Warped Spacelike Singularities and the $C^0$-Inextendibility of Birmingham-Kottler Spacetimes

We establish a local obstruction to continuous Lorentzian extensions at a class of warped spacelike singularities.

The criterion is expressed through two integrability conditions, a monotonicity condition on the relative warp factors, and divergence of the longitudinal factor; it does not require any symmetry of the closed fiber.

The main geometric step is a radial compression of terminal causal traces, which replaces the rotational deformation available in spherical symmetry.

The compression yields a compact chronological separator in an adapted boundary chart.

Longitudinal translations then produce radial-slice distances that diverge intrinsically while remaining uniformly controlled in the extension chart.

Next, we construct the canonical one-horizon Birmingham-Kottler spacetime in global Kruskal coordinates and classify all finite proper-time ends of its timelike geodesics.

Every boundary-approaching finite maximizer supplied by a putative extension is thereby forced to the singular end.

The local obstruction and the geodesic classification imply $C^0$-inextendibility for the one-horizon Birmingham-Kottler family with nonpositive cosmological constant and every closed connected fiber satisfying $\text{Ric}_{\gamma_\Sigma}=(n-2)k\gamma_\Sigma$, without assumptions of homogeneity, orientability, or simple connectivity.