The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space

By [F1] $d$ is future timelike and $T>0$; put $n:=d\mathrm{/}T$ and fix, by [F2], the inertial frame with origin $p$ and first vector $n$, with chart $\xi$ and coordinate path $y=\xi \circ x=(y^0,z)$; then $\xi (p)=(0,0,0,0)$ and $\xi (q)=(T,0,0,0)$, so $y(a)=(0,0,0,0)$ and $y(b)=(T,0,0,0)$.

On every open piece $(s_{i-1},s_i)$ the tangent is future timelike, so [F3] and [F4] give $y^0{}^{\mathrm{\prime }}>\mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }\ge 0$; the density is $g_i=\sqrt{(y^0{}^{\mathrm{\prime }}{)}^2-\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2}$ with finite positive improper integral, and setting $f_i:=g_i$ and $h_i:=y^0{}^{\mathrm{\prime }}-f_i$ one has $h_i\ge 0$ and $f_i\le y^0{}^{\mathrm{\prime }}$ pointwise on the piece, because $0\le (y^0{}^{\mathrm{\prime }}{)}^2-\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2\le (y^0{}^{\mathrm{\prime }}{)}^2$ and $f_i,y^0{}^{\mathrm{\prime }}\ge 0$ with monotonicity of the square root [L5].

Endpoint bound: $c\mathrm{\Delta }\tau (x)={\sum }_{i=1}^m{\int }_{(s_{i-1},s_i)}{f}_i\le {\sum }_{i=1}^m{\int }_{(s_{i-1},s_i)}{y}^0{}^{\mathrm{\prime }}=y^0(b)-y^0(a)=T$, where the first equality is the definition of $\mathrm{\Delta }\tau$ in [F3], the inequality is termwise monotonicity of the improper integrals from step 2.1 together with positivity of each term, the middle equality combines the endpoint increments of [F3] for $y^0{}^{\mathrm{\prime }}$ with additivity under the partition, and the last equality is $y^0(b)-y^0(a)={\xi }^0(q)-{\xi }^0(p)=T$ from step 1.1; this is (i).

Assume now $c\mathrm{\Delta }\tau (x)=T$. Then by step 3.1 and additivity, ${\sum }_{i=1}^m{\int }_{(s_{i-1},s_i)}{h}_i=0$ with every term $\ge 0$ by step 2.1, so by [L8] every term vanishes; since each $h_i$ is continuous and nonnegative on the open piece, $h_i\equiv 0$ there: if $h_i(u_0)>0$ at some point, continuity gives $h_i\ge h_i(u_0)\mathrm{/}2>0$ on a compact interval around $u_0$ inside the piece, whose ordinary integral is then positive by [L6], and the improper integral dominates it, a contradiction.

Consequently $f_i^2=(y^0{}^{\mathrm{\prime }}{)}^2$ on each piece, so $\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2=(y^0{}^{\mathrm{\prime }}{)}^2-f_i^2=0$ and hence $z^{\mathrm{\prime }}=0$ there; each coordinate of $z$ is continuous on the closed piece with vanishing derivative on its interior, so by [L7] each coordinate is constant on the closed piece, and since consecutive pieces share their corner values, $z\equiv z_0$ for a single $z_0\in {\mathbb{R}}^3$ on all of $[a,b]$.

Hence for every $s\in [a,b]$ the coordinates of $x(s)$ are $(y^0(s),z_0)$; by the inverse chart formula ${\xi }^{-1}(u)=p+E(u)$ with $E(u)={\sum }_a{u}^a{e}_a$ from [F2] and step 1.1, $x(s)=p+\frac{y^0(s)}{T}d+w_0$ with $w_0$ the vector whose coordinates are $(0,z_0)$, and evaluating at $s=a$, where $x(a)=p$ and $y^0(a)=0$, gives $w_0=0$; thus $x(s)=p+\sigma (s)d$ for $\sigma (s):=\frac{y^0(s)}{T}$, and $\sigma$ is continuous with $\sigma (a)=0$ and $\sigma (b)=1$.

Properties of $\sigma$: it is $C^1$ on every open piece with ${\sigma }^{\mathrm{\prime }}=y^0{}^{\mathrm{\prime }}\mathrm{/}T>0$ by step 2.1; its one-sided difference-quotient derivatives at the piece endpoints are the one-sided quotients of $y^0$ divided by $T$, hence finite and positive because the corresponding one-sided tangent of $x$ is future timelike, so its time component strictly dominates the spatial norm [F3, F4]; and it is strictly increasing: for $s<t$ in $[a,b]$ the restricted worldline $x{\mathrm{\mid }}_{[s,t]}$ is again weak piecewise future timelike, and its proper time satisfies $\mathrm{\Delta }{\tau }_{[s,t]}(x)=c^{-1}\sum {\int }_{(\text{induced pieces})}f$ with $f=y^0{}^{\mathrm{\prime }}$ on all open pieces by step 5.1, so by the endpoint increments and additivity of [F3] this equals $c^{-1}(y^0(t)-y^0(s))$, while the based proper-time function is strictly increasing by [F3]; hence $y^0(t)-y^0(s)>0$ and $\sigma (t)>\sigma (s)$.

