300
Black holes in string theory
As expected of D p-branes, the solitons we have found are non-perturbative
objects whose mass diverges as gs -+ O. However, the fact that the mass scales as
g; I rather than 8;2 shows that they are unconventional solitons, quite unlike the
sphalerons whose action is given in (4.172), for example. The electric charge Q p
associated with the C(p+l) form gauge potential is given in (10.123):
1 1.
~ ( r;)3- P
Qp = - 2
F(p+2) = qp- 2if'Va' .
(10.135)
2K 10 Ss-p
KIO
Thus, the soli tons have qp units of the fundamental Dp-brane R-R 'electric'
charge IIp == (21r)-P(a')-(p+I)/2/8s ' Further, Mp = Qp, so, as claimed, these
are extreme states. There is an exact cancellation between the attractive forces
from NS-NS fields due to the mass of the soliton and the repulsive Coulomb like
electrical forces from the R-R fields due to its charge. As in the four-dimensional
case, this signals the fact that one-half of the supersymmetry charges are preserved
by the solution, which, in this case, is invariant under the 16 supersymmetry
charges
Qa+ PQa
(10.136)
where Q and Q act, respectively, on the right- and left-movers and P represents
the operator that reflects the directions y' transverse to the brane.
The first question then is: do any of these solutions give us the metric of
a black hole? If, with the benefit of hindsight, we try to obtain the extreme
five-dimensional RN solution given in (10.101), (10.102) and (10.103), then the
obvious first attempt is to choose p = 5. (We are, therefore, considering type lIB
superstriog theory, since p is odd.) This gives four (transverse) spatial dimensions
y6, y7, y8, y9,
r2 == (y6)2 + (y 7)2 + (y8)2 + (y9)2
(10.137)
and the harmonic function Hs(r) varies as r- 2 • As previously noted, if the five
longitudinal coordinates x ) , x 2 • x 3 , x4, x S are compactified (on as-torus TS say),
the metric resembles the five-dimensional RN black hole (10.101). However, the
warp factors in both the longitudinal and transverse directions are wrong and,
further, the dilaton is singular on the event horizon at r = rH = O. (The
quantum states associated with a soli ton are found by identifying the zero modes
(or collective coordinates) and quantizing them. This is not possible if the soliton
is singular.) Now, because the solitons are extreme, we may combine solutions
with different values of p, provided that (some) supersymmetry is preserved.
This requires that some of the supersymmetry charges (10.136) preserved by one
solution are also preserved by the other. Thus, if PI and P2 represent, respectively,
reflection of the coordinates transverse to the PI- and P2-solitons, the preserved
supersymmetry charges satisfy
-
-
- 1
Qa + PIQa = Qa + l'2Qa = Qa + p)(P I P2)Qa
(10.138)
and we see that the unbroken supersymmetries correspond to + I eigenvalues of
p l
l 1'2. In general, this requires that the number of directions that are transverse
Black holes in string theory
As expected of D p-branes, the solitons we have found are non-perturbative
objects whose mass diverges as gs -+ O. However, the fact that the mass scales as
g; I rather than 8;2 shows that they are unconventional solitons, quite unlike the
sphalerons whose action is given in (4.172), for example. The electric charge Q p
associated with the C(p+l) form gauge potential is given in (10.123):
1 1.
~ ( r;)3- P
Qp = - 2
F(p+2) = qp- 2if'Va' .
(10.135)
2K 10 Ss-p
KIO
Thus, the soli tons have qp units of the fundamental Dp-brane R-R 'electric'
charge IIp == (21r)-P(a')-(p+I)/2/8s ' Further, Mp = Qp, so, as claimed, these
are extreme states. There is an exact cancellation between the attractive forces
from NS-NS fields due to the mass of the soliton and the repulsive Coulomb like
electrical forces from the R-R fields due to its charge. As in the four-dimensional
case, this signals the fact that one-half of the supersymmetry charges are preserved
by the solution, which, in this case, is invariant under the 16 supersymmetry
charges
Qa+ PQa
(10.136)
where Q and Q act, respectively, on the right- and left-movers and P represents
the operator that reflects the directions y' transverse to the brane.
The first question then is: do any of these solutions give us the metric of
a black hole? If, with the benefit of hindsight, we try to obtain the extreme
five-dimensional RN solution given in (10.101), (10.102) and (10.103), then the
obvious first attempt is to choose p = 5. (We are, therefore, considering type lIB
superstriog theory, since p is odd.) This gives four (transverse) spatial dimensions
y6, y7, y8, y9,
r2 == (y6)2 + (y 7)2 + (y8)2 + (y9)2
(10.137)
and the harmonic function Hs(r) varies as r- 2 • As previously noted, if the five
longitudinal coordinates x ) , x 2 • x 3 , x4, x S are compactified (on as-torus TS say),
the metric resembles the five-dimensional RN black hole (10.101). However, the
warp factors in both the longitudinal and transverse directions are wrong and,
further, the dilaton is singular on the event horizon at r = rH = O. (The
quantum states associated with a soli ton are found by identifying the zero modes
(or collective coordinates) and quantizing them. This is not possible if the soliton
is singular.) Now, because the solitons are extreme, we may combine solutions
with different values of p, provided that (some) supersymmetry is preserved.
This requires that some of the supersymmetry charges (10.136) preserved by one
solution are also preserved by the other. Thus, if PI and P2 represent, respectively,
reflection of the coordinates transverse to the PI- and P2-solitons, the preserved
supersymmetry charges satisfy
-
-
- 1
Qa + PIQa = Qa + l'2Qa = Qa + p)(P I P2)Qa
(10.138)
and we see that the unbroken supersymmetries correspond to + I eigenvalues of
p l
l 1'2. In general, this requires that the number of directions that are transverse