String Theory and M-Theory

580 Black holes in string theory
(2π) 4 V , and Q1 F1-branes wrapping the y 1 circle, which has radius R. This
leads to the masses
M1 = Q0
, M2 =
gsℓs
Q4
(2π) 4 gsℓ 5 s
(2π) 4 V, M3 = Q1
2πℓ2 2πR.
s
Inserting this into the expression for the entropy Eq. (11.54) gives
S = A
4G5
= 2πgsℓ4
s √ M1M2M3 = 2π
RV
Q0Q4Q1.
Comparison of this formula with Eq. (11.55) shows that the D0-D4-F1 system
gives the same entropy for Q0 = Q1, Q4 = Q5 and Q1 = n, which is
what we wanted to show.
Note that the various dualities that relate the different brane descriptions
of the black hole do not change the five-dimensional metric except by an
overall constant factor. Such a factor has no bearing on the computation of
the entropy, which is dimensionless.
✷
EXERCISE 11.7
Compute the area of the horizon of the rotating black hole described in
Section 11.3 and deduce its entropy.
SOLUTION
In the near-horizon limit r ≈ 0 and constant t the metric Eq. (11.74) reduces
to
ds 2 = R 2 dΩ 2 3 − (a/R 2 ) 2 (cos 2 θdψ − sin 2 θdφ) 2
= R 2 dθ 2 + R 2 (cos θ sin θ) 2 (dφ + dψ) 2 + (R 2 − (a/R 2 ) 2 )(cos 2 θdψ − sin 2 θdφ) 2 ,
where
R 2 = (r1r2r3) 2/3 .
The easiest way to compute the area of the horizon described by this metric
is to define the orthonormal one-forms
e1 = Rdθ,
e2 = R cos θ sin θ(dφ + dψ),
e3 = R 2 − (a/R 2 ) 2 (cos 2 θdψ − sin 2 θdφ).