6.3 The Ernst Equations6 Stationary and Axisymmetric Space-Times6.1 Integrability Properties of Killing

6.2 Boundary Value Formulation 

The vacuum and the EM equations in the presence of a Killing symmetry describe harmonic mappings into coset manifolds, effectively coupled to three-dimensional gravity (see Sect. 4). This feature is shared by a variety of other self-gravitating theories with scalar (moduli) and Abelian vector fields (see Sect. 5.2), for which the field equations assume the form (32Popup Equation):

  equation1303

The current one-form tex2html_wrap_inline4613 is given in terms of the hermitian matrix tex2html_wrap_inline4215, which comprises the norm and the generalized twist potential of the Killing field, the fundamental scalar fields and the electric and magnetic potentials arising on the effective level for each Abelian vector field. If the dimensional reduction is performed with respect to the axial Killing field tex2html_wrap_inline4617 with norm tex2html_wrap_inline4619, then tex2html_wrap_inline4621 is Ricci tensor of the pseudo-Riemannian three-metric tex2html_wrap_inline3837, defined by

  equation1322

In the stationary and axisymmetric case under consideration, there exists, in addition to m, an asymptotically time-like Killing field k . Since k and m fulfill the Frobenius integrability conditions, the spacetime metric can be written in the familiar tex2html_wrap_inline3633 -split. Popup Footnote Hence, the circularity property implies that

With respect to the resulting Papapetrou metric [144],

  equation1340

the field equations (54Popup Equation) become a set of partial differential equations on the two-dimensional Riemannian manifold tex2html_wrap_inline4645 :

  equation1351

  equation1355

  equation1373

as is seen from the standard reduction of the Ricci tensor tex2html_wrap_inline4621 with respect to the static three-metric tex2html_wrap_inline4641 . Popup Footnote

The last simplification of the field equations is due to the circumstance that tex2html_wrap_inline4675 can be chosen as one of the coordinates on tex2html_wrap_inline4645 . This follows from the facts that tex2html_wrap_inline4675 is harmonic (with respect to the Riemannian two-metric tex2html_wrap_inline4681) and non-negative, and that the domain of outer communications of a stationary black hole spacetime is simply connected [44]. The function tex2html_wrap_inline4675 and the conjugate harmonic function z are called Weyl coordinates. Popup Footnote With respect to these, the metric tex2html_wrap_inline4681 can be chosen to be conformally flat, such that one ends up with the spacetime metric

  equation1395

the tex2html_wrap_inline3625 -model equations

  equation1408

and the remaining Einstein equations

  equation1417

for the function tex2html_wrap_inline4695 . Popup Footnote Since Eq. (58Popup Equation) is conformally invariant, the metric function tex2html_wrap_inline4695 does not appear in the tex2html_wrap_inline3625 -model equation (61Popup Equation). Therefore, the stationary and axisymmetric equations reduce to a boundary value problem for the matrix tex2html_wrap_inline4215 on a fixed, two-dimensional background. Once the solution to Eq. (61Popup Equation) is known, the remaining metric function tex2html_wrap_inline4695 is obtained from Eqs. (62Popup Equation) by quadrature.



6.3 The Ernst Equations6 Stationary and Axisymmetric Space-Times6.1 Integrability Properties of Killing

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