Kerr--NUT--Levi-Civita geometries from Ernst inversion: axis structure, curvature singularities, and the Manko--Ruiz parameter

Abstract

We construct and analyze stationary, axisymmetric vacuum metrics obtained by magnetic Ernst inversion of Kerr-NUT with Manko-Ruiz parameter $C$ and pre-inversion twist constant $β$. The transformation lies in the Ehlers orbit; our contribution is the NUT-dependent geometry and the roles of $C$ and $β$ in its axis, horizon, singularity, and azimuthal-CTC structure. The inversion preserves the canonical Weyl radius, the signed WLP numerator $F$, and the sign of $g_{ϕϕ}$ on regular domains. At the selected pole $x=σ$, $C=-σ$, the local axis condition and conicity are controlled by $χ_σ^{(β)}=β-2m(2σa+3l)$. Exterior zeros of the chosen seed Ernst representative obey the exact criterion $-β\in\mathcal{R}_C$; for the sampled families numerical traces yield half-line ranges with closed-form corner endpoints. For $D=a^2+l^2-2alC\ne0$, $Λ_βΣ$ has a finite nonzero seed-ring limit. High-precision calculations find direction-independent finite limits of both quadratic Weyl invariants along the sampled rays, without establishing $C^2$-extendibility. At $β=0$, exterior simple Ernst zeros are found for the sampled $C=-1,0$ cases but not for $C=+1$, consistently with the computed ranges. Near one zero the Kretschmann scalar has a generic sixth-order blow-up; a numerical angular scan identifies exceptional directions of lower order. All sampled points in the regular ($g_{ϕϕ}>0$) exterior are Petrov type I. Candidate horizon locations remain those of Kerr-NUT, and the asymptotics are Levi-Civita type.