Gravity and Strings

10.3 Sources: the AS shock wave 289
In Chapter 3, Section 3.2.3, we found precisely the same equation and it has the same
solution, Eqs. (3.133) and (3.134). Thus, we have found the solution
K (u, xd−2) =
ds 2 = 2dudv + 2K (u, xd−2)du 2 − d x 2
√ 2 p8πG (d)
N
(d − 4)ω(d−3)
1
d−2 ,
δ(u), d ≥ 5,
|xd−2|
d−4
K (u, x2) =− √ 2 p4G (4)
N ln |x2| δ(u), d = 4.
(10.39)
The d = 4 solution is the AS shock wave found in [24]. Observe that this solution is exactly
the same as that which we obtained in Section 3.2.3 by solving the linear-order theory.
There are no higher-order corrections to the first-order solution which is not renormalized.
This is due to the special structure of the linear solution and can be related to supersymmetry
as well.
There is another useful way to rewrite the pp-wave metrics that we have found. Defining
the function
H ≡ 1 − K, (10.40)
the solution takes the form
ds2 = H −1dt2 − H dz − α(H −1 − 1)dt 2 2 − d x d−2 , α =±1,
√ (d)
2 p8πG N 1 1√2
H = 1 −
δ (t − αz) , d ≥ 5,
(d − 4)ω(d−3) |xd−2|
d−4
H = 1 + √ 2 p4G (4)
N ln |x2|
1√2
δ (t − αz) , d = 4,
(10.41)
where we have introduced the constant α =±1totake care of the two possible directions
of propagation toward z = α∞.
Had we tried to solve the vacuum Einstein equations with the Ansatz Eq. (10.35), we
would have arrived at the conclusion that any function K (or H) harmonic in (d − 2)dimensional
Euclidean space transverse to z provides a solution. Thus, we obtain a family
of pp-wave solutions of the form
ds2 = H −1dt2 − H[dz − α(H −1 − 1)dt] 2 − d x 2
d−2 ,
∂ 2
(d−2) H = 0, α =±1.
(10.42)