Gravity and Strings

20.2 Black holes from branes 579
simpler alternative to the original configuration proposed by Strominger and Vafa in [870],
and the T-dual configuration W D2 D6 proposed in [680]. We are going to study the
former.
In d = 4, a configuration that fits the requirements, W D2 S5A D6, was also proposed
in [680] and two T-dual alternatives D0 D4 and D1 D5 plus an F1 and a KK
monopole were proposed in [606]. We are going to study the configuration proposed in
[680].
We will not study rotating BHs, although we have mentioned them in several places, remarking,
in particular, on the non-existence of supersymmetric rotating BHs with a regular
horizon in d = 4. It is interesting to mention their existence in d = 5, where they can also
be modeled with string-theory extended objects [180, 183].
d = 5 Black holes from intersecting branes. To obtain regular extreme d = 5BHs,itis
necessary to construct them as intersections of at least three d = 10 extended objects. A
possible configuration that leads to regular extreme d = 5BHsis
0 1 2 3 4 5 6 7 8 9
D1 + + ∼∼∼∼−−−−
D5 + + ++++−−−−
W + →∼∼∼∼−−−−
(20.13)
where + signs stand for worldvolume dimensions (isometric in the solutions), − signs
stand for overall transverse directions on which the solution depends, ∼ signs stand for
transverse directions in which the solution has been smeared, and → indicates the direction
in which the wave propagates. The direction with +, ∼, or→ signs will be compactified
on a T 5 = S 1 × T 4 with volume V 5 = 2π R1V 4 , where R1 is the radius of the coordinate y 1
and V 4 = (2π) 4 R2 ···R5 is the volume of the T 4 on which the coordinates y 2 ,...,y 5 are
compactified. Then the solution will depend only on the overall transverse coordinates x4.
Our procedure will be to construct first the d = 10 solution of N = 2B, d = 10 SUEGRA
that describes this system and then reduce it to d = 5intwo steps (S 1 and T 4 ). Since we
will also be interested in non-extremal BHs, we are going to construct the black intersecting
solution first and then we will take the extreme limit.
The harmonic-superposition rule cannot be used directly in this black intersection. We
start from the non-extreme D1 D5 intersection (contained in Eq. (19.150):
d ˆs 2 s = H − 1 2
D1 H − 1 2
D5 [Wdt2 − (dy 1 ) 2 ] − H 1 2
D1 H 1 2
D5 [W −1 dr 2 + r 2 d 2 (3) ]
−H 1 2
D1H − 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.
2 ,
(20.14)