论文标题
零距离编码因果关系
The Null distance encodes causality
论文作者
论文摘要
具有时间函数$τ$的Lorentzian歧管可以使用NULL距离，$ \ hat {d}_τ$（由Sormani和Vega定义）转换为公制空间。我们表明，如果时间函数是安德森，加洛韦和霍华德以及瓦尔德和YIP所研究的适当的常规宇宙学时间函数，或者如果更一般地，它满足了Chruściel，Grant和Minguzzi的抗Lipschitz状态\ hat {d}_τ（p，q）=τ（q） - τ（p）\ iff q \ textrm {位于} p的因果未来。结果，$$在尺寸$ n+1 $，$ n \ ge 2 $中，我们证明，如果两个这样的空间之间存在一张二原始图，$ f：m_1 \至m_2 $，它可以保留宇宙学时间函数，$τ_2（f（p））=τ_1（p）$ for $ p \ p） $ \ hat {d} _ {τ_2}（f（p），f（q））= \ hat {d} _ {τ_1} _ {τ_1}（p，q）$ for noy $ p，q \ in m_1 $，然后在它们之间有lorentzian等轴测，$ f_*g_1 = g_1 = g_1 = g_2 $。这产生了一个规范的程序，使我们能够将这种空间转换为具有因果结构和时间功能的独特度量空间。这将在我们即将进行的工作中应用于定义时空内在的平面收敛。
A Lorentzian manifold endowed with a time function, $τ$, can be converted into a metric space using the null distance, $\hat{d}_τ$, defined by Sormani and Vega. We show that if the time function is a proper regular cosmological time function as studied by Andersson, Galloway and Howard, and also by Wald and Yip, or if, more generally, it satisfies the anti-Lipschitz condition of Chruściel, Grant and Minguzzi, then the causal structure is encoded by the null distance in the following sense: $$ \hat{d}_τ(p,q)=τ(q)-τ(p) \iff q \textrm{ lies in the causal future of } p. $$ As a consequence, in dimension $n+1$, $n\ge 2$, we prove that if there is a bijective map between two such spacetimes, $F: M_1\to M_2$, which preserves the cosmological time function, $ τ_2(F(p))= τ_1(p)$ for any $ p \in M_1$, and preserves the null distance, $\hat{d}_{τ_2}(F(p),F(q))=\hat{d}_{τ_1}(p,q)$ for any $p,q\in M_1$, then there is a Lorentzian isometry between them, $F_*g_1=g_2$. This yields a canonical procedure allowing us to convert such spacetimes into unique metric spaces with causal structures and time functions. This will be applied in our upcoming work to define Spacetime Intrinsic Flat Convergence.
