Considering two arbitrary solutions of the Ernst equations, Robinson was able to construct an identity , 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 [131, 133] and Bunting  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 (34) applies to the relative difference of two arbitrary hermitian matrices. If the latter are solutions of a -model with symmetric target space of the form , then the identity implies70
The reduction of the EM equations with respect to the axial Killing field yields the coset (see Section 4.5), which, reduces to the vacuum coset (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 defined by Eq. (55). Now using the existence of the stationary Killing symmetry and the circularity property, one has , which reduces Eq. (72) to an equation on . Integrating over the semi-strip and using Stokes’ theorem immediately yields
The RHS is non-negative because of the following observations: First, the inner product is definite, and is a positive volume-form, since is a Riemannian metric. Second, the factor is non-negative in , since is the image of the upper half-plane, . Last, the one-forms and are space-like, since the matrices depend only on the coordinates of .
In order to establish that on the boundary of the semi-strip, one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. A careful investigation71 then shows that 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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