Elements of gravitational waves

General relativity is a theory of gravity that is consistent with special relativity in many respects, and in particular with the principle that nothing travels faster than light. This means that changes in the gravitational field cannot be felt everywhere instantaneously: they must propagate. In general relativity they propagate at exactly 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, no sharp distinction between the part of the metric that represents the waves and the rest of the metric. Only in certain approximations can we clearly define gravitational radiation. Three interesting approximations in which it is possible to 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 are written and solved in a nearly flat spacetime. The static and wave parts of the field cleanly separate. We idealize gravitational waves as a 'ripple' propagating through 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 time dependent and whose radius of curvature is much larger than the wavelength of the waves. We will describe this in detail in chapter 5. This approximation describes wave propagation well, but it is inadequate for wave generation. The most useful approximation for sources is the post-Newtonian approximation, where waves arise at a high order in corrections that carry general relativity away from its Newtonian limit; we treat these in chapters 6 and 7.

For now we concentrate our attention on linearized theory. We follow the notation and conventions of Misner et al (1973) and Schutz (1985). In

particular we choose units in which c = G = 1; Greek indices run from 0 to 3; Latin indices run from 1 to 3; repeated indices are summed; commas in subscripts or superscripts denote partial derivatives; and semicolons denote covariant derivatives. The metric has positive signature. These above two textbooks and others referred to at the end of these chapters give more details on the theory that we outline here. For an even simpler introduction, based on a scalar analogy to general relativity, see [1].

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