Adopted ideal-clock and ideal-vacuum-light interpretation

Adopt as physical assumptions of the special-relativity framework the following two statements, with the SI coordinate and unit conventions of SI units and coordinate conventions of the special-relativity framework.

(i) Ideal clocks. Let $I\subseteq \mathbb{R}$ be a nondegenerate interval and $\gamma :I\to A$ a curve. Choose an inertial chart $\chi$ with $E(u)={\sum }_a{u}^a{e}_a$ as in Inertial frames and inertial charts of the affine Minkowski model. Require $y:=\chi \circ \gamma :I\to {\mathbb{R}}^4$ to be $C^1$ on the interior of $I$, with its derivative extending continuously to any included endpoints, where it is the one-sided derivative. Define the displacement-valued derivative by $\dot{\gamma }(s):=E(y^{\mathrm{\prime }}(s))\in V$ and require $\dot{\gamma }(s)\in C^{+}$ everywhere, including such endpoints. This derivative is chart-independent: a second chart has coordinates $\tilde{y}=M^{-1}(y-\chi (o^{\mathrm{\prime }}))$, so its parameter derivative is $M^{-1}\mathrm{d}y\mathrm{/}\mathrm{d}s$, while its basis map is $E^{\mathrm{\prime }}=EM$; their product is $E\mathrm{d}y\mathrm{/}\mathrm{d}s$.

An ideal clock on this curve has a reading map $\sigma :I\to \mathbb{R}$ in seconds, specified up to an additive constant, with, for all $a<b$ in $I$,

$\sigma (b)-\sigma (a)=c^{-1}{\int }_a^b\sqrt{-\eta (\dot{\gamma }(s),\dot{\gamma }(s))}\mathrm{d}s\mathrm{.}$

The integrand is positive and continuous on $[a,b]$: the coordinate derivatives are continuous, $\eta$ is a finite bilinear expression, and the square root is continuous at every positive argument, as follows from $\mathrm{\mid }\sqrt{r}-\sqrt{q}\mathrm{\mid }=\mathrm{\mid }r-q\mathrm{\mid }\mathrm{/}(\sqrt{r}+\sqrt{q})$ for $r,q>0$. Thus A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion applies to this compact interval and gives a well-defined Riemann integral. The reading rule contains no additional acceleration-dependent correction (clock hypothesis); it does not require $C^2$ regularity or an acceleration to exist. This is an adopted physical interpretation of the length integral, not a proof about real clocks.

(ii) Ideal light. In vacuum, the ideal light rays are the oriented straight lines parametrized by $\mathrm{\ell }:\mathbb{R}\to A$, $\mathrm{\ell }(\lambda )=p+\lambda v$, where $p\in A$ and $v\in V$ is future null; increasing $\lambda$ specifies their future direction. In an inertial chart $\chi$ their coordinate map is

$\chi (\mathrm{\ell }(\lambda ))=\chi (p)+\lambda (v^0,v^1,v^2,v^3)\mathrm{.}$

Since the chart's time-axis vector is future, the cone description of The timelike set has exactly two convex components, described in orthonormal coordinates gives $v^0=\mid v_{\mathrm{v}\mathrm{e}\mathrm{c}}\mid >0$. With $t={\chi }^0\mathrm{/}c$ the spatial coordinate velocity is $c(v^1,v^2,v^3)\mathrm{/}{v}^0$ and has Euclidean magnitude $c$ in every inertial chart. The conversion constant is the same across charts; its exact SI numerical value is a unit convention as specified in SI units and coordinate conventions of the special-relativity framework, while the ideal-vacuum-light interpretation is an adopted, empirically testable model assumption.

• Adopted, not proved. Both clauses are adopted assumptions of the framework, with the physical interpretation fixed by Adopted inertial covariance of the model equations. They are not experimentally proved here, and they carry no claim that real clocks or real light are exactly ideal.
• Ideal apparatus, complete vacuum, geometric ray regime. The light clause concerns ideal light in complete vacuum in the geometric (nondispersive) ray regime. No dispersive or refractive medium is described, and the ray model carries no wave-optics, coherence or intensity structure.
• No quantum photon model. No photon is assumed, and in particular the frequency of the ideal light is not identified with an energy by a factor $\mathrm{\hslash }$ anywhere in this framework's primitive layer.
• Later development. Parametrization invariance of the clock reading, the proper-time function along general worldlines, four-velocity and four-acceleration are developed on the later worldlines-and-clocks page from its own items; the present item states the reading rule and the ray interpretation for the model and does not derive that theory here.
• Empirical status. Agreement of an experiment with predictions of this model would be evidence about the specified model and its auxiliary assumptions; it would not be a proof of this postulate, and a finite dataset cannot establish its exact validity.