Gravity and Strings

10
Gravitational pp-waves
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 hµν satisfies
the wave equation (3.101) and, correspondingly, there are wave-like solutions of the weakfield
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.
10.1 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 ℓµ:
∇µℓν = 0, ℓ 2 = ℓµℓ µ = 0. (10.1)
The first spacetimes with this property were discovered by Brinkmann in [193]. To describe
pp-wave metrics, we define light-cone coordinates u and v in terms of the usual
Cartesian coordinates
u = 1
√ (t − z), v =
2 1
√ (t + z), (10.2)
2
which are related to the null Killing vector by
ℓµ = ∂µu, ℓ µ ∂µv = 1, (10.3)
i.e. v is the coordinate we can make the metric independent of, the only non-vanishing
components of ℓ are ℓu = ℓ v = 1, and the metric describes a gravitational wave propagating
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