Total proper time of a piecewise C^1 future timelike worldline

Let $c>0$ be the conversion constant of Proper time along a future timelike worldline and let $x:[a,b]\to A$, $a<b$, be a piecewise $C^1$ future timelike worldline in the weak sense of Curves, regular causal curves and future timelike worldlines in affine Minkowski space: $x$ is $C^1$ on each open piece of a finite partition $a=s_0<\cdots <s_m=b$, its interior tangents and finite one-sided difference-quotient endpoint tangents are future timelike, and continuous extension of the derivative is not assumed.

(i) The piecewise density. On each open piece let

$g_i(t)=\sqrt{-\eta (x^{\mathrm{\prime }}(t),x^{\mathrm{\prime }}(t))}>0.$

By clause (b) of Improper integrability and endpoint increments of weak piecewise future causal curves the function $g_i$ is continuous on $(s_{i-1},s_i)$ and its two-ended improper Riemann integral

${\int }_{(s_{i-1},s_i)}{g}_i:={\lim }_{u\downarrow s_{i-1}}{\int }_u^r{g}_i+{\lim }_{v\uparrow s_i}{\int }_r^v{g}_i$

(for any $r\in (s_{i-1},s_i)$) is finite and nonnegative; it is positive because $g_i$ is positive and continuous on a nondegenerate compact interior subinterval, the improper value dominates the ordinary integral over such a subinterval (If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$), and that ordinary integral is strictly positive: a continuous nonnegative function on a nondegenerate compact interval whose integral vanishes is identically zero there (A continuous $f\ge 0$ on $[a,b]$ with ${\int }_a^{b}f=0$ is identically $0$), whereas $g_i$ is strictly positive at every point of the piece.

(ii) Total proper time. Define

$\mathrm{\Delta }{\tau }_{[a,b]}(x):=c^{-1}{\sum }_{i=1}^m{\int }_{(s_{i-1},s_i)}{g}_i\mathrm{.}$

The finite sum is independent of the admissible partition, by subdivision additivity and a common refinement (clause (c) of Improper integrability and endpoint increments of weak piecewise future causal curves), and it is independent of the inertial chart: for every truncation $s_{i-1}<u<r<v<s_i$ the ordinary integral ${\int }_u^v{g}_i$ is intrinsic by Proper time along a future timelike worldline (applied to the $C^1$ restriction), and the improper value is the limit of these intrinsic truncation values, which is intrinsic.

(iii) Based proper-time function. For a distinguished $s_0\in [a,b]$ set ${\tau }_x(s):=\mathrm{\Delta }{\tau }_{[s_0,s]}(x)$ for $s\ge s_0$ and ${\tau }_x(s):=-\mathrm{\Delta }{\tau }_{[s,s_0]}(x)$ for $s<s_0$, with ${\tau }_x(s_0)=0$, using restrictions and oriented limits (The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). It is strictly increasing, and changing the distinguished base point changes it by an additive constant.

(iv) Corner hypothesis (H). Hypothesis (H) is the special case in which each open-piece derivative has finite limits at both ends of its piece and hence extends continuously to the closed piece. Under (H) the limiting values equal the finite one-sided quotient derivatives of the curve, each $g_i$ extends continuously and positively to the closed piece, its improper integral equals its ordinary Riemann integral there (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$), and the same finite-sum formula applies. In particular, if $x$ is globally $C^1$ on $[a,b]$ (the one-piece case of (H)), then $\mathrm{\Delta }{\tau }_{[a,b]}(x)=\tau (b)-\tau (a)$ of Proper time along a future timelike worldline.

(v) Straight pieces. If on the piece $(s_{i-1},s_i)$ the tangent is a constant future timelike vector $d_i$, the contribution of that piece is $c^{-1}(s_i-s_{i-1})\sqrt{-\eta (d_i,d_i)}$; a piecewise straight future timelike worldline therefore has total proper time equal to the finite sum of its straight-segment proper times. In every case $\mathrm{\Delta }{\tau }_{[a,b]}(x)>0$.

(vi) Additivity over concatenation. For $a<d<b$, the restrictions of a weak piecewise worldline $x$ to $[a,d]$ and $[d,b]$ satisfy $\mathrm{\Delta }{\tau }_{[a,b]}(x)=\mathrm{\Delta }{\tau }_{[a,d]}(x)+\mathrm{\Delta }{\tau }_{[d,b]}(x)$: insert $d$ into the partition and use (ii).

The definition supplies metric time only; identification with an ideal clock reading is a separate physical assumption. No global $C^1$ inverse, tangent at a kink, or four-acceleration at a kink is asserted.

• Well-definedness is the integrability lemma. The finiteness of each improper integral, the independence of $r$, the additivity under subdivision and the independence of the admissible partition are clauses (b) and (c) of Improper integrability and endpoint increments of weak piecewise future causal curves, not new hypotheses; the definition only packages the sums and the based function. Because the integrals are one-dimensional Riemann integrals of coordinate functions and no object is selected from a family, no choice principle is used.

• Why the chart does not enter. The two-ended improper integrals are formed in a fixed inertial chart, but each truncation integral is the intrinsic proper-time difference of a $C^1$ restriction, and the improper value is the limit of the truncation values along a monotone family; the limit of intrinsic numbers is intrinsic. So $\mathrm{\Delta }{\tau }_{[a,b]}(x)$ is a function of the parametrized worldline alone.

• Under (H) nothing improper remains. If the derivative extends continuously to the closed piece with endpoint value $w$, then for $u<v$ in the piece the truncated fundamental theorem gives $x(v)-x(u)={\int }_u^v{x}^{\mathrm{\prime }}$ (coordinatewise, and the value in $V$ is recovered), so the quotient $(x(s_{i-1}+h)-x(s_{i-1}))\mathrm{/}h$ is the average $\frac{1}{h}{\int }_{s_{i-1}}^{s_{i-1}+h}{x}^{\mathrm{\prime }}$ of the continuous extension over a shrinking interval and tends to $w$; the extended endpoint value is therefore the one-sided quotient tangent of the curve, and similarly at the right end (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$). The improper integral over the closed piece is then an ordinary Riemann integral by its definition as the common limit of the two truncations.

• Scope. The construction covers future timelike piecewise $C^1$ curves; tangent vectors at kinks and four-accelerations at kinks are not defined here, and the statement asserts no clock reading, no maximality property of straight worldlines and no empirical content. The factor $c^{-1}$ appears once, in (ii); sums of the functions of this definition must not receive a further factor.