Gravity and Strings

288 Gravitational pp-waves
The equations of motion for gµν(x), X µ (ξ), and γ(ξ)are, respectively,
16πG (d)
N
√ |g|
γ 1 2
p
δS
δg µν = Gµν + 8πG(d)
√
|g|
N p
dξ √ γγ −1 gµρ gνσ ˙X ρ ˙X σ δ (d) (x − X) = 0,
gσρ δS
δX ρ = ¨X σ + Ɣρν σ ˙X ρ ˙X ν − d
dξ (ln γ)1 2 ˙X σ = 0,
4γ 3 2
p
δS
δγ = gµν ˙X µ ˙X ν = 0.
(10.33)
Since the particle is massless, it must move at the speed of light (this is the content of the
equation of motion of γ ). If it moves in the direction of the x d−1 ≡ z axis, one can use the
light-cone coordinates u and v defined above.
If it moves in the sense of increasing z at the speed of light, its equation of motion is
U(ξ) = 0. We can set V (ξ) = √ 2 ξ. Thus, our Ansatz for the X µ (ξ) is
U(ξ) = 0, V (ξ) = √ 2ξ, X ≡ (X 1 ,...,X d−2 ) = 0. (10.34)
Agravitational wave moves at the speed of light, and thus our Ansatz for the spacetime
metric is that of a gravitational pp-wave moving in the same direction (i.e. with null Killing
vector ℓµ = δµu so, in particular, nothing depends on v):
ds 2 = 2dudv + 2K (u, xd−2)du 2 − d x 2
d−2 , xd−2 = x 1 ,...,x d−2 . (10.35)
Now we plug our Ansatz into the equation of motion above. First, we immediately see that
the equation for γ is satisfied because ˙X µ = √ 2δ µ v and gvv = 0. The equation of motion
for X µ is also satisfied by taking a constant worldline metric γ = 1 because Ɣvv σ = 0.
Only one equation remains to be solved. On substituting our Ansatz for the coordinates
and γ plus |g|=1 (which holds for the above pp-waves), we find
Gµν + 8πG (d)
N p
dξδµuδνuδ(u)δ(v − √ 2ξ)δ (d−2) (xd−2) = 0. (10.36)
For the pp-wave metric Eq. (10.35) we also have exactly (that is, without using any
property of the metric apart from the light-like character of ℓ µ )
Gµν =−δµuδνu ∂
2
d−2K (u, xd−2). (10.37)
Then, on integrating over ξ and substituting the above result, the Einstein equation reduces
to the following equation for K (u, xd−2):
∂ 2
d−2 K (u, xd−2) =− √ 28πG (d)
N pδ(u)δ(d−2) (xd−2). (10.38)