Miguel Prados-Abad | Searcharxiv

arXiv2026

Timelike ideal boundary of non-positively curved Lorentzian spaces

We introduce the notion of timelike ideal boundary of a Lorentzian length space as the set of asymptotic classes of future or past-directed timelike geodesic rays, a construction complementary to the causal boundary in the sense of Geroch-Kronheimer-Penrose and akin to the concept of ideal boundary of a metric space. We endow such a timelike ideal boundary with a natural cone topology and an angular metric, and establish upper curvature bounds for the resulting metric space. Finally, we consider generalized cones as a model and study the relation between the timelike ideal boundary and both the metric ideal boundary of the fiber and the asymptotic behaviour of the warping function.

math.MG↗

arXiv2026

Null distance on cosmological spacetimes and monotone convergence

The metric theory of spacetimes studies Lorentzian manifolds using tools of metric geometry. This is achieved via the null distance, which is a definite distance constructed from a time function on a spacetime. This enables the study of Gromov-Hausdorff-type convergence of spacetimes, a program recently initiated by Sakovich and Sormani. In this paper we study such notions of convergence for cosmological spacetimes with compact slices, i.e., $(a,b)\times M$ endowed with a Lorentzian metric $-dt^2+h_t$, where $h_t$ is a family of Riemannian metrics on the compact manifold $M$. Assuming mild extension properties of $h_t$, we first establish that these spacetimes are causally-null compactifiable and future developed. We then study monotone sequences with a uniform upper bound on the spatial diameter, obtaining uniform convergence of the null distances, as well as convergence of the associated timed metric spaces in the future developed Gromov-Hausdorff sense. Finally, we prove that causally-null compactifiable spacetimes satisfying a mild causal accessibility condition are causally-null, and relate the causally-null distance induced by the limit distance with the null distance induced by the (possibly non-smooth) limit metric tensor. Examples are provided to motivate the necessity of our hypotheses.

math.DG↗