The rational timelike exponential set and the rational exponential basis in the flat vacuum development

(One exponential step produces the rational points with nonzero time coordinate.) By the flat geodesic calculation in [L2], the exponential map is ${\exp }_d(v)=d+v$ for every tangent vector. For $d=(0,q)\in \{0\}\times {\mathbb{Q}}^3=D_0$ and a rational vector $v=(a,w)\in {\mathbb{Q}}^4$ the point $d+v=(a,q+w)$ lies in $D_1$ whenever $v$ is timelike, i.e. $a^2>\mathrm{\mid }w{\mathrm{\mid }}^2$. Given $(a,b)\in {\mathbb{Q}}^4$ with $a\mathrm{\ne }0$, choose $q:=b$ and $w:=0$; then $v=(a,0)$ satisfies $\eta (v,v)=-a^2<0$, so it is timelike, and $(a,b)\in D_1$. Every exponential step from a rational point using a rational vector remains rational, so induction gives $D\subseteq {\mathbb{Q}}^4$. Hence $\{r\in {\mathbb{Q}}^4:r^0\mathrm{\ne }0\}\subseteq D_1$, and $D_0$ supplies the points with $r^0=0$, so ${\mathbb{Q}}^4\subseteq D$.

(Countability and density.) $D$ is countable by the construction (A countable dense subset of the universal development by rational timelike exponential steps from a dense subset of the initial surface, Finite, countably infinite, countable, uncountable) and contains ${\mathbb{Q}}^4$, which is dense in ${\mathbb{R}}^{1,3}$ by the density of $\mathbb{Q}$ in $\mathbb{R}$ coordinatewise (Both $\mathbb{Q}$ and $\mathbb{R}\setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable); hence $D$ is dense and meets $j(\mathrm{\Sigma })=\{t=0\}$ in $\{0\}\times {\mathbb{Q}}^3$.

(The basis.) The repaired abstract family uses translated rational tangent balls: here ${\exp }_d(B(a,r))=B(d+a,r)$. Since $d,a$ are rational, its centers are rational; conversely every rational center is obtained by taking $a=0$ and that center as $d$. Thus it is exactly the Euclidean rational-center family declared above. For $d\in D$ and rational $r>0$ the set ${\exp }_d(B(0;r))=d+B(0;r)$ is the open Euclidean ball of radius $r$ around $d$ centred at a rational point with rational radius; the family of all such balls is a countable basis of the Euclidean topology of ${\mathbb{R}}^4$ (Basis and subbasis for a topology, and the topology generated by a family of sets, Second countability: an at most countable basis for the topology). In particular $\mathcal{B}$ as defined in the statement is a countable basis of the topology of $M$, so $M$ is second countable, in accordance with An explicit rational-exponential basis for the universal development quotient.