论文标题
一类具有空间恒定签名曲率的宇宙学模型
A class of cosmological models with spatially constant sign-changing curvature
论文作者
论文摘要
我们构建全球双曲线空间，以使通用时间$ t $的每个切片$ \ {t = t_0 \} $是恒定曲率$ k（t_0）$的模型空间，它不仅可能随$ t_0 \ in \ mathbb {r} $而变化，还可以改变其标志。指标平稳且与flrw空间略有不同，即$ g = -dt^2+ dr^2+ s_ {k（t）}^2（r）g _ {\ Mathbb {\ MathBB {s}^{n-1}}} $ $ s_ {k（t）}（r）= \ sin（\ sqrt {k（t）} \，r）/\ sqrt {k（t）} $时$ k（t）\ geq 0 $ and $ s_ {k（k（k（t）}（k（t）}（t）}（t）}（r）= \ sinh（r）= \ sinh（\ sinh） r）/\ sqrt {-k（t）} $当$ k（t）\ leq 0 $。在开放式情况下，$ t $ - slices是（非coct）曲率$ k（t）\ leq 0 $的cauchy hypersurfaces，因此同型到$ \ m athbb {r}^n $;一个典型的示例是$ k（t）= -t^2 $（即$ s_ {k（t）}（r）= \ sinh（tr）/t $）。在封闭情况下，$ k（t）> 0 $某处，班级的略有扩展显示了$ t $ - slices的拓扑如何变化。这使至少一个共同观察者在有限的时间$ t $中消失，表现出与通货膨胀的相似之处。无论如何，时空是由Cauchy Hypersurfaces同构为球体散发的，而不是全部。
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.
