7 Conclusion6 Stationary and Axisymmetric Space-Times6.3 The Ernst Equations

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 (67Popup Equation) are equal if they are subject to the same boundary and regularity conditions on tex2html_wrap_inline4841, where tex2html_wrap_inline4843 is the semi-strip tex2html_wrap_inline4745 (see Sect. 6.3 .) For infinitesimally neighboring solutions, Carter solved this problem for the vacuum case by means of a divergence identity [28], 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. Popup Footnote In fact, the problem was only solved when Mazur [131], [133] and Bunting [24] 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. Popup Footnote

The Mazur identity (34Popup Equation) applies to the relative difference tex2html_wrap_inline4847 of two arbitrary hermitian matrices. If the latter are solutions of a tex2html_wrap_inline3625 -model with symmetric target space of the form tex2html_wrap_inline4281, then the identity implies Popup Footnote

  equation1595

where tex2html_wrap_inline4289, and tex2html_wrap_inline4855 is the difference between the currents.

The reduction of the EM equations with respect to the axial Killing field yields the coset tex2html_wrap_inline4707 (see Sect. 4.5), which, reduces to the vacuum coset tex2html_wrap_inline4859 (see Sect. 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 tex2html_wrap_inline3837 defined by Eq. (55Popup Equation). Now using the existence of the stationary Killing symmetry and the circularity property, one has tex2html_wrap_inline4641, which reduces Eq. (72Popup Equation) to an equation on tex2html_wrap_inline4645 . Integrating over the semi-strip tex2html_wrap_inline4843 and using Stokes' theorem immediately yields

  equation1622

where tex2html_wrap_inline4869 and tex2html_wrap_inline4871 are the volume form and the Hodge dual with respect to tex2html_wrap_inline4681 . 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 tex2html_wrap_inline4869 is a positive volume-form, since tex2html_wrap_inline4681 is a Riemannian metric. Second, the factor tex2html_wrap_inline4675 is non-negative in tex2html_wrap_inline4843, since tex2html_wrap_inline4843 is the image of the upper half-plane, tex2html_wrap_inline4743 . Last, the one-forms tex2html_wrap_inline4887 and tex2html_wrap_inline4299 are space-like, since the matrices tex2html_wrap_inline4215 depend only on the coordinates of tex2html_wrap_inline4645 .

In order to establish that tex2html_wrap_inline4895 on the boundary tex2html_wrap_inline4841 of the semi-strip, one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. A careful investigation Popup Footnote then shows that tex2html_wrap_inline4899 vanishes on the horizon, the axis and at infinity, provided that the solutions have the same mass, charge and angular momentum.



7 Conclusion6 Stationary and Axisymmetric Space-Times6.3 The Ernst Equations

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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