Static axisymmetric Einstein spaces with a cosmological constant and the limitation of canonical Weyl coordinates
Résumé
The canonical Weyl form for four-dimensional static axisymmetric vacuum metrics is obtained by identifying the area function of the two Killing orbits with a harmonic coordinate on the twodimensional orbit space. This construction is valid in Ricci-flat vacuum, but it is no longer available in Einstein spaces with nonzero cosmological constant. In this paper, we consider the generalized orthogonally transitive static axisymmetric line element and derive the reduced Einstein- $Λ$ field equations. We show that the canonical Weyl choice $W=ρ$ is locally admissible if and only if $Λ=0$. The Kottler metric gives the simplest explicit example of the resulting equation for the area function. Thus, the statement that "Weyl metrics do not allow $Λ\neq 0$ " is precise only when the metric is assumed to be in canonical Weyl coordinates. The issue is not staticity or axisymmetry, but rather the fact that the area function is no longer harmonic.
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