6.4 The Uniqueness Theorem for 6 Stationary and Axisymmetric Space-Times6.2 Boundary Value Formulation

6.3 The Ernst Equations 

The Ernst equations [52], [53] - being the key to the Kerr-Newman metric - are the explicit form of the circular tex2html_wrap_inline3625 -model equations (61Popup Equation) for the EM system, that is, for the coset tex2html_wrap_inline4707 . Popup Footnote The latter is parametrized in terms of the Ernst potentials tex2html_wrap_inline4455 and tex2html_wrap_inline4711, where the four scalar potentials are obtained from Eqs. (26Popup Equation) and (27Popup Equation) with tex2html_wrap_inline4713 . Instead of writing out the components of Eq. (61Popup Equation) in terms of tex2html_wrap_inline4199 and tex2html_wrap_inline3933, it is more convenient to consider Eqs. (29Popup Equation), and to reduce them with respect to the static metric tex2html_wrap_inline4641 (see Sect. 6.2). Introducing the complex potentials tex2html_wrap_inline4721 and tex2html_wrap_inline4723 according to

  equation1461

one easily finds the two equations

  equation1468

where tex2html_wrap_inline4725 stands for either of the complex potentials tex2html_wrap_inline4721 or tex2html_wrap_inline4723, and where the Laplacian and the inner product refer to the two-dimensional metric tex2html_wrap_inline4681 .

In order to control the boundary conditions for black holes, it is convenient to introduce prolate spheroidal coordinates x and y, defined in terms of the Weyl coordinates tex2html_wrap_inline4675 and z by

  equation1479

where tex2html_wrap_inline4741 is a constant. The domain of outer communications, that is, the upper half-plane tex2html_wrap_inline4743, corresponds to the semi-strip tex2html_wrap_inline4745 . The boundary tex2html_wrap_inline4747 consists of the horizon (x = 0) and the northern (y =1) and southern (y = -1) segments of the rotation axis. In terms of x and y, the Riemannian metric tex2html_wrap_inline4681 becomes tex2html_wrap_inline4761, up to a conformal factor which does not enter Eqs. (64Popup Equation). The Ernst equations finally assume the form (tex2html_wrap_inline4763, etc.)

displaymath1492

  equation1502

where tex2html_wrap_inline4725 stand for tex2html_wrap_inline4721 or tex2html_wrap_inline4723 . A particularly simple solution to the Ernst equations is

  equation1517

with real constants p, q and tex2html_wrap_inline4777 . The norm X, the twist potential Y and the electro-magnetic potentials tex2html_wrap_inline4045 and tex2html_wrap_inline4151 (all defined with respect to the axial Killing field) are obtained from the above solution by using Eqs. (63Popup Equation) and the expressions tex2html_wrap_inline4787, tex2html_wrap_inline4789, tex2html_wrap_inline4791, tex2html_wrap_inline4793 . The off-diagonal element of the metric, tex2html_wrap_inline4795, is obtained by integrating the twist expression (10Popup Equation), where the twist one-form is given in Eq. (27Popup Equation). Popup Footnote Eventually, the metric function h is obtained from Eqs. (62Popup Equation) by quadrature.

The solution derived in this way is the ``conjugate'' of the Kerr-Newman solution [36]. In order to obtain the Kerr-Newman metric itself, one has to perform a rotation in the tex2html_wrap_inline4799 -plane: The spacetime metric is invariant under tex2html_wrap_inline4801, tex2html_wrap_inline4803, if X, tex2html_wrap_inline4649 and tex2html_wrap_inline4809 are replaced by kX, tex2html_wrap_inline4813 and tex2html_wrap_inline4815, where tex2html_wrap_inline4817 . This additional step in the derivation of the Kerr-Newman metric is necessary because the Ernst potentials were defined with respect to the axial Killing field tex2html_wrap_inline4819 . If, on the other hand, one uses the stationary Killing field tex2html_wrap_inline3879, then the Ernst equations are singular at the boundary of the ergo-region.

In terms of Boyer-Lindquist coordinates,

  equation1549

one eventually finds the Kerr-Newman metric in the familiar form:

  equation1552

where the constant tex2html_wrap_inline3851 is defined by tex2html_wrap_inline4827 . The expressions for tex2html_wrap_inline4829, tex2html_wrap_inline4831 and the electro-magnetic vector potential A show that the Kerr-Newman solution is characterized by the total mass M, the electric charge Q, and the angular momentum tex2html_wrap_inline4839 :

  equation1564


6.4 The Uniqueness Theorem for 6 Stationary and Axisymmetric Space-Times6.2 Boundary Value Formulation

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
http://www.livingreviews.org/lrr-1998-6
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