5.3 The Israel–Wilson class

A particular class of solutions to the stationary EM equations is obtained by requiring that the Riemannian manifold (Σ,g¯) is flat [102Jump To The Next Citation Point]. For ¯gij = δij, the three-dimensional Einstein equations obtained from variations of the effective action (28View Equation) with respect to ¯g become53
( )( ) 4σ Λ,i ¯Λ,j = E,i+2 ¯Λ Λ,i ¯E,j +2 ΛΛ¯,j . (43 )
Israel and Wilson [102Jump 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. (43View 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 electro-magnetic potentials and the twist potential vanish in the asymptotic regime, one has E = 1 ∞ and Λ ∞ = 0, and thus
iα Λ = e--(1 − E), where α ∈ IR. (44 ) 2
It is crucial that this ansatz solves both the equation for E and the one for Λ: One easily verifies that Eqs. (29View Equation) reduce to the single equation
¯ −1 Δ (1 + E) = 0, (45 )
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 (44View Equation), together with the definitions of the Ernst potentials, 2 E = σ − |Λ | + iY and Λ = − ϕ + iψ (see Section 4.5), yields

√-- √ -- 1 + E = 2 σ, and ϕ = 1 − σ. (46 )
Since σ = 1 ∞, the linear relation between ϕ and the gravitational potential √ σ- implies (dσ )∞ = − (2dϕ )∞. By virtue of this, the total mass and the total charge of every asymptotically flat, static, purely electric Israel–Wilson solution are equal:
1 ∫ 1 ∫ M = − --- ∗dk = − --- ∗F = Q, (47 ) 8π 4π
where the integral extends over an asymptotic two-sphere.54 The simplest nontrivial solution of the flat Poisson equation (45View Equation), ¯ −1∕2 Δ σ = 0, corresponds to a linear combination of n monopole sources ma located at arbitrary points xa,
−1∕2 ∑n --ma---- σ (x) = 1 + |x-− x-|. (48 ) a=1 a
This is the Papapetrou–Majumdar (PM) solution [143128], with spacetime metric g = − σdt2 + σ −1dx2 and electric potential √ -- ϕ = 1 − σ. The PM metric describes a regular black hole spacetime, where the horizon comprises n disconnected components.55 In Newtonian terms, the configuration corresponds to n arbitrarily located charged mass points with √ -- |qa| = Gma. The PM solution escapes the uniqueness theorem for the Reissner–Nordström metric, since the latter applies exclusively to space-times with M > |Q |.

Non-static members of the Israel–Wilson class were constructed as well [102145]. However, these generalizations of the Papapetrou–Majumdar multi black hole solutions share certain unpleasant properties with NUT spacetime [140] (see also [16136]). In fact, the work of Hartle and Hawking [81], and Chruściel and Nadirashvili [42] 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 M = Q 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 [90].


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