Type II super gravity
293
There is. thus. no binding energy between the individual black holes: the
gravitational binding is precisely cancelled by the electrostatic repulsion. Near
the horizon Ipl « Q, the metric is approximated by AdS2 x S2 (exercise 11).
Thus, the extreme black hole may also be regarded as a soli ton that interpolates
between the Minkowski vacuum when p » Q and AdS2 x S2 when Ipl « Q.
Generalizations of this special form. involving extra dimensions. extended objects
and other charges, are important in what follows.
In particular, the simplest example of a string-theory black hole for which a
microscopic description can be found is provided by a five-dimensional analogue
of the RN solution (10.33) and (10.34). The static, solution of the fivedimensional Einstein-Maxwell equations outside of a (S3)-spherically symmetric
body of mass M and charge Q is
2
2M
Q
2
2M
Q
2
2
2
ds = ( I - - + - 2) dt - ( I - - + - 2)-1 dr - r dS'2 3 (10.99)
r2
r4
r2
r4
and the gauge potential I-form
Q
A = l'dt.
(10.100)
r
(The ,-2 dependence of the potential arises. of course. because there are now
four spatial dimensions.) As before. the extreme case where M = Q. with the
horizon at r = ../Q. may be cast in an isotropic form by using the coordinate
p == ./r2 - Q. Then the horizon is at p = 0 and
ds 2 = H- 2 dt 2 - H(dp2 + p2dO~)
(10.101)
A = (I - H-1)dl
(10.102)
where now the harmonic function is
H= I+.fl.
(10.103)
p
2 .
To connect this black hole with string theory. we need to consider the
effective field theory describing string theory in the low-energy limit. This is
a generalization of Einstein-Maxwell theory both with respect to the number of
spacetime dimensions (ten) and the fields involved. The fields involved are just
the massless modes that arise in (perturbative) string theory and the field theory
that desribes them is type II supergravity.
10.6 Type 11 supergravity
The massless states that arise in superstring theory include both bosons and
fermions. Black holes are, of course. solutions of the classical bosonic field
293
There is. thus. no binding energy between the individual black holes: the
gravitational binding is precisely cancelled by the electrostatic repulsion. Near
the horizon Ipl « Q, the metric is approximated by AdS2 x S2 (exercise 11).
Thus, the extreme black hole may also be regarded as a soli ton that interpolates
between the Minkowski vacuum when p » Q and AdS2 x S2 when Ipl « Q.
Generalizations of this special form. involving extra dimensions. extended objects
and other charges, are important in what follows.
In particular, the simplest example of a string-theory black hole for which a
microscopic description can be found is provided by a five-dimensional analogue
of the RN solution (10.33) and (10.34). The static, solution of the fivedimensional Einstein-Maxwell equations outside of a (S3)-spherically symmetric
body of mass M and charge Q is
2
2M
Q
2
2M
Q
2
2
2
ds = ( I - - + - 2) dt - ( I - - + - 2)-1 dr - r dS'2 3 (10.99)
r2
r4
r2
r4
and the gauge potential I-form
Q
A = l'dt.
(10.100)
r
(The ,-2 dependence of the potential arises. of course. because there are now
four spatial dimensions.) As before. the extreme case where M = Q. with the
horizon at r = ../Q. may be cast in an isotropic form by using the coordinate
p == ./r2 - Q. Then the horizon is at p = 0 and
ds 2 = H- 2 dt 2 - H(dp2 + p2dO~)
(10.101)
A = (I - H-1)dl
(10.102)
where now the harmonic function is
H= I+.fl.
(10.103)
p
2 .
To connect this black hole with string theory. we need to consider the
effective field theory describing string theory in the low-energy limit. This is
a generalization of Einstein-Maxwell theory both with respect to the number of
spacetime dimensions (ten) and the fields involved. The fields involved are just
the massless modes that arise in (perturbative) string theory and the field theory
that desribes them is type II supergravity.
10.6 Type 11 supergravity
The massless states that arise in superstring theory include both bosons and
fermions. Black holes are, of course. solutions of the classical bosonic field