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)
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
The uniqueness of the Kerr–Newman metric should follow now from
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]):
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
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Living Rev. Relativity 15, (2012), 7
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