10 - Gravitational pp-waves

Summary

As we saw in Part I, the weak-field limit of GR is just a relativistic field theory of a massless spin-2 particle propagating in Minkowski spacetime. In the absence of sources, by choosing the De Donder gauge Eq. (3.100), it can be shown that the gravitational field h_μν satisfies the wave equation (3.101) and, correspondingly, there are wave-like solutions of the weak-field equations like the one we found in Section 3.2.3 associated with a massless pointparticle moving at the speed of light.

GR is, however, a highly non-linear theory and it is natural to wonder whether there are exact wave-like solutions of the full Einstein equations. The answer is definitely yes and in this chapter we are going to study some of them, the so-called pp-waves, which are especially interesting for us. In particular we are going to see that the linear solution we found in Section 3.2.3 is an exact solution of the full Einstein equations that has the same interpretation. We will use this solution many times in what follows to describe the gravitational field of Kaluza–Klein momentum modes, for instance.

pp-Waves

pp-waves (shorthand for plane-fronted waves with parallel rays) are metrics that, by definition, admit a covariantly constant null Killing vector field ℓ_μ:

The first spacetimes with this property were discovered by Brinkmann in [193]. To describe pp-wave metrics, we define light-cone coordinates u and ν in terms of the usual Cartesian coordinates

which are related to the null Killing vector by

i.e. ν is the coordinate we can make the metric independent of, the only non-vanishing components of ℓ are ℓ_u = ℓ^ν = 1, and the metric describes a gravitational wave propagating in the positive…