String Theory and M-Theory

11.3 Black holes in string theory 581
Then the area of the horizon obtained from this metric is given by
A = e1 ∧ e2 ∧ e3 = R 2 R2 − (a/R2 ) 2
cos θ sin θdθ ∧ dφ ∧ dψ
Using this result,
= 2π 2 (r1r2r3) 2 − a 2 .
S = A
4G5
= 2π Q1Q5n − J 2 ,
where the angular momentum J is related to the parameter a by
J = πa
.
4G5
If a > r1r2r3, the black hole is over-rotating, and the geometry has a naked
singularity, at least in the supergravity approximation. ✷
EXERCISE 11.8
Consider the dual configuration of the D = 4 extremal four-charge black
described in Section 11.3. Show that this gives the entropy in Eq. (11.84),
with the charges replaced by the charges of the dual configuration.
SOLUTION
The dual configuration has three sets of D2-branes and one set of D6-branes.
The associated masses are
M1 = (2πR2)(2πR3)TD2 = R2R3
gsℓ 3 s
M3 = R1R6
gsℓ 3 s
Therefore, the entropy is
S = A
4G4
Q1, M2 = R4R5
gsℓ3 Q2,
s
Q3, M4 = (2π) 6 (R1 · · · R6)TD6P0 = R1 · · · R6
gsℓ7 P0.
s
g
= 16π
2 sℓ8
s
M1M2M3M4 = 2π
8R1 · · · R6
Q1Q2Q3P0,
which reproduces Eq. (11.84). ✷
EXERCISE 11.9
Construct the nonextremal generalization of the four-charge black hole by
analogy with the construction given for nonextremal black holes in five dimensions.
Interpret the masses, charges, and entropy in terms of branes and
antibranes, as was done in the five-dimensional case.