### 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 Section 4).
This feature is shared by a variety of other self-gravitating theories with scalar (moduli) and
Abelian vector fields (see Section 5.2), for which the field equations assume the form (32):
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 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
Frobenius integrability conditions, the spacetime metric can be written in the familiar
(2+2)-split.
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 [144],

the field equations (54) 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
.
The last simplification of the field equations is due to the circumstance that can be chosen as one of the
coordinates on . This follows from the facts that is harmonic (with respect to the Riemannian
two-metric ) and non-negative, and that the domain of outer communications of a stationary black hole
spacetime is simply connected [44]. The function and the conjugate harmonic function are called Weyl
coordinates.
With respect to these, the metric can be chosen to be conformally flat, such that one ends up with the
spacetime metric

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