Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent

(Common refinement.) Let $P=\{s_i\}$ and $P^{\mathrm{\prime }}=\{s_j^{\mathrm{\prime }}\}$ be admissible partitions of $[a,b]$ for $\gamma$, and let $Q$ be the partition whose nodes are the union of the two node sets, ordered increasingly. Then $Q$ is an admissible partition: each interval of $Q$ lies in an interval of $P$ and in one of $P^{\mathrm{\prime }}$, hence inside one open piece of each of the two parametrizations, and $\gamma$ is $C^1$ there.

(The density is independent of the partition.) Define, for each interval $[u,v]$ contained in the closure of one open piece of $\gamma$, the number $H(u,v):={\int }_u^{v}h$, where $h$ is the continuous positive extension of the density to $[u,v]$ from [L1]; this is the Darboux integral of a continuous function (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion). If the nodes of a subpartition are inserted, [L3] gives additivity: $H(u,v)=H(u,w)+H(w,v)$.

(Strict positivity.) For $a<b$, choose an open piece $(s_{i-1},s_i)$ and a closed interval $[u,v]\subseteq (s_{i-1},s_i)$ with $u<v$. Fix the midpoint $t_{\ast }=(u+v)\mathrm{/}2$. By continuity and $h(t_{\ast })>0$, shrink the interval symmetrically about $t_{\ast }$ so that $h\ge m:=h(t_{\ast })\mathrm{/}2>0$ on the resulting $[u,v]$; every lower Darboux sum of $h$ on any partition refining $\{u,v\}$ is at least $m(v-u)$, since $h\ge m$ on $[u,v]$ and $h>0$ elsewhere. Hence the lower integral, which equals the integral because $h$ is integrable, is at least $m(v-u)>0$ (For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P)={\sum }_i{m}_i{\mathrm{\Delta }}_i$ and $U(f,P)={\sum }_i{M}_i{\mathrm{\Delta }}_i$, The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$). Dividing by $c>0$ gives $\tau (b)-\tau (a)>0$ when $a<b$.

(Reparametrization: reduction to pieces.) By [L7], the continuous nowhere-zero derivative of $\phi$ has one strict sign; consider first ${\phi }^{\mathrm{\prime }}>0$. The mean value theorem makes $\phi$ strictly increasing, and continuity makes its image exactly $[\alpha ,\beta ]$ with $\alpha =\phi (a^{\mathrm{\prime }})<\beta =\phi (b^{\mathrm{\prime }})$. Strict monotonicity gives continuity of its inverse: at each image value the images of nearby parameter endpoints bound a relative neighbourhood. Restrict $\gamma$ to $[\alpha ,\beta ]$ and let $P=\{s_i\}$ be an admissible partition of that restriction; the preimages ${\phi }^{-1}(s_i)$ form an admissible partition for $\tilde{\gamma }=\gamma \circ \phi$: on each preimage interval $(u,v)={\phi }^{-1}(s_{i-1},s_i)$ the curve $\tilde{\gamma }$ is the composition of $C^1$ maps and hence $C^1$, with ${\tilde{\gamma }}^{\mathrm{\prime }}={\phi }^{\mathrm{\prime }}\cdot ({\gamma }^{\mathrm{\prime }}\circ \phi )$ by [L7], a strictly positive multiple of a future timelike vector and therefore future timelike by [L2].

(Derivative of the proper-time function.) Let $\gamma :I\to M$ be a $C^1$ future timelike worldline and $\tau (t)=\frac{1}{c}{\int }_{t_0}^{t}h$ its proper-time function, $h$ continuous and positive on $I$. For each compact $[u,v]\subseteq I$, [L4] gives $\tau (v)-\tau (u)=\frac{1}{c}{\int }_u^{v}h$; since $h\mathrm{/}c$ is continuous at every $t\in I$, the first fundamental theorem in the form ${\tau }^{\mathrm{\prime }}(t)=h(t)\mathrm{/}c$ follows at every interior point, and one-sided at the endpoints. This is the displayed formula; it is $>0$ because $h>0$ and $c>0$.

(Partition independence.) Write the sum of the definition for $P$ as ${\sum }_{i}H(s_{i-1},s_i)$ and similarly for $P^{\mathrm{\prime }}$ and $Q$. Refining $P$ to $Q$ and applying additivity of step 1.2 to each interval of $P$ split by the nodes of $Q$, the sum for $P$ equals the sum over $Q$; the same holds for $P^{\mathrm{\prime }}$. Hence both sums equal ${\sum }_{Q}H$ and the increment is independent of the admissible partition.

(Reparametrization: the integral identity.) On a piece $(u,v)$ as in step 1.4, the density of $\tilde{\gamma }$ equals ${\phi }^{\mathrm{\prime }}\cdot \sqrt{-g({\gamma }^{\mathrm{\prime }}\circ \phi ,{\gamma }^{\mathrm{\prime }}\circ \phi )}={\phi }^{\mathrm{\prime }}(h\circ \phi )$ by bilinearity of the metric and ${\phi }^{\mathrm{\prime }}>0$; it is continuous and positive on the closed piece. By [L5], ${\int }_u^v{\phi }^{\mathrm{\prime }}(h\circ \phi )={\int }_{s_{i-1}}^{s_i}h$. Summing over the finitely many pieces and dividing by $c$ gives $\tilde{\tau }(b^{\mathrm{\prime }})-\tilde{\tau }(a^{\mathrm{\prime }})=\tau (\beta )-\tau (\alpha )$. At a piece endpoint the one-sided tangent of $\tilde{\gamma }$ is ${\phi }^{\mathrm{\prime }}{\gamma }^{\mathrm{\prime }}$ with the same one-sided value of ${\phi }^{\mathrm{\prime }}>0$, hence future timelike, so $\tilde{\gamma }$ is a future timelike worldline.

