### 7.3 The Israel–Wilson–Perjés class

A particular class of solutions to the stationary EM equations is obtained by requiring that
the Riemannian manifold is flat [179]. For , the three-dimensional Einstein
equations obtained from variations of the effective action (6.23) with respect to become
where, as we are considering stationary configurations, we use the dimensional reduction with respect to the
asymptotically–time-like Killing field with norm . Israel and Wilson [179] have
shown that all solutions of this equation fulfill . In fact, it is not hard to verify that this
ansatz solves Eq. (7.11), provided that the complex constants and are subject to
. Using asymptotic flatness, and adopting a gauge where the limits at infinity of the
electro-magnetic potentials and the twist potential vanish, one has and
, and thus
It is crucial that this ansatz solves both the equation for and the one for : One easily verifies that
Eqs. (6.24) reduce to the single equation
where is the three-dimensional flat Laplacian.
For static, purely electric configurations the twist potential and the magnetic potential vanish.
The ansatz (7.12), together with the definitions of the Ernst potentials, and
(see Section 6.4), yields

Since , the linear relation between and the gravitational potential implies
. By virtue of this, the total mass and the total charge of every asymptotically flat,
static, purely electric Israel–Wilson–Perjés solution are equal:
where the integral extends over an asymptotic two-sphere. Note that for purely electric configurations one
has ; also, staticity implies and thus . The simplest
nontrivial solution of the flat Poisson equation (7.13), , corresponds to a linear combination
of monopole sources located at arbitrary points ,
This is the MP solution [262, 220], with spacetime metric and electric
potential . The MP metric describes a regular black-hole spacetime, where the
horizon comprises disconnected components. Hartle and Hawking [139] have shown that all
singularities are “hidden” behind these null surfaces. In Newtonian terms, the configuration
corresponds to arbitrarily-located singularities are “hidden” behind these null surfaces. In
Newtonian terms, the configuration corresponds to arbitrarily-located charged mass points with
.
Non-static members of the Israel–Wilson–Perjés class were constructed as well [179, 267]. However,
these generalizations of the MP multi–black-hole solutions share certain unpleasant properties with NUT
spacetime [252] (see also [32, 237]). In fact, the results of [81] (see [139, 78, 154] for previous results)
suggest that – except the MP solutions – all configurations obtained by the Israel–Wilson–Perjés technique
either fail to be asymptotically flat or have naked singularities.