8.4 The uniqueness theorem for the Kerr–Newman solution

In order to establish uniqueness of the Kerr–Newman metric among the stationary and axisymmetric black-hole configurations, one has to show that two solutions of the Ernst equations (8.19View Equation) are equal if they are subject to black-hole boundary conditions on ∂๐’ฎ, where ๐’ฎ is the half-plane ๐’ฎ = {(ρ,z)|ρ ≥ 0}. Carter proved non-existence of linearized vacuum perturbations near Kerr by means of a divergence identity [45], which Robinson generalized to electrovacuum spacetimes [279].

8.4.1 Divergence identities

Considering two arbitrary solutions of the Ernst equations, Robinson was able to construct an identity [280], the integration of which proved the uniqueness of the Kerr metric. The complicated nature of the Robinson identity dashed the hope of finding the corresponding electrovacuum identity by trial and error methods (see, e.g., [47]). The problem was eventually solved when Mazur [228Jump To The Next Citation Point, 230] and Bunting [41] independently derived divergence identities useful for the problem at hand. Bunting’s approach, applying to a general class of harmonic mappings between Riemannian manifolds, yields an identity, which enables one to establish the uniqueness of a harmonic map if the target manifold has negative curvature. We refer the reader to Sections 3.2.5 and 8.4.2 (see also [49]) for discussions related to Bunting’s method.

So, consider two solutions of the Ernst equations associated to, a priori, distinct black-hole spacetimes, each endowed with its own metric. As discussed in Section 8.3, Weyl coordinates and conformal invariance allow us to view the Ernst equations as equations on a flat half-plane; alternatively, they may be seen as equations for an axisymmetric field on three-dimensional flat space. The Mazur identity (7.2View Equation) applies to the relative difference −1 Ψ = Φ2Φ 1 − ๐Ÿ™ of the associated Hermitian matrices and implies (see Section 7.1 for details and references)

Trace(Δ Ψ ) = TraceโŸจโ„ณ, โ„ณ †โŸฉ, (8.24 ) γ
where Δ γ is the Laplace–Beltrami operator of the flat metric γ = dρ2 + dz2 + ρ2dφ2; also recall that โ„ณ = g−1๐’ฅ †g 1 โ–ณ 2, with ๐’ฅ † โ–ณ the difference between the currents.

The reduction of the EM equations with respect to the axial Killing field yields σ-model equations with SU (2, 1)โˆ•S(U (2) × U (1)) target (see Section 6.4), in vacuum reduces to SU (2)โˆ•S (U (1) × U (1 )) (see Section 6.2). Hence, the above formula applies to both the stationary and axisymmetric vacuum or electrovacuum field equations. Now, relying on axisymmetry once more, we can reduce the previous Mazur identity to an equation on the flat half-plane (๐’ฎ,δ); integrating and using Stokes’ theorem leads to

∫ ∫ ρ∗Trace (dΨ ) = ρTraceโŸจโ„ณ, โ„ณ †โŸฉηδ, (8.25 ) ∂๐’ฎ ๐’ฎ
where the volume form ηδ and the Hogde dual ∗ are related to the flat metric δ = dρ2 + dz2.

The uniqueness of the Kerr–Newman metric should follow now from

In order to establish that ρTrace (dΨ ) = 0 on the boundary ∂ ๐’ฎ of the half-plane,11 one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. One expects that ρTrace (dΨ ) vanishes on the horizon, the axis and at infinity, provided that the solutions have the same mass, charge and angular momentum, but no complete analysis of this has been presented in the literature; see [318] for some partial results. Fortunately, the supplementary difficulties arising from the need to control the derivatives of the fields disappear when the distance-function approach described in the next Section 8.4.2 is used.

8.4.2 The distance function argument

An alternative to the divergence identities above is provided by the observation that the distance d(ฯ•1, ฯ•2) between two harmonic maps ฯ•a, a = 1,2, with negatively curved target manifold is subharmonic [182, Lemma 8.7.3 and Corollary 8.6.4] (see also the proof of Lemma 2 in [321] following results in [287]):

Δ d(ฯ• ,ฯ• ) ≥ 0; (8.26 ) δ 1 2
compare (4.2View Equation). Here d is the distance function between points on the target manifold and Δ δ the flat Laplacian on โ„3. It turns out that the Ernst equations for the Einstein–Maxwell equations fall in this category; in the vacuum case this is obvious, as the target space is then the two-dimensional hyperbolic space. This is somewhat less evident for the Einstein–Maxwell Ernst map, and can be checked by a direct calculation, or can be justified by general considerations about symmetric spaces; more precisely this follows from [144, Theorem 3.1] after noting that the target spaces of the maps under consideration are of non-compact type (see also [320]).

Using this observation, the key to uniqueness is provided by the following non-standard version of the maximum principle:

Proposition 8.1 [75, Appendix C] Let ๐’œ denote the z-axis in โ„3, and let f ∈ C0 (โ„3 โˆ– ๐’œ ) satisfy

3 Δ δf ≥ 0 in โ„ โˆ– ๐’œ , in the distributional sense, (8.27 )
3 0 ≤ f ≤ 1, on โ„ โˆ– ๐’œ , (8.28 )
and
(x,y,z)∈โ„3โˆ–li๐’œm,|(x,y,z)|→∞ f (x,y,z) = 0. (8.29 )
Then
3 f ≡ 0, on โ„ โˆ– ๐’œ .

Hence, to prove uniqueness it remains to verify that d (ฯ•1,ฯ•2) is bounded on 3 โ„ โˆ– ๐’œ, and that

--d-(ฯ•1-(x),ฯ•2(x))-- f(x) := supy d(ฯ•1(y),ฯ•2(y))
goes to zero at infinity. The latter property follows immediately from asymptotic flatness. The main work is thus to prove that f remains bounded near the axis. Here one needs to keep in mind that the (ρ,z) coordinate system is constructed in an implicit way by PDE techniques, and that the whole axis is singular from the PDE point of view because of factors of ρ and −1 ρ in the equations. In particular the associated harmonic maps tend to infinity in the target manifold when the axis of rotation ๐’œ is approached. So the proof of boundedness of f requires considerable effort, with the first complete analysis for non-degenerate horizons in [76]. The major challenge are points where the axes of rotation meet the horizons. The degenerate horizons, first settled in [79], provide supplementary difficulties. The proof of boundedness of f near degenerate horizons proceeds via Hájíฤek’s Theorem [135] (rediscovered independently by Lewandowski and Pawล‚owski [215], see also [202]), that the near-horizon geometry of degenerate axisymmetric Killing horizons with spherical cross-sections coincides with that of the Kerr–Newman solutions.


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