String Theory and M-Theory

572 Black holes in string theory
M-theory interpretation
Starting from the Type IIA configuration described above, one can carry out
two more T-duality transformations along the y 2 and y 3 directions to obtain
a type IIA configuration consisting of Q1 D2-branes wrapped on y 2 and y 3 ,
Q5 D2-branes wrapped on y 4 and y 5 and n fundamental strings wrapped on
y 1 . This configuration can be interpreted at strong coupling as M-theory
compactified on a 6-torus. Calling the M-theory circle coordinate y 6 , the n
fundamental type IIA strings are then identified as n M2-branes wrapped
on the y 1 and y 6 circles. The two sets of D2-branes are then identified as
sets of M2-branes. Altogether, there are three sets of M2-branes wrapped
on three orthogonal tori. This is a satisfying picture in that it puts the
three sources of charges on a symmetrical footing, which nicely accounts for
their symmetrical appearance in the entropy formula. The verification that
this brane configuration gives the same entropy as before is a homework
problem.
Nonextremal three-charge black holes for D = 5
The extremal three-charge black-hole solutions in five dimensions given
above have nonextremal generalizations, which describe nonsupersymmetric
black holes with finite temperature. These black holes are described by the
metric
where
and
λ =
3
i=1
1 +
ds 2 = −h λ −2/3 dt 2 + λ 1/3
dr2 h + r2dΩ 2
3 , (11.60)
ri
2
r
h = 1 − r2 0
r 2
(11.61)
with r 2 i = r 2 0 sinh 2 αi, i = 1, 2, 3. (11.62)
This reduces to the extremal metric in Eq. (11.46) in the limit r0 → 0
with ri held fixed. Moreover, the limit αi → 0 with r0 held fixed gives the
Schwarzschild metric in five dimensions given in Eq. (11.9).
The mass of this black hole can be read off using the same rules as before
resulting in
M = πr2 0
(cosh 2α1 + cosh 2α2 + cosh 2α3) . (11.63)
8G5