292
Black holes ;n string theory
charge which is, in fact, the central charge.) For massive representations, we may
work in the rest frame where P,.,. = (M, 0). Then it is easy to see (exercise 9) that
we may form two linear combinations tla and ba of the generators that satisfy
{aa, al} = 4(M + Z)da~
{ba, b~} = 4(M - Z)da~
{CIa, b;} = 0 = {CIa. bfi}
(10.92)
Up to a normalization factor, these are just the anticommutation relations obeyed
by two independent sets of fermion annihilation and creation operators. In
general, starting with a state It} that is annihilated by aa and ba, we can
construct a total of 16 states using the creation operators a! and b!. Since
( I{aa , a!) I} ~ 0 for any state I} and, similarly, for ba, the 'BPS bound'
M ~ IZI
(10.93)
follows. The inequality is saturated by representations (BPS states) for which It}
is also annihilated by one set of creation operators (b! if Z > 0). Thus, such states
are 'short' massive representations of the supersymmetry algebra. since they are
constructed using only the a; creation operators. They are invariant under half of
the supersymmetry algebra.
The extreme RN black hole, obtained from (10.33) and (10.34) by setting
M = Q. is part of such a short hypermultiplet [16]. In this case, the two horizons
are both at r = M = Q and, defining p == r - Q, we may write the solution in the
'isotropic" form in which the spatial part of the metric is conformal to flat space:
ds 2 = H- 2 dt 2 - H2(dp2 + p2 dn~)
(10.94)
A = (I - H-1)dr
(10.95)
where
Q
H == 1 + -.
(10.96)
P
Both the temporal and spatial 'warp' factors (the factors multiplying the two
parts of the metric), as well as the electromagnetic vector-potential I-form, are
detennined by a single (harmonic) function H. Extreme solutions have the
important property that they are easily generalized to a case representing N
extreme black holes with charges qi = mi (with i = 1, 2, ... , N) by replacing
the function H given in (10.96) by
N
H=l+"~ .
(10.97)
~ Ir -rll
1=1
By Gauss' law, the total charge is Q = 1:;:'1 qi which. by the BPS bound, is also
the total mass
N
N
M=L~=L~=~
(10.98)
1=1
1=1
Black holes ;n string theory
charge which is, in fact, the central charge.) For massive representations, we may
work in the rest frame where P,.,. = (M, 0). Then it is easy to see (exercise 9) that
we may form two linear combinations tla and ba of the generators that satisfy
{aa, al} = 4(M + Z)da~
{ba, b~} = 4(M - Z)da~
{CIa, b;} = 0 = {CIa. bfi}
(10.92)
Up to a normalization factor, these are just the anticommutation relations obeyed
by two independent sets of fermion annihilation and creation operators. In
general, starting with a state It} that is annihilated by aa and ba, we can
construct a total of 16 states using the creation operators a! and b!. Since
( I{aa , a!) I} ~ 0 for any state I} and, similarly, for ba, the 'BPS bound'
M ~ IZI
(10.93)
follows. The inequality is saturated by representations (BPS states) for which It}
is also annihilated by one set of creation operators (b! if Z > 0). Thus, such states
are 'short' massive representations of the supersymmetry algebra. since they are
constructed using only the a; creation operators. They are invariant under half of
the supersymmetry algebra.
The extreme RN black hole, obtained from (10.33) and (10.34) by setting
M = Q. is part of such a short hypermultiplet [16]. In this case, the two horizons
are both at r = M = Q and, defining p == r - Q, we may write the solution in the
'isotropic" form in which the spatial part of the metric is conformal to flat space:
ds 2 = H- 2 dt 2 - H2(dp2 + p2 dn~)
(10.94)
A = (I - H-1)dr
(10.95)
where
Q
H == 1 + -.
(10.96)
P
Both the temporal and spatial 'warp' factors (the factors multiplying the two
parts of the metric), as well as the electromagnetic vector-potential I-form, are
detennined by a single (harmonic) function H. Extreme solutions have the
important property that they are easily generalized to a case representing N
extreme black holes with charges qi = mi (with i = 1, 2, ... , N) by replacing
the function H given in (10.96) by
N
H=l+"~ .
(10.97)
~ Ir -rll
1=1
By Gauss' law, the total charge is Q = 1:;:'1 qi which. by the BPS bound, is also
the total mass
N
N
M=L~=L~=~
(10.98)
1=1
1=1