Worked two-leg itinerary against the straight clock at speed 3c/5

By [L1] the displacement columns are $d=q-p=(2cT,0,0,0)$, $r-p=(cT,vT,0,0)$ and $q-r=(cT,-vT,0,0)$; their interval values are $\eta (d,d)=-4c^2{T}^2$, $\eta (r-p,r-p)=-c^2{T}^2+v^2{T}^2=-c^2{T}^2(1-v^2\mathrm{/}{c}^2)$, $\eta (q-r,q-r)=-c^2{T}^2(1-v^2\mathrm{/}{c}^2)$; since $0<\mathrm{\mid }v\mathrm{\mid }<c$ gives $c-\mathrm{\mid }v\mathrm{\mid }>0$, hence $c>\mathrm{\mid }v\mathrm{\mid }$, the time component of each of the three columns strictly dominates the Euclidean norm of its spatial part, and all three displacements are future timelike by [L2].

For (a): the straight worldline from $p$ to $q$ runs over a parameter interval of length $2T$ with constant tangent $(c,0,0,0)$, whose interval value is $-c^2$, so by [L3] its elapsed proper time is $c^{-1}\cdot 2T\cdot \sqrt{c^2}=2T$, agreeing with the value $c^{-1}\sqrt{-\eta (d,d)}=c^{-1}2cT$ computed from the displacement.

For (b): each leg runs over a parameter interval of length $T$ with constant tangent $(c,\pm v,0,0)$, whose interval value is $-c^2+v^2=-c^2(1-v^2\mathrm{/}{c}^2)$; by [L3] each leg has proper time $c^{-1}T\sqrt{c^2-v^2}=T\sqrt{1-v^2\mathrm{/}{c}^2}=T\mathrm{/}\gamma$, the last equality because $\gamma =(1-v^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}>0$; the two-leg total is the sum $2T\mathrm{/}\gamma$ by the additivity clause of [L3].

For (d): $\mu (0)=0$, $\mu (1)=1$ and ${\mu }^{\mathrm{\prime }}(s)=s+1\mathrm{/}2>0$ on $[0,1]$. For $0\le s<t\le 1$, $\mu (t)-\mu (s)=(t-s)(t+s+1)\mathrm{/}2>0$, proving strict increase; and for each $\lambda \in [0,1]$ the value $s=(-1+\sqrt{1+8\lambda })\mathrm{/}2$ lies in $[0,1]$ and satisfies $\mu (s)=\lambda$, proving surjectivity by [L5]. Thus $\mu$ is a strictly increasing $C^1$ surjection of $[0,1]$ onto itself; the worldline $y(s)=x^{-1}(2cT\mu (s),0,0,0)$ has parameter path $(2cT\mu (s),0,0,0)$ with tangent $(2cT{\mu }^{\mathrm{\prime }}(s),0,0,0)$, whose time component $2cT{\mu }^{\mathrm{\prime }}(s)$ is positive and whose interval value is $-(2cT{\mu }^{\mathrm{\prime }}(s){)}^2$, so $y$ is a future timelike worldline by [L2] with $\sqrt{-\eta (y^{\mathrm{\prime }},y^{\mathrm{\prime }})}=2cT{\mu }^{\mathrm{\prime }}(s)$; its image is $\{x^{-1}(2cT\lambda ,0,0,0):\lambda \in [0,1]\}$, the straight segment from $p$ to $q$.

For (c): by [L7] $0<v^2\mathrm{/}{c}^2<1$, so $0<1-v^2\mathrm{/}{c}^2<1$ and strict monotonicity of squaring on the nonnegatives gives $\sqrt{1-v^2\mathrm{/}{c}^2}<1$; hence $1\mathrm{/}\gamma =\sqrt{1-v^2\mathrm{/}{c}^2}<1$ and $2T\mathrm{/}\gamma <2T$. This agrees with [L4]: the displacement $r-p=(cT,vT,0,0)$ has second coordinate $vT\mathrm{\ne }0$ because $v\mathrm{\ne }0$ and $T>0$, so $r$ is not on the straight segment from $p$ to $q$ (all of whose events have vanishing second coordinate), and the strictness clause of [L4] applies to the two-leg itinerary.

Its elapsed proper time is $c^{-1}{\int }_0^{1}2cT{\mu }^{\mathrm{\prime }}(s)\mathrm{d}s=2T(\mu (1)-\mu (0))=2T$ by [L6] applied to the continuous derivative ${\mu }^{\mathrm{\prime }}$, so equality in the endpoint bound is attained although $\mu$ is not affine; by [L4] this is consistent with the equality characterization, since $\mu$ is a strictly increasing future-directed reparametrization of the straight segment.

Clauses (a)-(d) are verified in steps 2.1, 2.2, 3.1, 3.2: the readings $2T$ (straight) and $2T\mathrm{/}\gamma <2T$ (two-leg), the strict comparison, the numerical values $8T\mathrm{/}5$ and $2T$ at $v=3c\mathrm{/}5$, and the non-affine equality case of (d) are all finite evaluations of the model's definitions.