String Theory and M-Theory

574 Black holes in string theory
rotating black holes, even extremal ones, are not supersymmetric. The key is
to note that the rotation group in five dimensions is SO(4) ∼ SU(2)×SU(2).
Supersymmetry requires restricting the rotation to one of the two SU(2) factors,
which corresponds to simultaneous rotation, with equal angular momentum,
in two orthogonal planes. There are more general ways in which a
five-dimensional black hole can rotate, of course, but this is the only one that
is supersymmetric. It preserves 1/8 of the original 32 supersymmetries, just
like the previous examples. To describe this case, let us introduce angular
coordinates as follows:
Then
describes Euclidean space for
x 1 = r cos θ cos ψ, x 2 = r cos θ sin ψ, (11.70)
x 3 = r sin θ cos φ, x 4 = r sin θ sin φ. (11.71)
dx i dx i = dr 2 + r 2 dΩ 2 3
(11.72)
dΩ 2 3 = dθ 2 +sin 2 θdφ 2 +cos 2 θdψ 2 , 0 ≤ θ ≤ π/2, 0 ≤ φ, ψ ≤ 2π. (11.73)
The metric of the desired supersymmetric rotating black hole is a relatively
simple generalization of Eq. (11.46)
ds 2
−2/3
= −λ dt − a
r2 sin2 θdφ + a
r2 cos2 2 θdψ + λ 1/3 dr 2 + r 2 dΩ 2 3 ,
(11.74)
where λ is again given by Eq. (11.47). This metric describes simultaneous
rotation in the 12 and 34 planes. The parameter a is related to J12 = J34 = J
by
J = πa
. (11.75)
4G5
The area of the horizon at r = 0, and hence the entropy, is computed in
Exercise 11.7 and shown to be
S = A
4G5
= 2π Q1Q5n − J 2 . (11.76)
Extremal four-charge black holes for D = 4
The metric and entropy
The construction of supersymmetric black holes in four dimensions is quite
similar to the five-dimensional case. Before proposing a specific brane realization,
let us write down the metric and explore its properties. The analog