(Four-velocity relation.) At every $t\in I$, by the definition of $U$ and step 1.5, ${\tau }^{\mathrm{\prime }}(t)U(t)=\frac{h(t)}{c}\cdot \frac{c{\gamma }^{\mathrm{\prime }}(t)}{h(t)}={\gamma }^{\mathrm{\prime }}(t)$ and $g(U(t),U(t))=\frac{c^{2}g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}{h(t{)}^2}=\frac{c^{2}g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}{-g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}=-c^2$. The same computation applies on each open piece and, by the continuous one-sided velocity and density extensions, at each included one-sided endpoint of a piecewise-$C^1$ worldline; a corner retains its two separate values.

(Reparametrization by proper time.) By step 1.5, $\tau$ is $C^1$, and its increments are strictly positive by step 1.3, so it is strictly increasing on $I$; let $J:=\tau (I)$ and $\psi :={\tau }^{-1}:J\to I$. At each interior $\sigma \in J$, [L7] applied to $\tau$ at $t=\psi (\sigma )$ gives a $C^1$ local inverse; injectivity identifies it with $\psi$. At an included endpoint, monotonicity first gives continuity of the inverse from that side: any relative neighbourhood of $t$ has a strictly separated inner endpoint value under $\tau$. For ${\sigma }^{\mathrm{\prime }}\to \sigma$ from that side, put $t^{\mathrm{\prime }}=\psi ({\sigma }^{\mathrm{\prime }})$; then $t^{\mathrm{\prime }}\to t$ and $(\psi ({\sigma }^{\mathrm{\prime }})-\psi (\sigma ))\mathrm{/}({\sigma }^{\mathrm{\prime }}-\sigma )=(t^{\mathrm{\prime }}-t)\mathrm{/}(\tau (t^{\mathrm{\prime }})-\tau (t))\to 1\mathrm{/}{\tau }^{\mathrm{\prime }}(t)$. Continuity of ${\tau }^{\mathrm{\prime }}>0$ makes the inverse derivative continuous up to that endpoint. Applying the same local inverse and endpoint reciprocal-quotient argument to $\phi$, whose derivative is continuous and nowhere zero, also proves the $C^1$ diffeomorphism-onto-image assertion in claim 2. Thus $\psi$ is $C^1$ on $J$ in the stated one-sided sense and ${\psi }^{\mathrm{\prime }}(\sigma )=1\mathrm{/}{\tau }^{\mathrm{\prime }}(\psi (\sigma ))$.

(Additivity over a splitting.) Fix any $u\in (a,b)$ and insert $u$ into an admissible partition, giving nodes $q_0=a<\cdots <q_j=u<\cdots <q_n=b$. By step 1.2 the total sum is ${\sum }_{i=1}^{j}H(q_{i-1},q_i)+{\sum }_{i=j+1}^{n}H(q_{i-1},q_i)$; after division by the fixed constant $c$, step 2.1 identifies these sums with the two subinterval increments. Thus $\tau (b)-\tau (a)=(\tau (u)-\tau (a))+(\tau (b)-\tau (u))$ for every such $u$.

(Reversing orientation.) For ${\phi }^{\mathrm{\prime }}<0$, use its image $[\alpha ,\beta ]=[\phi (b^{\mathrm{\prime }}),\phi (a^{\mathrm{\prime }})]$; the chain rule gives ${\tilde{\gamma }}^{\mathrm{\prime }}={\phi }^{\mathrm{\prime }}({\gamma }^{\mathrm{\prime }}\circ \phi )$, a negative multiple of a future timelike tangent, hence past timelike. The closed-piece $C^1$ extensions compose in the same way, including their distinct corner values. Bilinearity gives $\tilde{h}=\mathrm{\mid }{\phi }^{\mathrm{\prime }}\mathrm{\mid }(h\circ \phi )$. Apply [L5] to the reversed endpoints on each piece and sum to get ${\int }_{a^{\mathrm{\prime }}}^{b^{\mathrm{\prime }}}\tilde{h}={\int }_{\alpha }^{\beta }h>0$. Meanwhile the signed original difference is $\tau (\phi (b^{\mathrm{\prime }}))-\tau (\phi (a^{\mathrm{\prime }}))=\tau (\alpha )-\tau (\beta )<0$. If the image is the full interval $[a,b]$, this is $-(\tau (b)-\tau (a))$. Thus the unsigned integral and oriented difference have different signs, as asserted; no proper-time function for a future curve is assigned a negative elapsed value.

(Velocity of the reparametrized curve.) For ${\gamma }_{\tau }=\gamma \circ \psi$ and $\sigma \in J$, [L7] applied in a chart around $\gamma (\psi (\sigma ))$ gives ${\gamma }_{\tau }^{\mathrm{\prime }}(\sigma )={\psi }^{\mathrm{\prime }}(\sigma ){\gamma }^{\mathrm{\prime }}(\psi (\sigma ))={\gamma }^{\mathrm{\prime }}(\psi (\sigma ))\mathrm{/}{\tau }^{\mathrm{\prime }}(\psi (\sigma ))=U(\psi (\sigma ))$ by step 2.3. In particular $g({\gamma }_{\tau }^{\mathrm{\prime }},{\gamma }_{\tau }^{\mathrm{\prime }})=-c^2$ and ${\gamma }_{\tau }^{\mathrm{\prime }}$ is future timelike, so ${\gamma }_{\tau }$ is a $C^1$ future timelike worldline on $J$; for a piecewise $C^1$ worldline the same construction applies on each closed piece using its one-sided inverse derivatives, with separate left and right velocities at a corner.