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 (32View Equation):
¯ 1- Rij = 4 Trace{ 𝒥i𝒥j} , d∗¯ğ’¥ = 0, (54 )
The current one-form −1 𝒥 = Φ dΦ is given in terms of the hermitian matrix Φ, 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 m = ∂φ with norm X ≡ (m , m ), then ¯ Rij is Ricci tensor of the pseudo-Riemannian three-metric ¯g, defined by
2 -1- g = X (dφ + a) + X ¯g. (55 )

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 (2+2)-split.62 Hence, the circularity property implies that

With respect to the resulting Papapetrou metric [144],

( ) g = X (dφ + a dt )2 + -1- − ρ2dt2 + &tidle;g , (56 ) t X
the field equations (54View Equation) become a set of partial differential equations on the two-dimensional Riemannian manifold &tidle; (Σ, &tidle;g):
&tidle;Δ ρ = 0, (57 )
&tidle;R − 1-&tidle;∇ ∇&tidle; ρ = 1Trace {𝒥 𝒥 }, (58 ) ab ρ b a 4 a b
&tidle;∇a (ρ J ) = 0, (59 ) a
as is seen from the standard reduction of the Ricci tensor R¯ij with respect to the static three-metric ¯g = − ρ2dt2 + &tidle;g.63

The last simplification of the field equations is due to the circumstance that ρ can be chosen as one of the coordinates on (&tidle;Σ, &tidle;g). This follows from the facts that ρ is harmonic (with respect to the Riemannian two-metric g&tidle;) and non-negative, and that the domain of outer communications of a stationary black hole spacetime is simply connected [44]. The function ρ and the conjugate harmonic function z are called Weyl coordinates.64 With respect to these, the metric &tidle;g can be chosen to be conformally flat, such that one ends up with the spacetime metric

ρ2 2 1 ( ) g = − --dt2 + X (d φ + atdt) + ---e2h d ρ2 + dz2 , (60 ) X X
the σ-model equations
∂ρ(ρ 𝒥ρ) + ∂z (ρ 𝒥z) = 0, (61 )
and the remaining Einstein equations
ρ- ρ- ∂ρh = 8 Trace{ 𝒥ρ𝒥 ρ − 𝒥z 𝒥z} , ∂zh = 4 Trace {𝒥 ρ𝒥z} , (62 )
for the function h(ρ,z).65 Since Eq. (58View Equation) is conformally invariant, the metric function h(ρ, z) does not appear in the σ-model equation (61View Equation). Therefore, the stationary and axisymmetric equations reduce to a boundary value problem for the matrix Φ on a fixed, two-dimensional background. Once the solution to Eq. (61View Equation) is known, the remaining metric function h(ρ,z) is obtained from Eqs. (62View Equation) by quadrature.
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