= Two-sheeted extension of superextremal Kerr spacetime
{title2=$\rho^2/(r^2+a^2)+z^2/r^2=1$}
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 $\Sigma=a^2\cos^2\theta\ne0$ 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.
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