Classification of globally hyperbolic spacetimes with non‑proper isometry actions

Classify globally hyperbolic Lorentzian spacetimes (M, g) for which the isometry group Isom(M, g) acts non‑properly on M, i.e., the action map G × M → M × M given by (g, p) ↦ (g·p, p) is not a proper map. The goal is to identify and organize all such spacetimes beyond currently known examples, characterizing the geometric or causal features that permit non‑proper isometric actions.

Background

The paper proves that under the no observer horizons (NOH) condition the time‑orientation‑preserving isometry group acts properly, yielding strong structural consequences (existence of invariant Cauchy temporal functions and a semi‑direct product decomposition of the isometry group). This contrasts with Riemannian manifolds, where isometry actions are always proper.

In Lorentzian geometry, non‑proper actions are tied to relativistic phenomena and occur in important spacetimes such as Minkowski, de Sitter, and anti‑de Sitter. Despite their significance, the general landscape of globally hyperbolic spacetimes admitting non‑proper isometry actions remains largely uncharted; prior work is limited, notably to results in two dimensions for spatially compact globally hyperbolic spacetimes.