The spacelike restart level set of the time separation from a point in Minkowski spacetime

(The distance formula.) For $r\in J^{-}(q)$ the displacement $q-r=(\mathrm{\Delta }t,\mathrm{\Delta }x)$ is causal, and for $r\mathrm{\ne }q$ the unique maximizing geodesic image is the straight causal segment, whose length is $\sqrt{(\mathrm{\Delta }t{)}^2-\mathrm{\mid }\mathrm{\Delta }x{\mathrm{\mid }}^2}=\sqrt{(t-1{)}^2-\mathrm{\mid }x{\mathrm{\mid }}^2}$ when the displacement is timelike; for nonzero lightlike displacement the value is $0$; for $r=q$ the degenerate constant-geodesic convention also gives zero; this is the Minkowski instance of the attainability and geodesy of maximizers in Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions clause (a).

(Smoothness and spacality.) On the open set $U:=\{r:{\tau }_q(r)>0\}=I^{-}(q)$ the function $f(t,x):=(t-1{)}^2-\mathrm{\mid }x{\mathrm{\mid }}^2$ is smooth with Lorentzian gradient $(-2(t-1),-2x)$, which is timelike on $U$ (there $(t-1{)}^2>\mathrm{\mid }x{\mathrm{\mid }}^2$); hence its level set $\{f={\tau }_0^2\}$, which is $S$, is a smooth embedded hypersurface with timelike normal and is therefore spacelike (Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, $C^r$ and smooth maps between smooth manifolds).

(At most one meeting along a causal curve.) Since ${\tau }_q$ is continuous and $g(\mathrm{\nabla }{\tau }_q,\cdot )$ has no nonzero causal kernel on $U$, the function $t\mapsto {\tau }_q(\gamma (t))$ is strictly monotone along every causal curve segment contained in $U$; hence such a segment meets the level ${\tau }_0$ in at most one parameter value, and a causal curve segment with ${\tau }_q$-values straddling ${\tau }_0$ meets $S$ at least once by the intermediate value theorem (Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions clause (b), Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point). If a causal curve met $S$ in two parameter values, the subsegment between them would join two points of $S$ with equal ${\tau }_q$-value, contradicting strict monotonicity; if the subsegment left $U$ it would contain a point $p$ with ${\tau }_q(p)=0$, that is $p\ll ̸q$; but $p\le \gamma (t_2)\ll q$ and push-up gives $p\ll q$, a contradiction (The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion, Causal curves, chronological and causal reachability, and inextendibility).

(Radial description.) For $r=(1-\sqrt{{\tau }_0^2+\mathrm{\mid }{x}_s{\mathrm{\mid }}^2},x_s)\in S$ the future segment to $q$ is $s\mapsto r+\frac{s}{{\tau }_0}(\sqrt{{\tau }_0^2+\mathrm{\mid }{x}_s{\mathrm{\mid }}^2},-x_s)$, which is a geodesic with timelike velocity of constant norm $1$, hence a future-directed timelike geodesic segment of length ${\tau }_0$ ending at $q$. Conversely every past endpoint of a future-directed timelike geodesic segment of length ${\tau }_0$ ending at $q$ lies on $S$.