(ed.). Gravitational waves (IOP, 2001)(422s).

Chapter 2Elements of gravitational wavesGeneral relativity is a theory of gravity that is consistent with special relativity inmany respects, and in particular with the principle that nothing travels faster thanlight. This means that changes in the gravitational field cannot be felt everywhereinstantaneously: they must propagate. In general relativity they propagate atexactly the same speed as vacuum electromagnetic waves: the speed of light.These propagating changes are called gravitational waves.However, general relativity is a nonlinear theory and there is, in general, nosharp distinction between the part of the metric that represents the waves andthe rest of the metric. Only in certain approximations can we clearly definegravitational radiation. Three interesting approximations in which it is possibleto make this distinction are:• linearized theory;• small perturbations of a smooth, time-independent background metric;• post-Newtonian theory.The simplest starting point for our discussion is certainly linearized theory,which is a weak-field approximation to general relativity, where the equations arewritten and solved in a nearly flat spacetime. The static and wave parts of thefield cleanly separate. We idealize gravitational waves as a ‘ripple’ propagatingthrough a flat and empty universe.This picture is a simple case of the more general ‘short-wave approximation’,in which waves appear as small perturbations of a smooth background that is timedependent and whose radius of curvature is much larger than the wavelength of thewaves. We will describe this in detail in chapter 5. This approximation describeswave propagation well, but it is inadequate for wave generation. The most usefulapproximation for sources is the post-Newtonian approximation, where wavesarise at a high order in corrections that carry general relativity away from itsNewtonian limit; we treat these in chapters 6 and 7.For now we concentrate our attention on linearized theory. We followthe notation and conventions of Misner et al (1973) and Schutz (1985). In15