Gravity and Strings

9.3 Charged Taub–NUT solutions and IWP solutions 281
This big family of solutions is known as the Israel–Wilson–Perjés (IWP) solutions [597,
769], although they were first discovered by Neugebauer [721]. This family contains all the
“extreme” solutions (RN, charged Taub–NUT, and their multicenter generalizations) that
we have found so far, plus many others that may have mass, electric and magnetic charges,
NUT charge, and also angular momentum. In particular, the M 2 = 4q 2 Kerr–Newman
solutions, for arbitrary angular momentum, belong to this family: their complex harmonic
function is
H = 1 +
In terms of more suitable oblate spheroidal coordinates,
the function H takes the form
M
. (9.59)
x 2 + y2 + (z − ia) 2
x + iy= [(r − M) 2 + a 2 ] 1 2 sin θ e iϕ ,
z = (r − M) cos θ,
H = 1 +
and the Euclidean three-dimensional metric becomes
d x 2
3 = (r − M) 2 + a 2 cos 2 θ dr2 (r − M) 2
2
+ dθ
+ a2 Furthermore, the 1-form A is given by
(9.60)
M
, (9.61)
r − M − iacos θ
+ (r − M) 2 + a 2 sin 2 θ dϕ 2 .
(9.62)
A = (2Mr − M2 )a sin 2 θ
(r − M) 2 + a2 cos2 dϕ, (9.63)
θ
and
|H| 2 = (r − m)2 − a2 cos2 θ
r 2 + a2 cos2 , (9.64)
θ
and we recover the Kerr–Newman solutions with M2 = 4q2 . These solutions are not BHs
because they violate the bound M2 − 4q2 − a2 ≥ 0. In fact, it has been argued by Hartle
and Hawking that the only BH-type solutions in the IWP family of metrics are the multi-
ERN solutions.
For us, one of the main interests of this family is that it is electric–magnetic-dualityinvariant
and it is the most general family that we can have with the above charges always
satisfying the identity M2 = 4|q| 2 .Anelectric–magnetic-duality transformation is nothing
but achange in the phase of H. Non-extreme solutions can be constructed from the IWP
class, by adding a “non-extremality function” W ,asinthe RN case [665]. We will study
them as a subfamily of the most general BH-type solutions of pure N = 4, d = 4 SUEGRA.