SI units and coordinate conventions of the special-relativity framework

This framework fixes the following SI coordinate and unit conventions for the oriented time-oriented affine Minkowski space $(A,V,\eta ,C^{+})$ and its inertial charts (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model).

1. Adopted conversion constant. The framework uses a positive constant $c$ with unit metre per second, $\mathrm{m}\mathrm{/}\mathrm{s}$, interpreted as the speed of light in vacuum under the framework's light postulate. In SI its numerical value is exactly $299792458$ by the definition of the metre (BIPM, SI defining constants, table row $c$). This exact unit convention does not prove that the ideal-light model describes nature; measurements and realizations of units still have uncertainty.

2. Coordinates. In every inertial chart $x=(x^0,x^1,x^2,x^3)$ define the coordinate-time map $t:A\to \mathbb{R}$ by $t(p)=x^0(p)\mathrm{/}c$, with unit second $\mathrm{s}$, and assign unit metre $\mathrm{m}$ to every $x^a$. An at-rest line of this frame is $s\mapsto o+cs{e}_0+{\sum }_{i=1}^3{r}^i{e}_i$, $s\in \mathbb{R}$, for fixed spatial coordinates $(r^1,r^2,r^3)$ in metres; its coordinate time is $s$. Interpreting coordinate-time differences as readings of synchronized ideal clocks on these lines is a physical assumption stated in the clock postulate, not a consequence of assigning units.

3. Metric components. The matrix ${\eta }_{ab}=\eta (e_a,e_b)$ in the frame is $\mathrm{diag}(-1,1,1,1)$ with dimensionless entries. Thus $I(p,q)=\eta (q-p,q-p)=-(\mathrm{\Delta }{x}^0{)}^2+{\sum }_{i=1}^3(\mathrm{\Delta }{x}^i{)}^2$ carries unit ${\mathrm{m}}^2$.

4. Proper time of a straight segment. For a timelike displacement $q-p$, the affine straight segment is $s\mapsto p+s(q-p)$, $s\in [0,1]$. Define its positive proper-time length as

   $\tau :=c^{-1}\sqrt{-\eta (q-p,q-p)}\mathrm{.}$

   It has unit second, since $c^{-1}$ converts the square-root interval in metres into seconds. This length is positive even if the segment is oriented toward the past; a physical advancing clock uses future orientation.

5. Kinematic bookkeeping. Let $J\subseteq \mathbb{R}$ be an open interval with parameter $\tau$ in seconds, and let $\gamma :J\to A$ have $C^2$ coordinates $y=x\circ \gamma :J\to {\mathbb{R}}^4$ ($C^k$ maps and multi-index derivative notation in Euclidean space). With $E(u)={\sum }_a{u}^a{e}_a$, require $U(\tau ):=E(y^{\mathrm{\prime }}(\tau ))\in C^{+}$ and $\eta (U,U)=-c^2$. This is the proper-time normalization used here for units; the general construction of a proper-time parameter is developed later. The four-velocity is $U:J\to V$, with components $U^a=(y^a{)}^{\mathrm{\prime }}$ and unit $\mathrm{m}\mathrm{/}\mathrm{s}$. The four-acceleration is $\mathcal{A}=E(y^{\mathrm{\prime }\mathrm{\prime }}):J\to V$, with unit $\mathrm{m}\mathrm{/}{\mathrm{s}}^2$. For a positive constant mass parameter $m$ in kilograms, define four-momentum $P=mU:J\to V$, with unit $\mathrm{k}\mathrm{g}\mathrm{m}\mathrm{/}\mathrm{s}$, and energy measured by this frame $\mathcal{E}=c{P}^0=-c\eta (e_0,P):J\to \mathbb{R}$, with unit joule $\mathrm{J}=\mathrm{k}\mathrm{g}{\mathrm{m}}^2\mathrm{/}{\mathrm{s}}^2$. The equality follows from $\eta (e_0,P)=-P^0$. These are definitions for dimensional bookkeeping; the physical particle interpretation and dynamics belong to later pages.

6. Temporal-coordinate convention. Replacing $x^0$ by $t=x^0\mathrm{/}c$ pulls the interval expression back to $-c^2(\mathrm{\Delta }t{)}^2+{\sum }_i(\mathrm{\Delta }{x}^i{)}^2$. Thus $g_{tt}=-c^2$ has unit ${\mathrm{m}}^2\mathrm{/}{\mathrm{s}}^2$, the spatial components remain dimensionless, and mixed time-space components have unit $\mathrm{m}\mathrm{/}\mathrm{s}$ (and equal zero in this inertial chart). Formulas must be transformed when changing this convention.

7. Natural units. A presentation with $c=1$ identifies temporal and spatial units by this conversion factor; it requires an explicit restoration table for $c$. It does not assert that the SI numerical value of $c$ is $1$.

Status. These are conventions and dimensional bookkeeping for the model. The SI numerical value of $c$ is exact by unit definition; the physical interpretation and applicability of the model remain adopted and empirically testable. SI unit status does not make a quantity primitive in the theory.