Worldlines And Clocks

Let $(A,V,\eta ,C^{+})$ be an oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)) and let $I$ be an order-convex subset of $\mathbb{R}$ with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); a curve of the model is a map $x:I\to A$.

Derivative and regularity of the parametrization

For $s\in I$ and $h\mathrm{\ne }0$ with $s+h\in I$ the difference quotient $(x(s+h)-x(s))\mathrm{/}h$ is an element of $V$, the displacement from $x(s)$ to $x(s+h)$ divided by the real number $h$ (Oriented time-oriented affine Minkowski space (the spacetime primitives)). Convergence in $V$ is understood through an inertial frame: for the frame isomorphism $E:{\mathbb{R}}^4\to V$ of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model) put $\mathrm{\mid }v{\mathrm{\mid }}_E:=\mathrm{\mid }{E}^{-1}v{\mathrm{\mid }}_2$, the Euclidean norm transported by $E$. If $E^{\mathrm{\prime }}$ is the isomorphism of another inertial frame, then $E^{\mathrm{\prime }-1}E$ and its inverse are linear maps ${\mathbb{R}}^4\to {\mathbb{R}}^4$, so they are bounded with constants $K,K^{\mathrm{\prime }}\ge 0$ (Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$): $\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}\le K^{\mathrm{\prime }}\mathrm{\mid }v{\mathrm{\mid }}_E$ and $\mathrm{\mid }v{\mathrm{\mid }}_E\le K\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}$ for all $v\in V$. The two transported norms are therefore equivalent, give the same limits, and give the same notion of differentiability. The curve is differentiable at $s$ when the limit of the difference quotients as $h\to 0$ (over $h\mathrm{\ne }0$, $s+h\in I$; one-sided at the endpoints of $I$) exists in $V$; then this limit, written $x^{\mathrm{\prime }}(s)\in V$, is the tangent vector of $x$ at $s$, and it does not depend on the inertial frame used to describe the convergence.

If $x_A:A\to {\mathbb{R}}^4$ is the global chart of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model, The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4), then $x$ is differentiable at $s$ exactly when the ${\mathbb{R}}^4$-valued map $x_A\circ x$ is differentiable at $s$ in the sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, because $E^{-1}$ is linear and carries the difference quotients of $x$ to those of $x_A\circ x$; and then

$x^{\mathrm{\prime }}(s)=E((x_A\circ x{)}^{\mathrm{\prime }}(s)),$

where $E:{\mathbb{R}}^4\to V$ is the frame isomorphism $E(u)={\sum }_a{u}^a{e}_a$. This value does not depend on the inertial chart used: two inertial charts are related by an affine bijection with an invertible linear part (The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4), the chain rule transports the coordinate derivative accordingly, and all norms on the finite-dimensional space $V$ give the same limit.

The curve is $C^1$ when it is differentiable at every point of $I$ and the derivative map $s\mapsto x^{\mathrm{\prime }}(s)$, $I\to V$, is continuous, meaning that the ${\mathbb{R}}^4$-valued map $s\mapsto (x_A\circ x{)}^{\mathrm{\prime }}(s)$ is continuous (Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions); by the equivalence of transported norms this condition is independent of the inertial frame. It is $C^k$ for $k\ge 1$ when the derivative map is $C^{k-1}$, with the same transported reading. A curve on a compact interval $[a,b]$ contained in $I$ is piecewise $C^1$ when there is a finite partition $a=s_0<s_1<\cdots <s_m=b$ such that $x$ is $C^1$ on every open subinterval $(s_{i-1},s_i)$ and the finite one-sided difference-quotient derivatives at both ends of every piece exist in $V$. These are limits of $(x(t+h)-x(t))\mathrm{/}h$ over the relevant side of the piece, not assertions that $x^{\mathrm{\prime }}(s)$ has a limit as $s$ approaches a corner; their finiteness implies continuity of $x$ at the corners (for a one-sided quotient derivative $v$ at $t$ one has $x(t+h)-x(t)=h(v+o(1))$ as $h\to 0$ on that side). No continuous extension of the open-piece derivative is required. Derivatives at the endpoints of $I$ are the one-sided difference-quotient limits over $I$.

