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 (Σ,g¯) is flat [179Jump To The Next Citation Point]. For ¯gij = δij, the three-dimensional Einstein equations obtained from variations of the effective action (6.23View Equation) with respect to ¯g become
( )( ) 4V Λ,i ¯Λ,j = E,i+2 ¯Λ Λ,i E¯,j +2 Λ¯Λ,j , (7.11 )
where, as we are considering stationary configurations, we use the dimensional reduction with respect to the asymptotically–time-like Killing field k with norm V = − g(k,k) = − N. Israel and Wilson [179Jump To The Next Citation Point] have shown that all solutions of this equation fulfill Λ = c0 + c1E. In fact, it is not hard to verify that this ansatz solves Eq. (7.11View Equation), provided that the complex constants c0 and c1 are subject to c0¯c1 + c1¯c0 = − 1∕2. Using asymptotic flatness, and adopting a gauge where the limits at infinity of the electro-magnetic potentials and the twist potential vanish, one has E∞ := limr→ ∞ E = 1 and Λ ∞ := limr → ∞ Λ = 0, and thus
iα Λ = e-- (1 − E ), where α ∈ ℝ. (7.12 ) 2
It is crucial that this ansatz solves both the equation for E and the one for Λ: One easily verifies that Eqs. (6.24View Equation) reduce to the single equation
−1 ¯Δ (1 + E) = 0, (7.13 )
where ¯Δ is the three-dimensional flat Laplacian.

For static, purely electric configurations the twist potential Y and the magnetic potential ψ vanish. The ansatz (7.12View Equation), together with the definitions of the Ernst potentials, 2 E = V − |Λ | + iY and Λ = − ϕ + iψ (see Section 6.4), yields

√ -- √ -- 1 + E = 2 V , and ϕ = 1 − V . (7.14 )
Since V∞ = 1, the linear relation between ϕ and the gravitational potential √ -- V implies (dV )∞ = − (2d ϕ)∞. By virtue of this, the total mass and the total charge of every asymptotically flat, static, purely electric Israel–Wilson–Perjés solution are equal:
∫ ∫ M = − -1- ∗dk = − 1-- ∗F = Q, (7.15 ) 8π 4π
where the integral extends over an asymptotic two-sphere. Note that for purely electric configurations one has F = k ∧ dϕ∕V; also, staticity implies k = − V dt and thus dk = − k ∧ dV ∕V = − F. The simplest nontrivial solution of the flat Poisson equation (7.13View Equation), ¯ΔV − 1∕2 = 0, corresponds to a linear combination of n monopole sources m a located at arbitrary points x -a,
∑n m V −1∕2(x) = 1 + ----a--. (7.16 ) a=1 |x-− xa|
This is the MP solution [262, 220], with spacetime metric 2 −1 2 g = − V dt + V dx- and electric potential √ -- ϕ = 1 − V. The MP metric describes a regular black-hole spacetime, where the horizon comprises n disconnected components. Hartle and Hawking [139Jump To The Next Citation Point] have shown that all singularities are “hidden” behind these null surfaces. In Newtonian terms, the configuration corresponds to n arbitrarily-located singularities are “hidden” behind these null surfaces. In Newtonian terms, the configuration corresponds to n arbitrarily-located charged mass points with √ -- |qa| = Gma.

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.


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