For static, purely electric configurations the twist potential and the magnetic potential vanish. The ansatz (44), together with the definitions of the Ernst potentials, and (see Section 4.5), yields54 The simplest nontrivial solution of the flat Poisson equation (45), , corresponds to a linear combination of monopole sources located at arbitrary points , [143, 128], with spacetime metric and electric potential . The PM metric describes a regular black hole spacetime, where the horizon comprises disconnected components.55 In Newtonian terms, the configuration corresponds to arbitrarily located charged mass points with . The PM solution escapes the uniqueness theorem for the Reissner–Nordström metric, since the latter applies exclusively to space-times with .
Non-static members of the Israel–Wilson class were constructed as well [102, 145]. However, these generalizations of the Papapetrou–Majumdar multi black hole solutions share certain unpleasant properties with NUT spacetime  (see also [16, 136]). In fact, the work of Hartle and Hawking , and Chruściel and Nadirashvili  strongly suggests that – except the PM solutions – all configurations obtained by the Israel–Wilson technique are either not asymptotically Euclidean or have naked singularities. In order to complete the uniqueness theorem for the PM metric among the static black hole solutions with degenerate horizon, it basically remains to establish the equality under the assumption that the horizon has some degenerate components. Until now, this has been achieved only by requiring that all components of the horizon have vanishing surface gravity and that all “horizon charges” have the same sign .
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