Polarization of gravitational waves

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, $R 0i0j$ , via the equation of geodesic deviation, given, for a pair of freely falling particles by $i j ¨x = − R0i0jx$ , where $i x$ 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 $h+$ and $h×$ discussed earlier (note the $cos 2φ$ and $sin 2φ$ 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 $(λ∕λC )2$ , where $λ$ is the wavelength of the radiation, and $λC$ 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 ( $R0i0j$ ). 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.

Figure 8:

The six polarization modes for gravitational waves permitted in any metric theory of gravity. Shown is the displacement that each mode induces on a ring of test particles. The wave propagates in the +z direction. There is no displacement out of the plane of the picture. In (a), (b), and (c), the wave propagates out of the plane; in (d), (e), and (f), the wave propagates in the plane. In GR, only (a) and (b) are present; in massless scalar-tensor gravity, (c) may also be present.

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.