By [L9], continuity and the endpoint values $\sigma (a)=0$, $\sigma (b)=1$ imply that $\sigma$ assumes every value of $[0,1]$; strict monotonicity keeps its values in $[0,1]$. This proves the forward implication of (ii): the $\sigma$ of step 6.1 and step 7.1 is a continuous strictly increasing surjection $[a,b]\to [0,1]$ with $\sigma (a)=0$, $\sigma (b)=1$, $C^1$ on the open pieces of the partition of $x$, with ${\sigma }^{\mathrm{\prime }}>0$ there and finite positive one-sided quotient derivatives, and $x(s)=p+\sigma (s)d$. The tangent form of the statement is immediate in this direction, since $x^{\mathrm{\prime }}(s)={\sigma }^{\mathrm{\prime }}(s)d$ with ${\sigma }^{\mathrm{\prime }}(s)>0$ at every differentiable point of every open piece.

Conversely, suppose $x(s)=p+\sigma (s)(q-p)=p+\sigma (s)d$ for a $\sigma$ as in (ii) with respect to a finite weak piecewise partition of $[a,b]$; then on each open piece the tangent is ${\sigma }^{\mathrm{\prime }}(s)d$ with ${\sigma }^{\mathrm{\prime }}(s)>0$, so $x$ is again a weak piecewise future timelike worldline with that partition, its coordinate path in the chart of step 1.1 is $y^0=T\sigma$, $z=0$, and its density is $f=\sqrt{(y^0{}^{\mathrm{\prime }}{)}^2}=y^0{}^{\mathrm{\prime }}=T{\sigma }^{\mathrm{\prime }}$; therefore $c\mathrm{\Delta }\tau (x)=\sum \int f=\sum \int y^0{}^{\mathrm{\prime }}=y^0(b)-y^0(a)=T(\sigma (b)-\sigma (a))=T$, using for the middle equality the endpoint increments and additivity of [F3] applied to $x$ with that partition; this is the reverse implication of (ii), and the stated equivalence with 'every open-piece tangent is a positive multiple of $q-p$' holds because the representation gives such tangents, and conversely such tangents have vanishing spatial coordinates in the rest frame of step 1.1, so [L7] makes the continuous spatial coordinates constant on each closed piece; the shared corner values and $z(a)=0$ make $z=0$ throughout. Then $\sigma =y^0\mathrm{/}T$ has precisely the positivity, endpoint and surjectivity properties established in steps 7.1 and 8.1.

For (iii): by (ii), $c\mathrm{\Delta }\tau (x)=T$ forces $x$ to be a reparametrization with $\sigma$ valued in $[0,1]$, so the image of $x$ is contained in $\{p+\lambda d:0\le \lambda \le 1\}$; hence if the image is not contained in the straight segment the inequality is strict. If $x$ is piecewise inertial with constant piece tangents $d_1,\dots ,d_m$, then: if all $d_i$ are positive multiples of one another, say $d_i={\alpha }_{i}u$ with $u$ future timelike and ${\alpha }_i>0$, then all legs lie on the line through $p$ in direction $u$ and the parameter $\sigma$ defined by $x=p+\sigma d$ is continuous, piecewise affine and strictly increasing with image $[0,1]$, so the image is exactly the straight segment and the bound is attained by step 9.1; conversely, if the image is contained in the straight segment, each leg's tangent is parallel to $d$ and future timelike, hence a positive multiple of $d$ by [F4], so all $d_i$ are positive multiples of one another; this proves (iii).

For (iv): $x_s(s)=p+\frac{s-a}{b-a}d$ has constant tangent $\frac{1}{b-a}d$, a future timelike vector, so it is a weak piecewise (indeed $C^1$) future timelike worldline, and $\sigma (s)=\frac{s-a}{b-a}$ satisfies the conditions of (ii): it is continuous, strictly increasing, maps $[a,b]$ onto $[0,1]$, is $C^1$ with positive constant derivative, and $x_s=p+\sigma d$; by step 9.1, $c\mathrm{\Delta }\tau (x_s)=T$. More generally, any weak piecewise future timelike worldline whose image is the straight segment from $p$ to $q$ and which is traversed in the future direction satisfies the representation of (ii) by the argument in step 9.1 and hence attains the bound.

Clause (v) is a scope statement: every step above is an argument in the oriented time-oriented affine Minkowski model with the fixed positive constant $c$, within the weak piecewise $C^1$ future timelike class; the corner hypothesis (H) is the special case in which the improper values equal the ordinary Riemann integrals and the total proper time agrees with Proper time along a future timelike worldline for globally $C^1$ histories by [F3], so that case is included; no clock reading, curved-spacetime maximality or choice principle is introduced, the only analytic inputs being the compact-interval improper-integral and one-dimensional differentiability suppliers listed above.