Causal character

Let $x$ be differentiable at $s\in I$. Then $x$ is regular at $s$ if $x^{\mathrm{\prime }}(s)\mathrm{\ne }0$; $x$ is timelike (respectively null, spacelike, causal) at $s$ if the vector $x^{\mathrm{\prime }}(s)$ is timelike (respectively null, spacelike, causal) in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space, and $x$ is future-directed at $s$ if $x^{\mathrm{\prime }}(s)$ is future timelike or future null in the sense of Time orientation, future and past timelike/null/causal vectors, and the closed cones. The zero vector has no causal character, so only a regular curve has a causal character at $s$.

Worldlines

A future timelike worldline (equivalently, a massive worldline) is a $C^1$ curve $x:I\to A$ with $x^{\mathrm{\prime }}(s)$ future timelike for every $s\in I$; a future causal curve (respectively a future null curve) is a $C^1$ curve with $x^{\mathrm{\prime }}(s)$ future causal (respectively future null) for every $s\in I$. A piecewise $C^1$ worldline is a piecewise $C^1$ curve whose tangents at every point of each open smooth piece and whose finite one-sided difference-quotient tangents at both endpoints of every piece are future timelike; a $C^k$ worldline is a worldline whose parametrization is $C^k$. $C^2$ regularity is what makes the four-acceleration of Four-velocity and four-acceleration of a future timelike worldline available.

The parametrization is part of the curve: the image of a future timelike worldline is also the image of the past-directed curve $s\mapsto x(-s)$, which is not a worldline of this model, and no worldline or clock claim is attached to it. The definition asserts the existence of no worldline beyond those given by an explicit formula.

• Well-definedness. The difference quotient is an element of $V$ because the affine action assigns a displacement to every ordered pair of events. The limit in $V$ is taken in the transported Euclidean norm $\mathrm{\mid }v{\mathrm{\mid }}_E=\mathrm{\mid }{E}^{-1}v{\mathrm{\mid }}_2$ of an inertial frame isomorphism $E$, and the choice of frame does not matter: for inertial frames $E$ and $E^{\mathrm{\prime }}$ the maps $E^{\mathrm{\prime }-1}E$ and its inverse are Euclidean linear maps, so by Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$ they are bounded with constants $K,K^{\mathrm{\prime }}\ge 0$, which gives $\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}\le K^{\mathrm{\prime }}\mathrm{\mid }v{\mathrm{\mid }}_E$ and $\mathrm{\mid }v{\mathrm{\mid }}_E\le K\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}$ on $V$; equivalent norms have the same convergent sequences and the same limits, and hence the same derivative $x^{\mathrm{\prime }}(s)$. The identification with the coordinate calculus, the value $x^{\mathrm{\prime }}(s)=E((x_A\circ x{)}^{\mathrm{\prime }}(s))$, the independence of the class $C^k$ and the continuity of $x^{\mathrm{\prime }}$ are then read through the same linear isomorphism, with the published vector-valued derivative and continuity of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral and Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions as the coordinate calculus, and the one-sided endpoint derivatives are the one-sided limits over the domain $I$ used by that same definition. The affine chart transitions of The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4 are consistent with this dictionary: they are affine with an invertible linear part, so they compose the coordinate calculus with a bounded linear change of coordinates.
• Regularity is implicit in every causal character. Because the zero vector is excluded from timelike, null and spacelike alike (Timelike, null, spacelike and causal vectors of a Lorentz vector space), the future timelike condition $x^{\mathrm{\prime }}(s)\in C^{+}$ already contains $x^{\mathrm{\prime }}(s)\mathrm{\ne }0$, and each side of a corner in the piecewise clause is required to carry a nonzero future timelike one-sided tangent. Null and spacelike curves are distinguished from massive worldlines and are used later only to delimit the proper-time construction, which is stated for future timelike worldlines alone.
• No choice principle is used. All objects are finite-dimensional or explicitly parametrized; the definition selects nothing from a family.

