Contents

19Stationary axisymmetric fields: basicconcepts and field equations19.1 The Killing vectorsWe now consider physical systems which, in addition to being stationary(with Killing vector ξ) possess a further symmetry: axial symmetry(Killing vector η).Independent of the field equations and of the presence of a timelikeKilling vector, axial symmetry is defined as an isometric SO(2) mappingof space-time such that the set of fixed points forms a (regular) twodimensionalsurface W 2 which is usually called the axis of rotation. Marsand Senovilla (1993a) collected some properties of axisymmetric gravitationalfields. In particular, it turns out that W 2 is timelike and that theKilling vector field η describing axial symmetry must be spacelike inaneighbourhood of the axis and zero only at points q on the axis. OutsideW 2 the Killing trajectories are closed (compact) curves. The Killing vectorη = x 1 ∂/∂ 2 − x 2 ∂/∂ 1 = ∂ ϕ vanishes on the rotation axis (x 1 =0=x 2 ).The tensor fieldH ab =(∇ a η c )(∇ b η c ) (19.1)isatanypointofW 2 the projection tensor to the space orthogonal to W 2 .To ensure Lorentzian geometry (‘elementary flatness’) in the vicinityof the rotation axis, the length of an orbit which passes through a pointp in some neighborhood of q ∈ W 2 should be, at first relevant order, 2πtimes the distance from p to the axis. This can be achieved (by scalingthe group parameter ϕ along the trajectories of η to have the standardperiodicity 2π) if the norm X of the Killing vector η is proportional tothe square of that distance. In this case (for points q ∈ W 2 and p with292