Counting the microstates
303
Bekenstein-Hawking fonnula (10.75), the entropy Sbh associated with this
solution is
AH
Sbh = - - = 2lr Jq\qsn
(10.149)
4GIO
independently of the size of the compact dimensions and of the string coupling
8s. Thus, the entropy is detennined entirely by the integers q\, qS and n and it is
this feature that allows the identification of the associated microstates. The fact
that the entropy of a black hole at zero temperature is non-zero does not imply
a violation of the third law of thennodymamics, if indeed the analogy between
black-hole dynamics and thennodynamics is exact. The version of the third law
that suggests that there is, is a statement about the equations of state for ordinary
matter.
10.9 Counting the microstates
We have already argued that D p-branes are sources of R-R charge and we have
also shown in (10.135) that the soli ton solution (10.129) has q p units of p-brane
R-R charge. Thus, the obvious interpretation of the black-hole solution that we
have just constructed is that it is a bound state made out of q5 D5-branes and
ql DI-branes (D-strings) with some momentum nj R. However, Dp-branes are
defined as pointlike objects in their transverse dimensions in an otherwise flat
spacetime. The R-R solitons that we have derived are only asymptotically fiat, so
why do we believe that they are made of D-branes? We have noted previously that
the effective action from which these solitons derive is a good approximation so
long as the curvature is small lR( G)a' « I. The length scale defined in (10.131),
associated with the solution (10.129), is given by Lp '" (gsqp)1/(7- p )..j(ii. Thus,
when gsq p > I, the curvature is small and the soliton solutions are valid. In
fact, the effective supergravity equations were derived using string perturbation
theory, which is valid only when gs < l. Consequently the soliton solutions
apply only when q p is large, so that the curvature is small. When the string
coupling gs is very small, the R-R soli tons are very massive, as is apparent from
(10.134). However, their gravitational field is proportional to G IOM p and, since
G IO ex g;, the associated spacetime becomes flat as gs -+ O. Also, the horizon
area AH = 4G IOSbh approaches zero in this limit. When it is smaller than the
string scale I; = 4lr 2 a'. the higher-order curvature tenns become important
and the Hat space description becomes valid. Provided that gsql « I and that
gsqS « I, we are considering weakly-coupled D-branes in a flat spacetime. In
this case, it is straightforward to count the number of configurations. We shall
return shortly to the case where gsqp > I in which our black-hole solutions are
valid.
The configuration in question (10.145) breaks the IO-dimensional Lorentz
symmetry SO(I, 9) -+ SO(l, 1) x SO(4)11 x SO(4)1. The first factor acts on
the D-string world sheet (xo, x I), the second factor on the rest of the D5-brane
303
Bekenstein-Hawking fonnula (10.75), the entropy Sbh associated with this
solution is
AH
Sbh = - - = 2lr Jq\qsn
(10.149)
4GIO
independently of the size of the compact dimensions and of the string coupling
8s. Thus, the entropy is detennined entirely by the integers q\, qS and n and it is
this feature that allows the identification of the associated microstates. The fact
that the entropy of a black hole at zero temperature is non-zero does not imply
a violation of the third law of thennodymamics, if indeed the analogy between
black-hole dynamics and thennodynamics is exact. The version of the third law
that suggests that there is, is a statement about the equations of state for ordinary
matter.
10.9 Counting the microstates
We have already argued that D p-branes are sources of R-R charge and we have
also shown in (10.135) that the soli ton solution (10.129) has q p units of p-brane
R-R charge. Thus, the obvious interpretation of the black-hole solution that we
have just constructed is that it is a bound state made out of q5 D5-branes and
ql DI-branes (D-strings) with some momentum nj R. However, Dp-branes are
defined as pointlike objects in their transverse dimensions in an otherwise flat
spacetime. The R-R solitons that we have derived are only asymptotically fiat, so
why do we believe that they are made of D-branes? We have noted previously that
the effective action from which these solitons derive is a good approximation so
long as the curvature is small lR( G)a' « I. The length scale defined in (10.131),
associated with the solution (10.129), is given by Lp '" (gsqp)1/(7- p )..j(ii. Thus,
when gsq p > I, the curvature is small and the soliton solutions are valid. In
fact, the effective supergravity equations were derived using string perturbation
theory, which is valid only when gs < l. Consequently the soliton solutions
apply only when q p is large, so that the curvature is small. When the string
coupling gs is very small, the R-R soli tons are very massive, as is apparent from
(10.134). However, their gravitational field is proportional to G IOM p and, since
G IO ex g;, the associated spacetime becomes flat as gs -+ O. Also, the horizon
area AH = 4G IOSbh approaches zero in this limit. When it is smaller than the
string scale I; = 4lr 2 a'. the higher-order curvature tenns become important
and the Hat space description becomes valid. Provided that gsql « I and that
gsqS « I, we are considering weakly-coupled D-branes in a flat spacetime. In
this case, it is straightforward to count the number of configurations. We shall
return shortly to the case where gsqp > I in which our black-hole solutions are
valid.
The configuration in question (10.145) breaks the IO-dimensional Lorentz
symmetry SO(I, 9) -+ SO(l, 1) x SO(4)11 x SO(4)1. The first factor acts on
the D-string world sheet (xo, x I), the second factor on the rest of the D5-brane