Two nonparallel future null legs compose to a future timelike displacement

Given: An inertial frame with chart $x$ and ${\eta }_0=\mathrm{diag}(-1,1,1,1)$, the event $p=x^{-1}(0,0,0,0)$, the piecewise straight path $x_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}}$ with legs $x^{-1}(s,s,0,0)$ on $[0,1]$ and $x^{-1}(s,1,s-1,0)$ on $[1,2]$, and the endpoint $q=x^{-1}(2,1,1,0)$.

Diagonal form and classification: for coordinates $v=(v^0,v_s)$, $\eta (v,v)=-(v^0{)}^2+\mathrm{\mid }{v}_s{\mathrm{\mid }}^2$; a nonzero vector is null when $\eta (v,v)=0$, spacelike when $\eta (v,v)>0$, and timelike when $\eta (v,v)<0$; future null means nonzero null with positive time coordinate (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones).

Coordinate criterion and sums: $q\in I^{+}(p)$ iff $D^0>\mathrm{\mid }{D}_s\mathrm{\mid }$ for $D=x(q)-x(p)$, and the sum of finitely many future causal vectors is future causal, strictly timelike if one summand is future timelike (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability). The nonparallel null case is a local calculation, not an extra conclusion of that supplier: write the vectors $(a,p)$ and $(b,q)$ with $a=\mathrm{\mid }p\mathrm{\mid }>0$ and $b=\mathrm{\mid }q\mathrm{\mid }>0$. If they are not parallel, $p\mathrm{/}a\mathrm{\ne }q\mathrm{/}b$, so $\mathrm{\mid }p\mathrm{/}a-q\mathrm{/}b{\mathrm{\mid }}^2=2-2(p\cdot q)\mathrm{/}(ab)>0$ by the spatial Euclidean structure. Thus $\eta ((a,p)+(b,q),(a,p)+(b,q))=2(p\cdot q-ab)=-ab\mathrm{\mid }p\mathrm{/}a-q\mathrm{/}b{\mathrm{\mid }}^2<0$, and the positive time component $a+b$ makes the sum future timelike. The spatial norm and scalar product are those of the named inertial frame (The Euclidean spatial structure and cross product of an inertial frame, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

Straight paths and reachability: the straight path $s\mapsto p+s(q-p)$ has constant tangent $q-p$, and for $q\mathrm{\ne }p$ one has $q\in J^{+}(p)$ iff that straight path is a future causal path, with $I^{+}(p)$ characterized by the straight path being future timelike; part (a) of the theorem also says that a future causal path with a future timelike one-sided tangent has future timelike displacement (Future causal reachability, absence of causal loops, time monotonicity and the straight-worldline proper-time maximum, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).

Piecewise $C^1$ curves: the concatenation of the two straight legs is a piecewise $C^1$ curve with constant tangents $(1,1,0,0)$ and $(1,0,1,0)$ on the open pieces and one-sided difference-quotient tangents equal to those vectors at every point of $[0,2]$ (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, The Euclidean spatial structure and cross product of an inertial frame, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).