Pre-Phase A Report - Lisa - Nasa

14 Chapter 1 Scientific Objectives
'+'
'×'
0.2
h/2 h
-0.2
Figure 1.1 Illustration of the polarisation of a gravitational wave.
Two linearly independent polarisations of a gravitational wave are illustrated
by displaying their effect on a ring of free particles arrayed in a plane perpendicular
to the direction of the wave. The wave-form is shown between the
two sequences, for a wave with the (large) dimensionless amplitude h =0.2 .
Shown to scale are the distortions in the original circle that the wave produces
if it carries the +-polarisation (above) and the ×-polarisation (below).
The motion of each particle can be discovered by comparing it to its original
position, shown as the “shadow” circles. In general relativity, there are only
two independent polarisations. The ones shown here are orthogonal to one
another — notice that individual particles move in orthogonal directions in
the two illustrations. These polarisations are transverse to the direction of the
wave.
It follows that there are only two independent linear polarizations. It is conventional
to take them as the two area-preserving distortions illustrated in Figure 1.1, which are
called “+” and “×”. The rotation by 45 ◦ from one polarisation to the other makes them
orthogonal: notice that for each particle the motion in one diagram is perpendicular to
its motion in the other. In the language of quantum field theory, one expects only two
independent polarisations for a pure spin-2 massless graviton, because such a particle
has only two independent helicity states. But note that, despite this language, observable
gravitational waves are not quantum fields: they contain such enormous numbers of
“gravitons” (10 80 or more for some sources) that they are completely classical.
Radiation and antenna patterns. We shall turn in the next section to the way waves
are generated by source motions. But again we will not get directional information from
our approach. We fill this gap by noting here that, happily, the directions of polarization
follow closely the mass motions in the source. Suppose for simplicity that the source
consists of two masses moving back and forth along a given line, as if on a spring; then
the polarization ellipse of the waves will align its major axis with this line. Thus, two
detector masses separated along a direction parallel to the separation of the source masses
move back and forth in synchronisation with the source masses, at the same retarded time
(i.e. allowing for the travel time of the wave from source to detector). It follows that the
two oscillating source masses emit no radiation along the direction of the line joining
them, because when seen from this direction they have no transverse motion at all.
It is possible from this information to build up the radiation patterns and antenna pat-
3-3-1999 9:33 Corrected version 2.08
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