### 8.2 Two-dimensional elliptic equations

The vacuum and the EM equations in the presence of a Killing symmetry describe harmonic maps into
coset manifolds, coupled to three-dimensional gravity (see Section 6). This feature is shared by a variety of
other self-gravitating theories with scalar (moduli) and Abelian vector fields (see Section 7.2), for which the
field equations assume the form (6.26):
The current one-form is given in terms of the Hermitian matrix , 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 with norm
, then is the Ricci tensor of the pseudo-Riemannian three-metric , defined by
In the stationary and axisymmetric case under consideration, there exists, in addition to , an
asymptotically–time-like Killing field . Since and fulfill the orthogonal-integrability conditions,
the spacetime metric can locally be written in a (2+2)-block diagonal form. Hence, the circularity property
implies that

- is a static pseudo-Riemannian three-dimensional manifold with metric
;
- the connection is orthogonal to the two-dimensional Riemannian manifold , that
is, ;
- the functions and do not depend on the coordinates and .

With respect to the resulting Papapetrou metric [263],

the field equations (8.6) become a set of partial differential equations on the two-dimensional Riemannian
manifold :
as is seen from the standard reduction of the Ricci tensor with respect to the static three-metric
. Further and .
The last simplification of the field equations is obtained by choosing as one of the coordinates on
. Roughly speaking, this follows from the fact that is non-negative, that its
square root is harmonic (with respect to the Riemannian two-metric ), and that the domain of outer
communications of a stationary black-hole spacetime is simply connected; see [79, 76, 64] for details. The
function and the conjugate harmonic function are called Weyl coordinates. With respect to these,
the metric becomes manifestly conformally flat, and one ends up with the spacetime metric

the -model equations
and the remaining Einstein equations
for the function . It is not hard to verify that Eq. (8.13) is the integrability condition for
Eqs. (8.14). Since Eq. (8.10) is conformally invariant, the metric function does not appear in the
-model equation (8.13). Taking into account that is non-negative, the stationary and axisymmetric
equations reduce to an elliptic system for a matrix on a flat half-plane. Once the solution to
Eq. (8.13) is known, the remaining metric function is obtained from Eqs. (8.14) by
quadrature.