be written as a sum of di erent contributions:
V (r) =Vmon (r)+Vdip (r)+Vqua (r) + (17.48)
G Z (r0)dV 0
with
Vmon (r) = r (17.49)
;
G Z (r0 ) ;^ r0 dV 0
r
Vdip (r) = r2 (17.50)
;
G Z (r0) 3 ;^ r0 2 r02 dV 0 :
r
Vqua(r) = 2r3 (17.51)
; ;
It follows that the gravitational potential can be written as sum of terms that decay with
increasing powers of r;n . Let us analyze these terms in turn. The term Vmon (r) in (17.49)
is the simplest since the volume integral of the mass density is simply the total mass of
R
the body: (r0 )dV 0 = M. This means that this term is given by
Vmon(r) = GM : (17.52)
r;
This is the potential generated by a point mass M. To leading order, the gravitational
eld is the same as if all the mass of the body would be concentrated in the origin. The
mass distribution within the body does not a ect this part of the gravitational eld at all.
Because the resulting eld is the same as for a point mass, this eld is called the monopole
eld.
For the analysis of the term Vdip (r) in (17.50) is us useful to de ne the center of gravity
rg of the body: R (r0 )r0dV 0
rg R (r0 )dV 0 (17.53)
This is simply a weighted average of the position vector with the mass density as weight
functions. Note that the word \weight" here has a double meaning!
Problem d: Show that Vdip(r) is given by:
Vdip (r) = GM (^ rg ) :
r2 r (17.54)
;
Note that this potential has exactly the same form as the potential (17.44) for an electric
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