The existence of a Cauchy set implies strong causality and compact causal diamonds

(Preliminaries.) By [L1] there is no closed causal curve, $S$ is closed, and $M=I^{-}(S)\bigsqcup S\bigsqcup I^{+}(S)$. Consequently, for an inextendible future causal curve $\gamma :I\to M$ we may speak of the parameter $t_0\in I$ with $\gamma (t_0)\in S$.

(The crossing property.) Let $\gamma :I\to M$ be an inextendible future causal curve and $t_0\in I$ with $\gamma (t_0)\in S$. Then $t>t_0$, $t\in I$ implies $\gamma (t)\in I^{+}(S)$, and $t<t_0$ implies $\gamma (t)\in I^{-}(S)$. Indeed, suppose $t>t_0$ and $\gamma (t)\mathrm{\notin }{I}^{+}(S)$; by step 1.1 $\gamma (t)\in I^{-}(S)$ (it is not in $S$, as $t_0$ is the unique element of ${\gamma }^{-1}[S]$). Choose $s\in S$ with $\gamma (t)\ll s$. If $s=\gamma (t_0)$, the concatenation of the causal segment $\gamma {\mathrm{\mid }}_{[t_0,t]}$ with a future timelike curve from $\gamma (t)$ to $\gamma (t_0)$ is a nonconstant closed future causal curve through $\gamma (t_0)$ (nonconstant because $\gamma (t)\ll \gamma (t_0)$ forces $\gamma (t)\mathrm{\ne }\gamma (t_0)$), contradicting step 1.1. If $s\mathrm{\ne }\gamma (t_0)$, the same concatenation is a future causal curve from $\gamma (t_0)$ to $s$ with both endpoints in $S$; by [L2] it lies on an inextendible causal curve whose parameter set over $S$ therefore contains the two distinct parameters corresponding to $\gamma (t_0)$ and $s$, contradicting the Cauchy property. Hence $\gamma (t)\in I^{+}(S)$. The case $t<t_0$ is dual.

(The contradiction package from a closure relation.) Let ${\alpha }^{+}:[0,\mathrm{\infty })\to M$ be a future-inextendible future causal curve and ${\alpha }^{-}:(-\mathrm{\infty },0]\to M$ a past-inextendible future causal curve such that for all $t_{+}\ge 0$, $t_{-}\le 0$ there are $u_k\le v_k$ with $u_k\to {\alpha }^{+}(t_{+})$ and $v_k\to {\alpha }^{-}(t_{-})$. Then this situation is impossible. Indeed, by [L2] there is an inextendible future causal curve ${\gamma }^{+}$ extending ${\alpha }^{+}$ with ${\gamma }^{+}{\mathrm{\mid }}_{[0,\mathrm{\infty })}={\alpha }^{+}$ and an inextendible future causal curve ${\gamma }^{-}$ whose restriction to $(-\mathrm{\infty },0]$ is ${\alpha }^{-}$. By the crossing property, step 2.1, there are $t_{+}^{\mathrm{\prime }}\in [0,\mathrm{\infty })$ with ${\alpha }^{+}(t_{+}^{\mathrm{\prime }})\in I^{+}(S)$ (take $t_{+}^{\mathrm{\prime }}$ larger than the unique $S$-parameter of ${\gamma }^{+}$ and than $0$) and $t_{-}^{\mathrm{\prime }}\le 0$ with ${\alpha }^{-}(t_{-}^{\mathrm{\prime }})\in I^{-}(S)$. Choose $s_1\in S$ with $s_1\ll {\alpha }^{+}(t_{+}^{\mathrm{\prime }})$ and $s_2\in S$ with ${\alpha }^{-}(t_{-}^{\mathrm{\prime }})\ll s_2$. Let $u_k\le v_k$ be the corresponding sequences. Since $I^{+}(s_1)$ is open and contains ${\alpha }^{+}(t_{+}^{\mathrm{\prime }})$, for large $k$ we have $s_1\ll u_k$; since $I^{-}(s_2)$ is open and contains ${\alpha }^{-}(t_{-}^{\mathrm{\prime }})$, for large $k$ we have $v_k\ll s_2$. Fix such a $k$. Then $s_1\ll u_k\le v_k$, so $s_1\ll v_k$ by [L4]; and $s_1\ll v_k\ll s_2$, so $s_1\ll s_2$ by transitivity of $\ll$ [L4]. If $s_1\mathrm{\ne }{s}_2$, this contradicts the acausality of $S$ [L1]. If $s_1=s_2=:s$, then the concatenation of a future timelike curve from $s$ to $u_k$, the future causal curve from $u_k$ to $v_k$ and a future timelike curve from $v_k$ to $s$ is a nonconstant closed future causal curve through $s$ (nonconstant because $s\ll u_k$), contradicting step 1.1.

