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 (6.26View Equation):
¯ 1- Rij = 4Trace (𝒥i𝒥j) , d¯∗𝒥 = 0, (8.6 )
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 e− 2λ := g(m, m ), then R¯ ij is the Ricci tensor of the pseudo-Riemannian three-metric ¯g, defined by
g = e−2λ(dφ + a)2 + e2λ¯g. (8.7 )

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 orthogonal-integrability conditions, the spacetime metric can locally be written in a (2+2)-block diagonal form. Hence, the circularity property implies that

With respect to the resulting Papapetrou metric [263],

−2λ 2 2λ( 2 2 ) g = e (dφ + atdt) + e − ρ dt + &tidle;g , (8.8 )
the field equations (8.6View Equation) become a set of partial differential equations on the two-dimensional Riemannian manifold &tidle; (Σ, &tidle;g):
Δ&tidle;gρ= 0, (8.9 )
1 1 &tidle;Rab − -∇&tidle;b &tidle;∇a ρ = --Trace(𝒥a 𝒥b), (8.10 ) ρ 4
∇&tidle;a (ρJa ) = 0, (8.11 )
as is seen from the standard reduction of the Ricci tensor R¯ij with respect to the static three-metric ¯g = − ρ2dt2 + &tidle;g. Further 𝒥t = 0 and ¯∗𝒥 = − ρdt ∧ &tidle;∗𝒥.

The last simplification of the field equations is obtained by choosing ρ as one of the coordinates on (&tidle;Σ, &tidle;g). Roughly speaking, this follows from the fact that ρ2 := g2 − g g tφ tt φφ is non-negative, that its square root ρ is harmonic (with respect to the Riemannian two-metric &tidle;g), and that the domain of outer communications of a stationary black-hole spacetime is simply connected; see [79Jump To The Next Citation Point, 76Jump To The Next Citation Point, 64] for details. The function ρ and the conjugate harmonic function z are called Weyl coordinates. With respect to these, the metric &tidle;g becomes manifestly conformally flat, and one ends up with the spacetime metric

2 −2λ 2 −2λ 2 2λ 2h( 2 2) g = − ρ e dt + e (dφ + atdt) + e e d ρ + dz , (8.12 )
the σ-model equations
∂ρ (ρ𝒥ρ) + ∂z (ρ𝒥z ) = 0, (8.13 )
and the remaining Einstein equations
ρ ρ ∂ρh = -Trace (𝒥ρ𝒥 ρ − 𝒥z𝒥z ), ∂zh = -Trace (𝒥ρ𝒥z ), (8.14 ) 8 4
for the function h (ρ, z). It is not hard to verify that Eq. (8.13View Equation) is the integrability condition for Eqs. (8.14View Equation). Since Eq. (8.10View Equation) is conformally invariant, the metric function h (ρ, z) does not appear in the σ-model equation (8.13View Equation). Taking into account that ρ is non-negative, the stationary and axisymmetric equations reduce to an elliptic system for a matrix Φ on a flat half-plane. Once the solution to Eq. (8.13View Equation) is known, the remaining metric function h(ρ,z) is obtained from Eqs. (8.14View Equation) by quadrature.
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