Gravity and Strings

286 Gravitational pp-waves
The specific properties of each pp-wave solution depend on the form of the function K . 3
K has two different terms. The first is independent of the electromagnetic field; only the
second depends on it. The first term (the real part of the analytic f (u,ξ))isjust any harmonic
function H(u, x2) in the wavefront Euclidean two-dimensional space and it provides
a purely gravitational solution. It represents a sort of perturbation of the electromagnetic
and gravitational background described by the second term of K .
A particularly interesting type of pp-waves is shock or impulse waves with the first term
of K given by
K (u,ξ,¯ξ)= δ(u)K (ξ, ¯ξ). (10.25)
An example of a gravitational shock wave is provided by the purely gravitational
Aichelburg–Sexl solution [24]
K = H(u, x2) = δ(u) ln |ξ|, (10.26)
which describes the gravitational field of a massive point-like particle boosted to the speed
of light. In [24] this metric was obtained by performing an infinite boost in the direction z
to a Schwarzschild black hole. This method for generating impulsive waves also works in
(anti-)de Sitter spacetimes [565] using the Schwarzschild–(anti-)de Sitter solution and has
also been applied to the Kerr–Newman solution [73, 74, 385, 661] and to Weyl’s axisymmetric
vacuum solutions.[775]. 4 However, in Section 10.3 we will identify d-dimensional
Aichelburg–Sexl-type (AS) shock waves as the gravitational field produced by a massless
particle moving at the speed of light, checking explicitly that (AS) shock waves satisfy the
equations of motion of Einstein’s action coupled to a massless particle.
This interpretation will later turn out to be very useful. In Chapter 11 we will be interested
in the gravitational field produced by massless particles moving at the speed of light in
compact dimensions. These particles appear as massive and charged in the non-compact
dimensions and their gravitational field (a charged extreme black hole) can be derived from
the massless-particle gravitational field. Then, we will simply have to adapt the AS shockwave
solution to a spacetime with compact dimensions.
Another example, this time with the first term of K vanishing, is provided by a solution
with Hpp-wave-type metrics (10.18). A particular case is the four-dimensional Kowalski–
Glikman solution KG4 [637],
ds2 = 2du(dv + 1
8λ2 |x2| 2du) − d x 2
Fu1 = λ,
2 ,
(10.27)
which is a maximally supersymmetric solution of the d = 4 Einstein–Maxwell theory that
is the Penrose limit of the RB solution. We will study the (super)symmetries of these vacua
in Chapter 13.
Before studying shock wave sources, we consider the higher-dimensional generalization
of the pp-wave solutions Eq. (10.24).
3 A detailed classification and description of metrics of this kind that are solutions of the Einstein–Maxwell
equations can be found in [640].
4 For further results and references on impulse waves see e.g. [774, 865].