Let $x:[a,b]\to A$, $a<b$, be a piecewise $C^1$ curve in the sense of Curves, regular causal curves and future timelike worldlines in affine Minkowski space, with a finite partition $a=s_0<\cdots <s_m=b$, $C^1$ on each open piece and finite one-sided difference-quotient derivatives at both ends of each piece. Assume its tangents are nonzero future causal at every open-piece point and at each one-sided endpoint.

In any inertial chart (Inertial frames and inertial charts of the affine Minkowski model) write $y=(y^0,z)$, $z\in {\mathbb{R}}^3$, for the coordinate path. On a piece $(\alpha ,\beta )$, choose $r\in (\alpha ,\beta )$; for a function $g$ continuous there, write

${\int }_{(\alpha ,\beta )}g:={\lim }_{u\downarrow \alpha }{\int }_u^{r}g+{\lim }_{v\uparrow \beta }{\int }_r^{v}g$

when both limits are finite. Then:

(a) $y$ is continuous on $[\alpha ,\beta ]$, $y^{0^{\mathrm{\prime }}}\ge \mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }\ge 0$ on $(\alpha ,\beta )$, and $y^{0^{\mathrm{\prime }}}>0$ there.

(b) Every coordinate derivative is absolutely improperly Riemann integrable (its absolute value has finite improper integral), and its signed improper integral equals the corresponding endpoint increment. The continuous functions $\mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }$ and $f=\sqrt{(y^{0^{\mathrm{\prime }}}{)}^2-\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2}=\sqrt{-\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})}$ have finite nonnegative improper integrals, with ${\int }_{(\alpha ,\beta )}f\le {\int }_{(\alpha ,\beta )}{y}^{0^{\mathrm{\prime }}}=y^0(\beta )-y^0(\alpha )$.

(c) The improper values are independent of $r$, additive under subdivision, and the finite piece sums are independent of the admissible partition.

(d) $D=x(b)-x(a)$ is nonzero future causal: $D^0>0$ and $\mathrm{\mid }{D}_{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}\mathrm{\mid }\le D^0$ in every inertial chart. If some interior or one-sided endpoint tangent is future timelike, then $D$ is future timelike. Coordinate time is strictly increasing on every nondegenerate subinterval. The integrals need not be ordinary closed-piece Riemann integrals. No derivative-limit hypothesis, Lebesgue-integration premise, clock identification, or choice principle is used.

• The improper integrals need not be closed-piece Riemann integrals. The derivative is not assumed to extend continuously to the corners, and no bound on $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }$ near a corner is assumed; the only estimate used is the coordinate domination $0\le \mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }\le y^{0^{\mathrm{\prime }}}$, which turns the (possibly unbounded) spatial derivatives into functions dominated by the milder quantity $y^{0^{\mathrm{\prime }}}$, itself controlled by the endpoint increment through the truncated fundamental theorem.

• Where the causal hypothesis is spent. The sign of $\eta$ alone never enters the integrability conclusions: $\int y^{0^{\mathrm{\prime }}}$ is a finite increment because $y^{0^{\mathrm{\prime }}}$ is a derivative, and domination by $y^{0^{\mathrm{\prime }}}$ needs only $\mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }\le y^{0^{\mathrm{\prime }}}$, which is the future causal inequality. The strict positivity $y^{0^{\mathrm{\prime }}}>0$ uses in addition that the tangent is nonzero; the strict timelikeness conclusions of clause (d) are the only place where a future timelike (not merely causal) tangent is used.

• The statement is chart-relative for the improper integrals and chart-independent for the displacement. The improper integrals are formed in a fixed inertial chart; the conclusions $D\in \mathrm{cl}(C^{+})\setminus \{0\}$, strict coordinate-time monotonicity and the timelikeness criterion are statements about the intrinsic displacement $D$ and the intrinsic tangents, since the coordinate criterion of [L3] describes the same cone in every inertial chart.

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.

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.