(The causal relation is closed.) Let $p_k\le q_k$ with $p_k\to p$ and $q_k\to q$. If infinitely many $p_k=q_k$, then $p=q$ (by uniqueness of limits) and $p\le q$. Otherwise, for large $k$ choose a future causal curve ${\gamma }_k$ with ${\gamma }_k(0^{\mathrm{\prime }})=p_k$, ${\gamma }_k(1^{\mathrm{\prime }})=q_k$ and reparametrize it by $h$-arclength: ${\beta }_k:[0,L_k]\to M$, ${\beta }_k(0)=p_k$, ${\beta }_k(L_k)=q_k$; the $L_k$ are positive. If the sequence $L_k$ is bounded, clause (i) of [L3] (applied to a subsequence with $L_k\to b$) gives a future causal curve $\beta :[0,b]\to M$ from $p$ to $q$, and either $b>0$ and $p\le q$ via $\beta$, or $b=0$, in which case $p=q$ by clause (i) and $p\le p$. If $L_k\to \mathrm{\infty }$ along a subsequence, clause (ii) of [L3] gives future-inextendible ${\alpha }^{+}$ from $p$ and past-inextendible ${\alpha }^{-}$ to $q$ with the closure relation, which is impossible by step 3.1. Hence $p\le q$.

(Strong causality.) Suppose strong causality fails at $p\in M$ with neighbourhood $U$. By [L5] there are an open $W\ni p$ with $\bar{W}\subseteq U$ and future causal curves ${\sigma }_k:[0,1]\to M$ with endpoints $x_k,y_k\in W$, $x_k\to p$, $y_k\to p$, and ${\sigma }_k([0,1])\subseteq ̸W$ for every $k$. Fix a coordinate ball $B=x^{-1}(B_{r_0})\subseteq W$ around $p$ with $x(p)=0$ and let $c_1:=\sqrt{c}{r}_0\mathrm{/}2>0$, where $c>0$ is a lower bound for the $h$-length density on the closed coordinate ball against the Euclidean coordinate speed (Local comparison of a riemannian metric with the euclidean metric, The auxiliary Riemannian metric h=g+2n-flat tensor n-flat is complete after a proper-exhaustion conformal factor, and its closed bounded sets are compact clause (a)). For all sufficiently large $k$, $\mathrm{\mid }x(x_k){\mathrm{\mid }}_{\mathrm{e}}<r_0\mathrm{/}2$. The first exit point $w_k$ of ${\sigma }_k$ from $B$ then satisfies $\mathrm{\mid }x(w_k)-x(x_k){\mathrm{\mid }}_{\mathrm{e}}\ge r_0-\mathrm{\mid }x(x_k){\mathrm{\mid }}_{\mathrm{e}}>r_0\mathrm{/}2$. The local metric lower bound and absolute-continuity calculus give ${\mathrm{\ell }}_h({\sigma }_k{\mathrm{\mid }}_{[0,{\tau }_k]})\ge \sqrt{c}{\int }_0^{{\tau }_k}\mathrm{\mid }(x\circ {\sigma }_k{)}^{\mathrm{\prime }}{\mathrm{\mid }}_{\mathrm{e}}ds\ge \sqrt{c}\mathrm{\mid }x(w_k)-x(x_k){\mathrm{\mid }}_{\mathrm{e}}>c_1$, where ${\tau }_k$ is the first-exit parameter. Reparametrize each ${\sigma }_k$ by $h$-arclength: ${\alpha }_k:[0,L_k]\to M$, with ${\alpha }_k(0)=x_k$, ${\alpha }_k(L_k)=y_k$; the same first-exit estimate gives $L_k\ge c_1$ for all large $k$. If $L_k$ is bounded along a subsequence, clauses (i) and (iii) of [L3] (with $x=z=p$ and $b=\lim L_k\ge c_1>0$) produce a nonconstant closed future causal curve through $p$, contradicting step 1.1. If $L_k\to \mathrm{\infty }$, clause (ii) of [L3] produces the impossible configuration of step 3.1. Hence strong causality holds at every $p$.

(The causal diamonds are compact.) Let $p,q\in M$ and $A:=J^{+}(p)\cap J^{-}(q)$; it is closed in $M$ because $J^{+}(p)$ and $J^{-}(q)$ are closed by step 4.1. With the metric $d_h$ inducing the manifold topology [L6], if $A$ were not compact then [L7] would give a sequence $r_k\in A$ with no convergent subsequence. For each $k$ choose a future causal curve from $p$ to $q$ through $r_k$ and reparametrize by $h$-arclength as ${\gamma }_k:[0,M_k]\to M$, with ${\gamma }_k(s_k)=r_k$. If $(M_k)$ is bounded, pass to $M_k\to b<\mathrm{\infty }$. If $b=0$, then $d_h(r_k,p)\le M_k\to 0$, contradiction. If $b>0$, clause (i) of [L3] gives the common-domain curves ${\tilde{\gamma }}_k(t)={\gamma }_k((M_k\mathrm{/}b)t)$ converging uniformly on $[0,b]$ to $\gamma$. Put $t_k:=b{s}_k\mathrm{/}{M}_k\in [0,b]$ and pass to a subsequence with $t_k\to t_{\ast }$. Then $r_k={\tilde{\gamma }}_k(t_k)\to \gamma (t_{\ast })$ by uniform convergence and continuity, again a contradiction. Hence $M_k\to \mathrm{\infty }$ along a subsequence, and clause (ii) of [L3] produces the impossible configuration of step 3.1. Therefore $A$ is compact.

(Conclusion.) Clause (a) is step 1.1, clause (b) is step 4.2, and clause (c) is step 5.1; strong causality and compactness of all causal diamonds are exactly the definition of global hyperbolicity in Chronology, causality, strong causality, global hyperbolicity and time functions clause (c).