The Israel–Wilson class

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 [102

]. For $¯gij = δij$ , the three-dimensional Einstein equations obtained from variations of the effective action (28

) with respect to $¯g$ become⁵³

Israel and Wilson [102

] 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. (43

), 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

It is crucial that this ansatz solves both the equation for $E$ and the one for $Λ$ : One easily verifies that Eqs. (29

) reduce to the single equation

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 (44 ), together with the definitions of the Ernst potentials, $2 E = σ − |Λ | + iY$ and $Λ = − ϕ + iψ$ (see Section 4.5), yields

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:

where the integral extends over an asymptotic two-sphere.⁵⁴ The simplest nontrivial solution of the flat Poisson equation (45

), $¯ −1∕2 Δ σ = 0$ , corresponds to a linear combination of $n$ monopole sources $ma$ located at arbitrary points $xa$ ,

This is the Papapetrou–Majumdar (PM) solution [143 , 128 ], 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 [102 , 145 ]. However, these generalizations of the Papapetrou–Majumdar multi black hole solutions share certain unpleasant properties with NUT spacetime [140 ] (see also [16 , 136 ]). 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 ].