276
Black holes in string theory
mass and charge (or, more generally, charges). These and their generalization
to five dimensions are discussed in the following section. To get black holes
from string theory, we need to find analogous solutions to the underlying classical
field theory. This is type II supergravity, which is described in section 10.6. It
involves the field strengths of certain (antisymmetric) 'Ramond-Ramond' gauge
fonn-fields. These lead to a generalized notion of electric charge, which. in
turn, indicates the existence of (non-perturbative) extended objects called 'Dbranes' which have non-zero Ramond-Ramond charges. This is described in
section 10.7. In section 10.8 we construct the (five-dimensional, extreme) blackhole solutions, with three charges, that have an event horizon with non-zero area.
If we use this area in the Bekenstein-Hawking fonnula, the entropy of the black
hole is detennined entirely by the (product of the) charges used. The non-zero
charges have an immediate interpretation in terms of underlying microstates and
the counting of these is done in section 10.9. The number of microstates obtained
agrees precisely with that predicted from the calculated Bekenstein-Hawking
entropy.
10.2 Black-hole event horizons
It is convenient to use mass units in which the Planck mass m p and, hence,
Newton's constant GN. are unity: mp = G;I/2 = 1. The most well-known
black-hole solution of general relativity is the Schwarzchild solution which gives
in spherical polar coordinates the line element outside of a spherical body of mass
M:
&2 == glLII dx lL dx ll
(10.1 )
2M) 2 (
2
2 2
2M)-1
= ( 1--;- dt - 1--;dr -r d0 2
(10.2)
where
dO~ == d0 2 + sin 2 8 dt/J2
(10.3)
is the line element on the unit two-sphere S2. The metric is singular at r = 2M
but this is merely a coordinate singularity. For example, a particle on a radial
timelike geodesic r = R(t) falls from its starting position at r = R(O) > 2M,
through R = 2M. and reaches R = 0 in a finite proper time (exercise 1). On a
radial null geodesic,
dt2 = (I _ 2~)-2 dr2 == (dr",)2
(l0.4)
where
rIll == r + In I r - 2M I
(10.5)
2M .
Black holes in string theory
mass and charge (or, more generally, charges). These and their generalization
to five dimensions are discussed in the following section. To get black holes
from string theory, we need to find analogous solutions to the underlying classical
field theory. This is type II supergravity, which is described in section 10.6. It
involves the field strengths of certain (antisymmetric) 'Ramond-Ramond' gauge
fonn-fields. These lead to a generalized notion of electric charge, which. in
turn, indicates the existence of (non-perturbative) extended objects called 'Dbranes' which have non-zero Ramond-Ramond charges. This is described in
section 10.7. In section 10.8 we construct the (five-dimensional, extreme) blackhole solutions, with three charges, that have an event horizon with non-zero area.
If we use this area in the Bekenstein-Hawking fonnula, the entropy of the black
hole is detennined entirely by the (product of the) charges used. The non-zero
charges have an immediate interpretation in terms of underlying microstates and
the counting of these is done in section 10.9. The number of microstates obtained
agrees precisely with that predicted from the calculated Bekenstein-Hawking
entropy.
10.2 Black-hole event horizons
It is convenient to use mass units in which the Planck mass m p and, hence,
Newton's constant GN. are unity: mp = G;I/2 = 1. The most well-known
black-hole solution of general relativity is the Schwarzchild solution which gives
in spherical polar coordinates the line element outside of a spherical body of mass
M:
&2 == glLII dx lL dx ll
(10.1 )
2M) 2 (
2
2 2
2M)-1
= ( 1--;- dt - 1--;dr -r d0 2
(10.2)
where
dO~ == d0 2 + sin 2 8 dt/J2
(10.3)
is the line element on the unit two-sphere S2. The metric is singular at r = 2M
but this is merely a coordinate singularity. For example, a particle on a radial
timelike geodesic r = R(t) falls from its starting position at r = R(O) > 2M,
through R = 2M. and reaches R = 0 in a finite proper time (exercise 1). On a
radial null geodesic,
dt2 = (I _ 2~)-2 dr2 == (dr",)2
(l0.4)
where
rIll == r + In I r - 2M I
(10.5)
2M .