### 6.2 Polarization of gravitational waves

A laser interferometric or resonant bar gravitational wave detector whose scale is small compared to the
gravitational wavelength measures the local components of a symmetric 3 3 tensor which is composed
of the “electric” components of the Riemann curvature tensor, , via the equation of geodesic
deviation, given, for a pair of freely falling particles by , where denotes the spatial
separation. In general there are six independent components, which can be expressed in terms of
polarizations (modes with specific transformation properties under rotations and boosts). Three are
transverse to the direction of propagation, with two representing quadrupolar deformations and one
representing a monopolar “breathing” deformation. Three modes are longitudinal, with one an axially
symmetric stretching mode in the propagation direction, and one quadrupolar mode in each of the two
orthogonal planes containing the propagation direction. Figure 8 shows the displacements induced on a ring
of freely falling test particles by each of these modes. General relativity predicts only the first two
transverse quadrupolar modes (a) and (b) independently of the source; these correspond to
the waveforms and discussed earlier (note the and dependences
of the displacements). Massless scalar-tensor gravitational waves can in addition contain the
transverse breathing mode (c). In massive scalar-tensor theories, the longitudinal mode (d)
can also be present, but is suppressed relative to (c) by a factor , where is the
wavelength of the radiation, and is the Compton wavelength of the massive scalar. More
general metric theories predict additional longitudinal modes, up to the full complement of six
(TEGP 10.2 [281]).
A suitable array of gravitational antennas could delineate or limit the number of modes present
in a given wave. The strategy depends on whether or not the source direction is known. In
general there are eight unknowns (six polarizations and two direction cosines), but only six
measurables (). If the direction can be established by either association of the waves with
optical or other observations, or by time-of-flight measurements between separated detectors,
then six suitably oriented detectors suffice to determine all six components. If the direction
cannot be established, then the system is underdetermined, and no unique solution can be
found. However, if one assumes that only transverse waves are present, then there are only three
unknowns if the source direction is known, or five unknowns otherwise. Then the corresponding
number (three or five) of detectors can determine the polarization. If distinct evidence were found
of any mode other than the two transverse quadrupolar modes of GR, the result would be
disastrous for GR. On the other hand, the absence of a breathing mode would not necessarily
rule out scalar-tensor gravity, because the strength of that mode depends on the nature of the
source.

Some of the details of implementing such polarization observations have been worked out for arrays of
resonant cylindrical, disk-shaped, spherical, and truncated icosahedral detectors (TEGP 10.2 [281], for
recent reviews see [169, 266]); initial work has been done to assess whether the ground-based or
space-based laser interferometers (or combinations of the two types) could perform interesting polarization
measurements [267, 47, 177, 117, 273]. Unfortunately for this purpose, the two LIGO observatories (in
Washington and Louisiana states, respectively) have been constructed to have their respective arms as
parallel as possible, apart from the curvature of the Earth; while this maximizes the joint sensitivity
of the two detectors to gravitational waves, it minimizes their ability to detect two modes of
polarization.