Gravity and Strings

580 String black holes in four and five dimensions
Now, to add a wave, we use the procedure studied on page 327 and boost the above
solution in the direction y 1 , obtaining [208, 550]
d ˆs 2 s = H − 1 2
D1 H − 1
2 −1
D5 HW Wdt2 − HW[dy1 + αW(H −1
W − 1)dt]2
− H 1 2
D1 H 1 2
D5 [W −1 dr 2 + r 2 d 2 (3) ]
− H 1 2
D1 H − 1 2
Ĉ (2) ty 1 = αD1e − ˆφ0
D5 [(dy2 ) 2 + (dy3 ) 2 + (dy4 ) 2 + (dy5 ) 2 ],
−1
HD1 − 1 , Ĉ (6) ty1 ···y5 = αD5e− ˆφ0 −1
HD5 − 1 ,
e−2 ˆφ −2 ˆφ0 = e HD5/HD1, Hi = 1 + r 2 i
,
r 2
ω
W = 1 +
r
ω = r 2 i (1 − α2 i ), i = D1, D5, W.
2 ,
(20.15)
In the extreme limit ω = 0werecover a D1 D5 W that could have been constructed
using the harmonic-superposition rule.
Let us now dimensionally reduce this solution in the direction y 1 .Inthis reduction a
modulus field that measures the size of that direction arises, 3
k1
k10
=|ˆg y 1 y 1| 1 2 = H 1 2
W
H 1 4
D1 H 1 4
D5
, (20.16)
and k10 = R1/ℓs. Inthe reduction on T 4 we obtain another modulus field associated with
its volume, 4
k V 4 =|ˆg y 2 y 2 ˆg y 3 y 3 ˆg y 4 y 4 ˆg y 5 y 5| 1 2 = HD1/HD5. (20.17)
The d = 5 dilaton is given by
e −2φ = e −2 ˆφ k1 = e −2φ0
H 1 2
W
H 1 4
D1 H 1 4
D5
, e −2φ0 = e −2 ˆφ0 k1,0. (20.18)
3 Before reducing, we can rescale y 1 so that it takes values in [0, 2πℓs]. We will do the same systematically
in common worldvolume directions, but not in relative transverse directions, since, in order to apply
Eqs. (11.124), the coordinates have to take values in [0, 2π R]. The value at infinity of the corresponding
modulus is 1.
4 All we are doing here is a standard toroidal compactification of the kind we have performed in Section 16.5
and studied in general in Section 11.4. In general we would obtain a bunch of moduli fields coming from the
internal metric. For this and the solutions that will follow, the internal metric is proportional to the identity
and there is only one non-trivial modulus: its determinant. Its square root is kv. Wehave not performed the
toroidal reduction of RR fields, but it is clear that they give rise to a series of form potentials of equal and
lower ranks. Since the potentials in this and the other solutions that we are going to study have components
only in compact directions (plus time) only the time component of the vector fields that originate from the
reduction will be non-trivial and have the obvious value.