Proper time along a future timelike worldline

Let $c>0$ be a fixed real number, the conversion constant of the model, and let $x:I\to A$ be a $C^1$ future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space).

The integrand. The function

$g_x:I\to \mathbb{R},g_x(s):=-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s)),$

is continuous and strictly positive. For continuity, fix an inertial frame and write $y:=x_A\circ x$ for the coordinate path; by the derivative and regularity definition in Curves, regular causal curves and future timelike worldlines in affine Minkowski space the ${\mathbb{R}}^4$-valued map $y^{\mathrm{\prime }}$ is continuous, so each component $y^{\mathrm{\prime }a}$, $a\in \{0,1,2,3\}$, is a continuous real function (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions). In the $\eta$-orthonormal basis underlying the frame one has $\eta (e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ (Inertial frames and inertial charts of the affine Minkowski model), whence

$g_x(s)=-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))=(y^{\mathrm{\prime }0}(s){)}^2-(y^{\mathrm{\prime }1}(s){)}^2-(y^{\mathrm{\prime }2}(s){)}^2-(y^{\mathrm{\prime }3}(s){)}^2,$

a finite sum of products of continuous real functions, hence continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)). Strict positivity $g_x(s)=-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))>0$ is exactly the future timelike hypothesis on $x^{\mathrm{\prime }}(s)$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

The proper-time function. Hence $u\mapsto \sqrt{g_x(u)}$ is continuous on $I$: the square root is the power $t^{1\mathrm{/}2}$ of Rational powers $a^r$ of a positive base, which is continuous and differentiable on $(0,\mathrm{\infty })$ (Continuity and derivatives of positive-base real powers), and it is composed with the continuous positive function $g_x$ (A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs); consequently it is Riemann integrable on every compact subinterval of $I$ (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). For $s_0,s\in I$ the proper time from $s_0$ to $s$ is

$\tau (s):=c^{-1}{\int }_{s_0}^s\sqrt{-\eta (x^{\mathrm{\prime }}(u),x^{\mathrm{\prime }}(u))}du,$

the integral being taken with oriented limits as in The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$; for $a<b$ in $I$ this reads

$\tau (b)-\tau (a)=c^{-1}{\int }_a^b\sqrt{-\eta (x^{\mathrm{\prime }}(u),x^{\mathrm{\prime }}(u))}du,$

by oriented additivity (For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$); equivalently, the increment is the integral function with base point $a$ of the density $c^{-1}\sqrt{g_x}$, evaluated at $b$ on $[a,b]$ (The integral function $F(x):={\int }_a^{x}f$ of an integrable $f$). Changing the base point $s_0$ changes $\tau$ by an additive constant, so the elapsed times between parameter values are intrinsic. $\tau$ depends on the parametrized history $x$ and on the parameter values; its invariance under increasing $C^1$ reparametrizations with everywhere positive derivative is proved in Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps.

Straight segments. For the straight future timelike segment $x(s)=p+(s-s_0)(q-p)$ of Oriented time-oriented affine Minkowski space (the spacetime primitives) the difference quotients are identically the future timelike displacement $q-p$, so $x^{\mathrm{\prime }}(s)=q-p$ and

$\tau (b)-\tau (a)=c^{-1}\sqrt{-\eta (q-p,q-p)}(b-a),$

the straight-segment proper time of the model's coordinate conventions.

Scope. The construction is restricted to future timelike worldlines: on a future null curve the length density vanishes identically, so its integral is constant; a zero density at one isolated point does not imply constancy; for a spacelike curve $-\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})$ is negative and the square root is not a real number; and for a past-directed timelike curve the same integral is defined but is not attached to a worldline of this model. The construction asserts nothing about physical clock readings.

• Well-definedness. (i) $g_x$ is the composition of the continuous $V$-valued derivative $x^{\mathrm{\prime }}$, the quadratic form $v\mapsto -\eta (v,v)$ of Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space and the continuous square root; the coordinate computation displayed above evaluates the quadratic form in an $\eta$-orthonormal basis and needs no basis-free topology on $V$. (ii) The integrand is defined on all of $I$ because strict positivity of $g_x$ is the future timelike hypothesis, so no branch of the square root and no punctured domain is needed. (iii) Existence of the integral on every compact subinterval is A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, and the oriented convention of The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$ makes $\tau (s_0)=0$ and reverses the sign when the endpoints are exchanged, so $\tau$ is a genuine function on $I$ for every choice of base point. (iv) The base point enters only additively: for $s_1,s_2,s\in I$ the additivity identity of For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$ gives $c^{-1}{\int }_{s_1}^s=c^{-1}{\int }_{s_1}^{s_2}+c^{-1}{\int }_{s_2}^s$, so replacing the base point $s_0$ by $s_1$ subtracts the constant $c^{-1}{\int }_{s_0}^{s_1}$ from the whole function. (v) The straight-segment value follows by evaluating a constant integrand: $x^{\mathrm{\prime }}=q-p$ is future timelike, $\sqrt{-\eta (q-p,q-p)}$ is its positive length, and the integral of a constant over $[a,b]$ is the value times $b-a$.
• What is deliberately not asserted. The definition does not assert that $\tau$ is differentiable or invertible (that is The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse) and does not assert that a physical clock reads $\tau$ (that identification is the adopted assumption that ideal clocks indicate proper time, an assumption of the framework, not a mathematical consequence). No choice principle is used: the construction selects no object from any family and all integrals are one-dimensional.