Existence of spacetimes with Isom^↑(M,g) ≅ R ⋉ N not isomorphic to a direct product

Determine whether there exist Lorentzian spacetimes (M, g) whose time‑orientation‑preserving isometry group Isom^↑(M, g) is a semi‑direct product R ⋉ N that is not isomorphic to the direct product R × N. Ascertain whether such examples can occur, noting that this would require N to have more than one connected component.

Background

Within the paper’s main results under the NOH assumption, the isometry group splits as Isom^↑(M, g) = L ⋉ N with L ∈ {trivial, Z, R} and N compact. Moreover, when L ≅ R, the connected component of the identity is shown to split as a direct product R × N^0, so any failure of a direct product decomposition must come from disconnectedness of N.

Examples constructed in the paper realize Z ⋉ N (not necessarily as a direct product), and a lemma shows that for compact connected N and L ≅ R the semi‑direct product must in fact be a direct product. The remaining question is whether one can realize R ⋉ N that is genuinely non‑direct by allowing N to have multiple connected components.