Worked boost family on the spacelike pair $(11,13,0,0)$: simultaneity at $b=11\mathrm{/}13$ and both time orders

Worked computation in the chart $x$ of an inertial frame of an affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model) with ${\eta }_0=\mathrm{diag}(-1,1,1,1)$. Let $p:=x^{-1}(0,0,0,0)$ and $q:=x^{-1}(11,13,0,0)$; then $D:=x(q)-x(p)=(11,13,0,0)$ has $\eta (D,D)=-{11}^2+{13}^2=-121+169=48>0,$ so $q-p$ is spacelike and $\mathrm{\mid }{D}^0\mathrm{\mid }=11<13=\mathrm{\mid }{D}_s\mathrm{\mid }$, in accordance with clause (a) of Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign.

(a) Simultaneity at $b_0=11\mathrm{/}13$. With $e:=D_s\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }=e_1$ and $b_0:=D^0\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }=11\mathrm{/}13$, so that $\mathrm{\mid }{b}_0\mathrm{\mid }<1$, the chart $x_{b_0}$ of the inertial frame built from the passive boost $B(b_0{e}_1)$ by clause (d) of the same lemma satisfies $x_{b_0}^0(q)-x_{b_0}^0(p)={\gamma }_{b_0}(11-\frac{11}{13}\cdot 13)=0,{\gamma }_{b_0}=(1-\frac{121}{169}{)}^{-1\mathrm{/}2}=\frac{13}{\sqrt{48}}:$ the two events are simultaneous in that frame.

(b) Below $b_0$: positive time difference. With $b=3\mathrm{/}5$ one has ${\gamma }_b=(1-9\mathrm{/}25{)}^{-1\mathrm{/}2}=5\mathrm{/}4$ and $x_b^0(q)-x_b^0(p)=\frac{5}{4}(11-\frac{3}{5}\cdot 13)=\frac{5}{4}\cdot \frac{16}{5}=4>0.$

(c) Above $b_0$: negative time difference. With $b=12\mathrm{/}13$ one has ${\gamma }_b=(1-144\mathrm{/}169{)}^{-1\mathrm{/}2}=13\mathrm{/}5$ and $x_b^0(q)-x_b^0(p)=\frac{13}{5}(11-\frac{12}{13}\cdot 13)=\frac{13}{5}(-1)=-\frac{13}{5}<0.$

(d) The spatial separation in the three frames. The boost in the $(0,1)$-plane acts by $x^{\mathrm{\prime }0}=\gamma (x^0-b{x}^1)$, $x^{\mathrm{\prime }1}=\gamma (x^1-b{x}^0)$ (The passive Lorentz boost with parameter beta), so the spatial difference of the pair in the boosted chart is $x_b^1(q)-x_b^1(p)={\gamma }_b(13-11b)$. It equals $\sqrt{48}$ for $b=b_0$, $8$ for $b=3\mathrm{/}5$ and $37\mathrm{/}5$ for $b=12\mathrm{/}13$; in each case it is nonzero and $\mathrm{\mid }{x}_b^0(q)-x_b^0(p)\mathrm{\mid }<\mathrm{\mid }{x}_b^1(q)-x_b^1(p)\mathrm{\mid }$. The displacement stays spacelike because a boost is a Lorentz map and preserves $\eta$ (Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups, Linear and affine Lorentz maps, and the Lorentz group), so the three charts are charts of three inertial frames related to $x$ by passive boosts and the time-order sign of a spacelike-separated pair is an artifact of the chart, not of the pair. No physical synchronization procedure is asserted; simultaneity means equality of the chart's time coordinate.