Chapter 4: The Tensor Modes Curved Space–Times

As established in the two previous chapters, in general relativity the gravitational waves are associated with a symmetric, solenoidal and traceless rank-two tensor in three (spatial) dimensions. When the gravity theory does not coincide with general relativity, up to four supplementary polarizations may arise in flat space–time and this happens, for instance, for the non-Einsteinian theories of gravity (e.g. scalar–vector, scalar–tensor, scalar–vector–tensor). If the Minkowski space–time is a solution of the field equations in the absence of matter sources, it is often plausible to ignore the fluctuations of the energy–momentum tensor but such a simplifying assumption is only justified far from the matter sources that are ultimately responsible for the curvature of the underlying space–time. The inhomogeneities of the matter sources in curved backgrounds entail the presence of supplementary polarizations not only in non-Einsteinian theories (as we saw in Chapter 2) but also in general relativity. The approach pioneered by Lifshitz [1], Grishchuk [2] and others considers the propagation of the tensor modes within a covariant description where the tensor indices are all kept four-dimensional but a complementary strategy, particularly useful in cosmological backgrounds, is to decompose the fluctuations of the metric in a non-covariant language by explicitly separating, form the very beginning, the spatial from the temporal components of the perturbed metric [3]. As in the case of non-Einsteinian theories of gravity in flat space–time (see Chapter 2), in four-dimensional curved backgrounds the 10 independent components of the perturbed metric can be reduced to six either by selecting a specific coordinate system or by defining an appropriate set of gauge-invariant fluctuations, as suggested in the context of the Bardeen approach [3]…