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.62 Hence, the circularity property implies that
With respect to the resulting Papapetrou metric ,63
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 . The function and the conjugate harmonic function are called Weyl coordinates.64 With respect to these, the metric can be chosen to be conformally flat, such that one ends up with the spacetime metric65 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.
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