Two-sheeted extension of superextremal Kerr spacetime (${\rho }^2/(r^2+a^2)+z^2/r^2=1$)

For $0<M<a$, $\Delta =(r-M{)}^2+a^2-M^2>0$, so no horizon blocks need adjoining. The oblate Cartesian coordinates of the Kerr metric satisfy the displayed relation, where ${\rho }^2=x^2+y^2$. Keeping only the positive root $r$ leaves a cut along the disk $z=0$, $\rho <a$. Continue $r$ through zero away from its boundary, the Kerr ring singularity, onto a negative-radius sheet. Equivalently cut two Cartesian sheets along the disk and join opposite banks, so $r$ changes sign continuously. The metric is analytic at the disk interior in signed oblate coordinates since $\Sigma =a^2{\cos }^2\theta \ne 0$ there. The extension has asymptotically flat ends at $r\to +\infty$ and $r\to -\infty$; only the singular ring remains inextendible by an analytic Lorentz metric.