Gravitational radiation from a rotating

magnetic dipole
arXiv:1609.06296v2 [gr-qc] 13 Jan 2017

S. Hacyan
February 20, 2018

Instituto de Fı́sica, Universidad Nacional Autónoma de México,

A. P. 20-364, Cd. de México, 01000, Mexico.

Abstract
The gravitational radiation emitted by a rotating magnetic dipole
is calculated. Formulas for the polarization amplitudes and the radi-
ated power are obtained in closed forms. A comparison is made with
other sources of gravitational and electromagnetic radiation, particu-
larly neutron stars with extremely powerful magnetic fields.

PACS: 04.30.Db; 04.30.Tv

Key words: gravitational waves; neutron stars

1 Introduction
Gravitational radiation is an important source of energy in many astrophys-
ical phenomena. A neutron star, for instance, radiates both electromagnetic
[1, 2] and gravitational waves [3, 4, 5]; the main sources of radiation are the
interior of the star (behaving as a magnetized fluid), the external magnetic
field and the corotating magnetosphere.
In the present article, we study the gravitational waves (GWs) generated
by one of these possible sources: a rotating magnetic dipole. In Section 2, the
radiated energy and the polarization amplitudes of the GWs are calculated
using the quadrupole formula and considering the electromagnetic field in

1
the near zone of the dipole, where most of the energy of the field is located.
The results are discussed in Section 3 and compared with other sources of
radiation, electromagnetic or gravitational, with a focus on stars such as
magnetars[6] that possess extremely powerful magnetic fields.

2 Near zone
Our starting point is the formula for the metric hTijT in the TT gauge (see
Maggiore [7] for notation and details),

1 2G
hTijT = Λij,kl(n̂)Q̈kl (t − r/c), (2.1)
r c4
where dots represent derivation with respect to time t,
1
Λij,kl (n̂) = Pik Pjl − Pij Pkl , (2.2)
2
and
Pij (n̂) = δij − n̂i n̂j (2.3)
is the projection tensor with respect to the unit vector n̂. The quadrupole is
defined as Qij = M ij − 13 δ ij M nn in terms of

1
Z
ij
M = 2 T 00 xi xj dV, (2.4)
c

where T 00 is the 00 component of the energy-momentum tensor.

The energy radiated in the form of GWs in the direction of n̂ is
Z ∞
dE r 2 c3
= dt ḣTijT ḣTijT . (2.5)
dΩ 32πG −∞

2.1 Magnetic dipole

Consider the field of a magnetic dipole of magnitude m. In the near zone,
the electric field can be neglected and the energy density of the magnetic
field is
1 m2  2

T00 = 1 + 3(r̂ · û(t)) , (2.6)
8π r 6

2
where û(t) is the unit vector in the direction of the dipole, and r̂ is a unit
radial vector. Further corrections to the electromagnetic field are of order
ωr/c with respect to Bi (where ω is the rotation frequency of the dipole).
For neutron stars of radius R ∼ 10 km, the approximation is valid for ω ≪
c/R ∼ 3 × 104 s−1 .
It follows with some straightforward algebra that

m2  i 1 ij 
Qij (t) = û (t)û j
(t) − δ , (2.7)
5Rc2 3
where R is a lower cut-off that can be identified with the radius of the star.
It is understood that the volume integral (2.4) covers the region r ≥ R.
The metric hTijT follows from the above formula and Eq. (2.1):

1 2Gm2 d2  k 
hTijT = Λ ij,kl û (tr ) û l
(tr ) , (2.8)
r 5Rc6 dt2
where tr = t − r/c.
Let us now take a coordinate system in which the rotation axis of the
dipole is in the z direction. Thus

û(t) = (u⊥ cos(ωt), u⊥ sin(ωt), uk), (2.9)

where uk is the constant component of û(t) along the rotation axis and u2⊥ =
1−u2k . In this same system of coordinates we can define the three orthonormal
vectors

n̂ = (sin θ cos φ, sin θ sin φ, cos θ), (2.10)

θ̂ = (cos θ cos φ, cos θ sin φ, − sin θ),
φ̂ = (− sin φ, cos φ, 0),

together with the useful formulas

1
θ̂i θ̂j Λij,kl = θ̂k θ̂l − Pkl ,
2
1
φ̂i φ̂j Λij,kl = φ̂k φ̂l − Pkl ,
2
φ̂i θ̂j Λij,kl = φ̂k θ̂l . (2.11)

