A class of cosmological models with spatially constant sign-changing curvature (Q6062708)

Summary: We construct globally hyperbolic spacetimes such that each slice \(\{t=t_0\}\) of the universal time \(t\) is a model space of constant curvature \(k(t_0)\) which may not only vary with \(t_0 \in\mathbb{R}\) but also change its sign. The metric is smooth and slightly different to FLRW spacetimes, namely, \(g=-dt^2 +dr^2 + S_{k(t)}^2 (r) g_{\mathbb{S}^{n-1}}\), where \(g_{\mathbb{S}^{n-1}}\) is the metric of the standard sphere, \(S_{k(t)}(r)=\sin (\sqrt{k(t)} r)/\sqrt{k(t)}\) when \(k(t)\geq 0\) and \(S_{k(t)}(r)=\sinh (\sqrt{-k(t)} r)/\sqrt{-k(t)}\) when \(k(t)\leq 0\). In the open case, the \(t\)-slices are (non-compact) Cauchy hypersurfaces of curvature \(k(t)\leq 0\), thus homeomorphic to \(\mathbb{R}^n\); a typical example is \(k(t)=-t^2\) (i.e., \(S_{k(t)}(r)=\sinh (tr)/t)\). In the closed case, \(k(t)>0\) somewhere, a slight extension of the class shows how the topology of the \(t\)-slices changes. This makes at least one comoving observer to disappear in finite time \(t\) showing some similarities with an inflationary expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces homeomorphic to spheres, not all of them \(t\)-slices.