6.4 The uniqueness theorem for the Kerr–Newman solution

In order to establish the 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 (67View Equation) are equal if they are subject to the same boundary and regularity conditions on ∂𝒮, where 𝒮 is the semi-strip 𝒮 = {(x,y)|x ≥ 1 ,|y| ≤ 1} (see Section 6.3.) For infinitesimally neighboring solutions, Carter solved this problem for the vacuum case by means of a divergence identity [29], which Robinson generalized to electrovac space-times [151].

Considering two arbitrary solutions of the Ernst equations, Robinson was able to construct an identity [152], 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 electrovac identity by trial and error methods.68 In fact, the problem was only solved when Mazur [131133] and Bunting [25] independently succeeded in deriving the desired divergence identities by using the distinguished structure of the EM equations in the presence of a Killing symmetry. 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.69

The Mazur identity (34View Equation) applies to the relative difference Ψ = Φ2 Φ −11− 𝟙 of two arbitrary hermitian matrices. If the latter are solutions of a σ-model with symmetric target space of the form SU (p,q )∕S (U (p) × U (q)), then the identity implies70

{ } Trace Δ¯Ψ = Trace ⟨ℳ , ℳ †âŸ©, (72 )
where † ℳ ≡ g−11𝒥△g2, and † 𝒥△ is the difference between the currents.

The reduction of the EM equations with respect to the axial Killing field yields the coset SU (2, 1)∕S (U (2) × U (1)) (see Section 4.5), which, reduces to the vacuum coset SU (2)∕S (U (1) × U (1)) (see Section 4.3). Hence, the above formula applies to both the axisymmetric vacuum and electrovac field equations, where the Laplacian and the inner product refer to the pseudo-Riemannian three-metric ¯g defined by Eq. (55View Equation). Now using the existence of the stationary Killing symmetry and the circularity property, one has 2 2 ¯g = − ρ dt + &tidle;g, which reduces Eq. (72View Equation) to an equation on (&tidle;Σ, &tidle;g). Integrating over the semi-strip 𝒮 and using Stokes’ theorem immediately yields

∫ ∫ ρ &tidle;∗Trace {dΨ } = ρ Trace⟨ℳ , ℳ †âŸ© &tidle;η, (73 ) ∂𝒮 𝒮
where &tidle;η and &tidle;∗ are the volume form and the Hodge dual with respect to &tidle;g. The uniqueness of the Kerr–Newman metric follows from the facts that

The RHS is non-negative because of the following observations: First, the inner product is definite, and &tidle;η is a positive volume-form, since &tidle;g is a Riemannian metric. Second, the factor ρ is non-negative in 𝒮, since 𝒮 is the image of the upper half-plane, ρ ≥ 0. Last, the one-forms 𝒥△ and ℳ are space-like, since the matrices Φ depend only on the coordinates of &tidle; (Σ, &tidle;g).

In order to establish that ρTrace {d Ψ} = 0 on the boundary ∂𝒮 of the semi-strip, one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. A careful investigation71 then shows that ρ Trace {dΨ } vanishes on the horizon, the axis and at infinity, provided that the solutions have the same mass, charge and angular momentum.

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