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

Abstract

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.

29 pages, no figure. Comments are welcome. During preparation of this paper, we become aware of K. Mosani Arxiv 2606.25755 which may have overlap results