String Theory and M-Theory

568 Black holes in string theory
The five-dimensional metric describing this black-hole can be obtained
from the ten-dimensional type IIB theory by wrapping the corresponding
branes as described above, or it can be constructed directly. In either case,
the resulting metric can be written in Einstein frame in the form
ds 2 = −λ −2/3 dt 2 + λ 1/3 dr 2 + r 2 dΩ 2 3 , (11.46)
where
λ =
3
i=1
1 +
ri
2
. (11.47)
r
The relation between the parameters ri and the charges Qi is derived below.
This solution describes an extremal three-charge black hole with a vanishing
temperature T = 0. Note that this formula reduces to the extremal
Reissner–Nordström black-hole metric given in Eq. (11.32) in the special
case r1 = r2 = r3, that is, when the three charges are equal. The dilaton
is a constant, so there is a globally well defined string coupling constant gs.
Thus, the string-frame metric differs from the one given above only by a
constant factor.
The horizon of the black hole in Eq. (11.46) is located at r = 0, and its
area is
A = 2π 2 r1r2r3. (11.48)
This vanishes when any of the three charges vanishes, which is the reason
that three charges have been considered in the first place. Put differently,
one needs to break 7/8 of the supersymmetry in order to form a horizon
that has finite area in the supergravity approximation, and this requires
introducing three different kinds of excitations. When there is only one or
two nonzero charges, there still is a horizon of finite area, but its dependence
on the string scale is such that its area vanishes in the supergravity
approximation. For the supergravity approximation to string theory to be
valid, it is necessary that the geometry is slowly varying at the string scale.
This requires ri ≫ ℓs.
The black hole mass
Using Eq. (11.11) one can read off the mass of the black hole M to be
M = M1 + M2 + M3 where, Mi = πr2 i
4G5
. (11.49)
The fact that the masses are additive in this way is a consequence of the
form of the metric. However, this had to be the case, because the BPS
condition is satisfied, and the charges are additive.