3
2.2 Metric
The two metric potentials of the GW can be calculated from Eq. (2.8) and
the formulas (2.11). The result is
1 Gm2 d2 2
h+ ≡ hTijT φ̂i φ̂j = −hTijT θ̂i θ̂j = (u − u2θ )
r 5Rc6 dt2 φ
TT 1 2Gm2 d2
h× ≡ −hij θ̂i φ̂j = − (uθ uφ ), (2.12)
r 5Rc6 dt2
where uθ = û · θ̂ and uφ = û · φ̂, and (û · n̂)2 + u2θ + u2φ = 1. The above two
formulas can be written as
1 Gm2 d2
h+ + ih× = (uφ − iuθ )2 . (2.13)
r 5Rc dt
6 2

Explicitly
uφ = u⊥ sin(ωt′ )
uθ = u⊥ cos θ cos(ωt′) − uk sin θ, (2.14)
with ωt′ = ωtr − φ, from where it follows that
1 Gm2 2 n h
2 ′ ′
i
h+ + ih× = ω u⊥ 2u⊥ (1 + cos θ) cos(2ωt ) + 2i cos θ sin(2ωt )
r 5Rc6
h io
− 2uk sin(θ) cos(θ) cos(ωt′ ) + i sin(ωt′ ) . (2.15)
Accordingly, the spectrum of the GW has two lines, one at ω correspond-
ing to the uk component, and one at 2ω corresponding to the u⊥ component
(only the latter is present for a GW propagating along the rotation axis).
The amplitude of the wave is of order Gm2 ω 2 u⊥ /(Rc6 r).

2.3 Radiated energy

The radiated power can be calculated noticing that ḣTijT ḣTijT = |ḣ+ + iḣ× |2 .
The energy radiated per unit time follows from Eq. (2.5) performing the
integration over one period T = 2π/ω and dividing by T . The result is
dP Gm4 2 6 h 4 2 4 2
i
= u ω 1 − cos θ + u⊥ (5 cos θ + 24 cos θ + 3) . (2.16)
dΩ 200R2 c9 ⊥
Finally, an integration over solid angles yields the total power radiated:
πGm4 2 6
P = u⊥ ω (1 + 18u2⊥ ). (2.17)
75R c
2 9

4
3 Comparisons and conclusions
For the dipole field, we can set m = B0 R3 , where B0 is the average strength
of the magnetic field at the surface of the star. If B0 ∼ 1012 G and ω ∼ 1 s−1 ,
the power radiated in the form of gravitational radiation is
 B 4  R 10
0
P ∼ 106 (ω s)6 u2⊥ ergs/s
10 G
12 10 km
according to formula (2.17). Of course, for average pulsars, this is many
orders of magnitude below the power emitted in the form of electromagnetic
waves, which is typically 1028 ergs/s [1]. Nevertheless, for a millisecond
magnetar with B0 ∼ 1014 G, the power of the GWs could be of the order of
1032 ergs/s.
It is also instructive to compare our results with those obtained by Bonaz-
zola and Gourgoulhon [3] for the emission of GWs from the interior of a
rotating neutron star. These authors obtained a value for the amplitudes of
GWs
1 4G
|h+ + ih× | ∼ Iǫω 2 , (3.18)
r c4
where I is the moment of inertia and ǫ is the ellipticity of the star; typical
values of these parameters are ǫ ∼ 10−6 or smaller, and I ∼ 1045 g cm2 . If
we compare their result with our Eq. (2.15), we see that the amplitudes of
the GWs produced by the rotating fluid are larger by a factor

1013 ǫ (B0 /1012 G)−2

than the amplitudes of GWs produced by the rotation of the magnetic field.
Thus, for a usual neutron star with B0 ∼ 1012 G, the contribution of the
rotating dipole is comparatively negligible. However, it is not negligible
for magnetars having fields B0 ∼ 1014 G [6] and rather small deformations
ǫ < 10−6 .
In conclusion, the external magnetic field of a magnetar can make a sig-
nificant correction to the gravitational radiation produced by the internal
magnetized fluid.

References
[1] F. Pacini, Nature 216, 567 (1967).

5
[2] S. Hacyan, Phys. Rev. D 93, 044066 (2016).

[3] S. Bonazzola and E. Gourgoulhon, Astron. Astrophys. 312, 675 (1996).

[4] C. Cutler, Phys. Rev. D 66, 084025 (2002).

[5] D. I. Jones, Class. Quan. Grav. 19, 1255 (2002).

[6] S. A. Olausen and V. M. Kaspi, Astrophys. J., Supp. Series 212, 1

(2014).

[7] M. Maggiore, Gravitational Waves (Oxford, Oxford U. Press, 2008),

Vol. 1. Sect. 3.1.