Gravity and Strings

10.3 Sources: the AS shock wave 287
10.2.1 Higher-dimensional pp-waves
Ageneral pp-wave solution of the d-dimensional Einstein–Maxwell theory Eq. (8.217) is
given by [664]
ds 2 = 2du(dv + Kdu) −˜gi j(xd−2)dx i dx j ,
Fui = Ci,
˜∇ 2 K = 1
4 ˜Ci ˜C i ,
˜dC = ˜
d ⋆ C = 0,
˜Ri j = 0,
(10.28)
i.e. C(u, xd−2)idx i is a harmonic 1-form in the Ricci-flat wavefront space and K satisfies
the above differential equation that can be integrated if the Green function for the Laplacian
on the wavefront space is known. If the wavefront space is flat, ˜gi j =−δi j,andwetake the
xd−2-independent harmonic 1-form Ci(u), K is given by
K = H(u, xd−2) + 1
4 CiC i (u)Mij(u)x i x j , Tr(M) = 1, ∂i∂i H = 0. (10.29)
Again, K consists of two terms: the first is a harmonic function on the Euclidean wavefront
space H(u, xd−2). This is the part of K that can be related to singular sources (massless
particles), as we are going to see in the next section. The second term in K describes
the gravitational and electromagnetic background. The solutions with H = 0 and Ci and
Mij constant have, again, Hpp-wave metrics:
ds2 = 2du(dv + Aijx i x jdu) − d x 2
d−2 ,
Fui = Ci, Tr(A) = 1
4CiC i .
(10.30)
One particular case is the KG4 solution Eq. (10.27). Another interesting case is the fivedimensional
Kowalski–Glikman solution KG5 [690], which is also maximally supersymmetric
in N = 1, d = 5 SUGRA [261]:
ds2
= 2du dv + λ2 5
24 (4z2 + x 2 + y2
)du − dx2 − dy2 − dz2 ,
F = λ5du ∧ dz.
10.3 Sources: the AS shock wave
(10.31)
We consider a massless particle moving in d-dimensional curved space coupled to the Einstein
action for the gravitational field. This coupled system is described by the following
action (see Section 7.2, where, in particular, the action for a massless particle Eq. (3.258)
was derived) with c = 1:
1
S =
16πG (d)
N
d d x |g| R − p
2
dξ √ γγ −1 gµν(X) ˙X µ ˙X ν . (10.32)