10.7 Gravitational waves and the compass of inertia 255
compensating effect makes the spin contribution to the clock effect equal to zero;
this case therefore appears indistinguishable from that of spinless particles.
10.7 Gravitational waves and the compass of inertia
The existence of gravitational waves as predicted by general relativity has been
indirectly ascertained by observation of the binary pulsar PSR 1913+16 (Taylor
and Weisberg, 1989). A large body of literature is now available on the properties
of gravitational waves, and ways to detect them. Hence we shall confine our
attention to a particular aspect of this problem which has been considered only
recently, namely the dragging of inertial frames by a gravitational wave (Bini and
de Felice, 2000; Sorge, Bini, and de Felice, 2001; Biˇcak, Katz, and Lynden-Bell,
2008). We then ask the question: what observable effects might be produced by
a plane gravitational wave acting on a test gyroscope? It is well known that
in the absence of significant coupling between the background curvature and the
multipole moments of the energy-momentum tensor of an extended body, its spin
vector is Fermi-Walker transported along its own trajectory (see de Felice and
Clarke, 1990, and references therein); for measurable effects in a gravitational
wave background, see also (Cerdonio, Prodi, and Vitale, 1988; Mashhoon, Paik,
and Will, 1989; Krori, Chaudhury, and Mahanta, 1990; Fortini and Ortolan, 1992;
Herrera, Paiva, and Santos, 2000).
The effects of a plane gravitational wave on a frame which is not Fermi-Walker
transported are best appreciated by studying the precession of a gyroscope at
rest in that frame (de Felice, 1991). The task, then, is to find a frame which
is not Fermi-Walker transported and is also operationally well-defined so that,
monitoring the precession of a gyroscope with respect to that frame, we can study
the dragging induced on it by a plane gravitational wave. Our main purposes are:
(i) to establish the existence of a relativistic effect which is measurable, namely
the gyroscopic precession induced by a plane gravitational wave;
(ii) to show that a frame can be selected with respect to which the precession is
due to one polarization state only.
In the latter case we will have constructed a gravitational polarimeter.
The metric of a plane monochromatic gravitational wave, elliptically polarized
and propagating along a direction which we fix as the x coordinate direction,
can be written in transverse-traceless (TT) gauge, as in Chapter 8, Eq. (8.195).
The time-like geodesics of this metric, deduced in de Felice (1979), are described
by (8.196).
Test gyroscopes in motion along a geodesic
We consider a test gyroscope moving along a geodesic described by a tangent
vector field U = U
(g)
given by (8.196). The spin vector S(U ) satisfies the equation