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 [228, 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 (7.2) applies to the relative difference of the associated Hermitian matrices and implies (see Section 7.1 for details and references)

where is the Laplace–Beltrami operator of the flat metric ; also recall that , with the difference between the currents.The reduction of the EM equations with respect to the axial Killing field yields -model equations with target (see Section 6.4), in vacuum reduces to (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 ; integrating and using Stokes’ theorem leads to

where the volume form and the Hogde dual are related to the flat metric .The uniqueness of the Kerr–Newman metric should follow now from

- the fact that the integrand on the right-hand side is non-negative, and
- the fact that the boundary at infinity on the left-hand side vanishes for two solutions with the same mass, electric charge and angular momentum, and
- the expectation that the integral over the axis and horizons, where the integrand becomes singular, vanishes for black-hole configurations with the same quotient-space structure.

In order to establish that on the boundary of the
half-plane,^{11}
one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. One
expects that 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.

An alternative to the divergence identities above is provided by the observation that the distance between two harmonic maps , , 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]):

compare (4.2). Here is the distance function between points on the target manifold and the flat Laplacian on . 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:

Hence, to prove uniqueness it remains to verify that is bounded on , and that

Living Rev. Relativity 15, (2012), 7
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