Boundary value formulation

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 $−1 𝒥 = Φ dΦ$ 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

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

$(Σ,¯g )$ is a static pseudo-Riemannian three-dimensional manifold with metric $¯g = − ρ2dt2 + &tidle;g$ ;
the connection $a$ is orthogonal to the two-dimensional Riemannian manifold $&tidle; (Σ, &tidle;g)$ , that is, $a = atdt$ ;
the functions $at$ and $&tidle;gab$ do not depend on the coordinates $t$ and $φ$ .

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 $&tidle; (Σ, &tidle;g)$ :

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

the $σ$ -model equations

and the remaining Einstein equations

for the function $h(ρ,z)$ .⁶⁵ Since Eq. (58

) is conformally invariant, the metric function $h(ρ, z)$ does not appear in the $σ$ -model equation (61

). 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. (61

) is known, the remaining metric function $h(ρ,z)$ is obtained from Eqs. (62

) by quadrature.