The flat model is globally hyperbolic and its inertial level sets are Cauchy hypersurfaces

(Reachability is the closed cone.) If $q=p$ then $q-p=0\in \overline{C^{+}}$ and $q\in J^{+}(p)$, so both displays of (a) hold. If $q\mathrm{\ne }p$ and $q-p\in \overline{C^{+}}$, then by [L2] the straight path $s\mapsto p+s(q-p)$, $s\in [0,1]$, has the nonzero future causal tangent $q-p$ and is a future causal curve from $p$ to $q$; conversely, if a future causal curve joins $p$ to $q$, then its coordinate displacement $(x^0(q)-x^0(p),z(q)-z(p))$ satisfies the Lipschitz bound of [L3], which is precisely $q-p\in \overline{C^{+}}$ in the chart. This proves the second display of (a). For the first display: if $q-p\in C^{+}$, the same straight path has tangent $q-p$ in the open cone, hence is a future timelike curve from $p$ to $q$, so $q\in I^{+}(p)$; conversely, let $\gamma :[a,b]\to M$ be a future timelike curve from $p$ to $q$ and put $c(t):=x(\gamma (t))=(c^0(t),c_s(t))$. At almost every $t$ one has $c^0{}^{\mathrm{\prime }}(t)>\mathrm{\mid }{c}_s^{\mathrm{\prime }}(t){\mathrm{\mid }}_{\mathrm{e}}$, and the set of such $t$ has full measure in $[a,b]$; by the fundamental theorem of calculus of [L3], $c^0(b)-c^0(a)={\int }_a^b{c}^0{}^{\mathrm{\prime }}$ and $c_s(b)-c_s(a)={\int }_a^b{c}_s^{\mathrm{\prime }}$, so $c^0(b)-c^0(a)-\mid c_s(b)-c_s(a){\mid }_{\mathrm{e}}\ge {\int }_a^b(c^0{}^{\mathrm{\prime }}-\mathrm{\mid }{c}_s^{\mathrm{\prime }}{\mathrm{\mid }}_{\mathrm{e}})>0$, because the integrand is nonnegative almost everywhere by the Lipschitz bound and strictly positive on a set of full measure. Hence $x^0(q)-x^0(p)>\mathrm{\mid }z(q)-z(p){\mathrm{\mid }}_{\mathrm{e}}$, that is $q-p\in C^{+}$. This proves the first display of (a).

(The level sets are Cauchy hypersurfaces.) By [L4] each ${\mathrm{\Sigma }}_c$ is a spacelike embedded hypersurface. Let $\gamma :I\to M$ be an inextendible causal curve; by Monotonicity of inertial time and the Lipschitz graph property for causal curves in the flat model clause (c), $x^0\circ \gamma$ is a strictly increasing surjection onto $\mathbb{R}$, so for the value $c$ the parameter set $\{t:x^0(\gamma (t))=c\}$ is a singleton. Thus every inextendible causal curve meets ${\mathrm{\Sigma }}_c$ in exactly one parameter value, and ${\mathrm{\Sigma }}_c$ is a Cauchy hypersurface. This proves (b).

(Compactness of the diamonds.) By step 1.1, $J^{+}(p)\cap J^{-}(q)=\{r:r-p\in \overline{C^{+}}\text{ and }q-r\in \overline{C^{+}}\}$. This set is closed in ${\mathbb{R}}^4$: the maps $r\mapsto r-p$ and $r\mapsto q-r$ are continuous and $\overline{C^{+}}$ is closed, being described by the non-strict inequality $v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }$. It is bounded: if $r$ lies in it then $x^0(r)-x^0(p)\ge 0$ and $x^0(q)-x^0(r)\ge 0$, so $x^0(r)\in [x^0(p),x^0(q)]$, and $\mathrm{\mid }z(r)-z(p){\mathrm{\mid }}_{\mathrm{e}}\le x^0(r)-x^0(p)\le x^0(q)-x^0(p)$; in particular $r$ lies in a closed bounded box. By [L5] the diamond is compact. This proves (c).

(Strong causality.) Let $p\in M$, write $e_0:=(1,0,0,0)\in M$ for the first coordinate vector of the chart $x$, and let $U\ni p$ be open. Choose $\epsilon >0$ so small that the double cone $V:=I^{+}(p-\epsilon e_0)\cap I^{-}(p+\epsilon e_0)$ is contained in $U$: in the coordinates of $x$, $V$ is a neighborhood of $p$ contained in the coordinate box of radius $2\epsilon$ around $p$, and $\epsilon$ can be taken below half the Euclidean distance from $p$ to the complement of $U$. Then $V$ is open and $p\in V$. To see that $V$ is causally convex for causal curves that meet it twice, let $\gamma :[a,b]\to M$ be a future causal curve with $\gamma (a),\gamma (b)\in V$ and let $a<t<b$. Since $\gamma {\mathrm{\mid }}_{[a,t]}$ and $\gamma {\mathrm{\mid }}_{[t,b]}$ are future causal curves, step 1.1 gives $\gamma (t)-\gamma (a)\in \overline{C^{+}}$ and $\gamma (b)-\gamma (t)\in \overline{C^{+}}$. Since $\gamma (a)\in V$, step 1.1 gives $\gamma (a)-(p-\epsilon e_0)\in C^{+}$; by [L6], $\gamma (t)-(p-\epsilon e_0)=(\gamma (a)-(p-\epsilon e_0))+(\gamma (t)-\gamma (a))\in C^{+}$, that is $p-\epsilon e_0\ll \gamma (t)$ by step 1.1. Similarly $\gamma (b)\in V$ gives $(p+\epsilon e_0)-\gamma (b)\in C^{+}$, and [L6] applied to $((p+\epsilon e_0)-\gamma (b))+(\gamma (b)-\gamma (t))$ gives $(p+\epsilon e_0)-\gamma (t)\in C^{+}$, that is $\gamma (t)\ll p+\epsilon e_0$. Hence $\gamma (t)\in V$: no causal curve leaves $V$ and later returns to it, and the model is strongly causal. This proves (d).

(Conclusion.) By step 1.2 the inertial level sets are Cauchy hypersurfaces and by Chronology, causality, strong causality, global hyperbolicity and time functions clause (e) the coordinate $x^0$ is temporal by Monotonicity of inertial time and the Lipschitz graph property for causal curves in the flat model clause (d); step 2.2 gives strong causality and step 2.1 gives compactness of all causal diamonds, so the model is globally hyperbolic in the sense of clause (c) of that definition. With steps 1.1, 2.1 and 2.2 this proves all the claims.