PACS: 04.30.Db; 04.30.Tv

1. Introduction

Gravitational radiation is an important source of energy in many astrophysical 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 Sec. 2, the radiated energy and the polarization amplitudes of the GWs are calculated using the quadrupole formula and considering the electromagnetic field in the near zone of the dipole, where most of the energy of the field is located. The results are discussed in Sec. 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. Radiation from the near zone

Our starting point is the formula for the metric ^{7} for notation and details),

where dots represent derivation with respect to time *t*,

and

is the projection tensor with respect to the unit vector

Where *T*
^{00} is the 00 component of the energy-momentum tensor.

The energy radiated in the form of GWs in the direction of

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

where

It follows with some straightforward algebra that

where *R* is a lower cut-off that can be identified with the radius of the star. It is understood that the volume integral (4) covers the region

The metric

where *t*
_{
r
} = *t* - *r*/*c*.

Let us now take a coordinate system in which the rotation axis of the dipole is in the *z* direction. Thus

where

together with the useful formulas

2.2. Metric

The two metric potentials of the GW can be calculated from Eq. (8) and the formulas (11). The result is

where

Explicitly

with

Accordingly, the spectrum of the GW has two lines, one at

2.3. Radiated energy

The radiated power can be calculated noticing that *T*. The result is

Finally, an integration over solid angles yields the total power radiated:

3. Comparisons and conclusions

For the dipole field, we can set *B*
_{0} is the average strength of the magnetic field at the surface of the star. If

according to formula (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 10^{28} ergs/s ^{1}. Nevertheless, for a millisecond magnetar with ^{32} ergs/s.

It is also instructive to compare our results with those obtained by Bonazzola 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

where *I* is the moment of inertia and ^{2}. If we compare their result with our Eq. (15), we see that the amplitudes of the GWs produced by the rotating fluid are larger by a factor

than the amplitudes of GWs produced by the rotation of the magnetic field. Thus, for a usual neutron star with ^{6} and rather small deformations

In conclusion, the external magnetic field of a magnetar can make a significant correction to the gravitational radiation produced by the internal magnetized fluid.