Analytic extension of a spacetime

An analytic extension embeds a spacetime isometrically as a proper open region in a larger analytic Lorentzian manifold with a nondegenerate metric tensor. A divergence of coordinate components does not obstruct extension if a new chart makes all metric coefficients analytic and the determinant nonzero. For a static spherical metric with an analytic lapse $f$ and a positive simple horizon radius, ingoing coordinates replace $-fd{t}^2+d{r}^2/f$ by $-fd{v}^2+2dvdr$. This removes the coordinate singularity at $f=0$; transverse metric coefficients must also remain regular. An extension across such a horizon is distinct from extending through a curvature singularity.

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.