Units, coordinate conventions and c restoration for the wave and Doppler formulas

Remark on the conventions of SI units and coordinate conventions of the special-relativity framework and Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave, for use with Doppler shift and aberration for an ideal plane wave between inertial descriptions. The frame data and wave data of a future null wave covector are related as in Frame data of a future null wave covector: the unique pair $(\varpi ,s)$ with $\mathrm{\mid }s\mathrm{\mid }=1$, and the symbols below have the meanings fixed there and in Affine phase functions and wave covectors on affine Minkowski space.

(a) Units in the adopted coordinates $x^0=ct$. The phase $\theta$ is dimensionless; the coordinates $x^a$, the wavelength $\lambda$ and all distances are in metres; the components $k_a$ of a wave covector, the wave vector $\kappa$ and $\varpi =-k(e_0)$ are in ${\mathrm{m}}^{-1}$; the angular frequencies $\omega$ and ${\omega }_n$ and the linear frequencies $\nu$ are in ${\mathrm{s}}^{-1}$ ($\nu$ also in hertz); $c$ is in $\mathrm{m}{\mathrm{s}}^{-1}$; and $s$, $\beta$, ${\beta }_w$ and $\gamma$ are dimensionless. These assignments are consistent with the identities $\omega =c\varpi$, $\kappa =(\omega \mathrm{/}c)s$, $\lambda =2\pi c\mathrm{/}\omega =c\mathrm{/}\nu$ and phase $=\theta$ dimensionless.

(b) Two coordinate forms. In the adopted coordinates $\theta =-\omega t+\kappa \cdot x_{\mathrm{v}\mathrm{e}\mathrm{c}}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ and $k_a=(-\omega \mathrm{/}c,(\omega \mathrm{/}c)s_1,(\omega \mathrm{/}c)s_2,(\omega \mathrm{/}c)s_3)$; in a chart whose time coordinate is $t$ in seconds directly, so that the metric component $g_{tt}=-c^2$ carries ${\mathrm{m}}^2{\mathrm{s}}^{-2}$ and the time-space components carry $\mathrm{m}{\mathrm{s}}^{-1}$, the same phase is $\theta =-\omega t+\kappa \cdot x_{\mathrm{v}\mathrm{e}\mathrm{c}}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ with the same $\omega$ and $\kappa$, and the component tuple $(-\omega ,(\omega \mathrm{/}c)s)$ carries mixed units $({\mathrm{s}}^{-1},{\mathrm{m}}^{-1})$ because it is read in the coordinate basis $({\mathrm{\partial }}_t,{\mathrm{\partial }}_{x^i})$ rather than in the dimensionless frame basis $(e_0,e_1,e_2,e_3)$. Formulas derived in one convention are not reused unchanged in the other; this is the general warning of SI units and coordinate conventions of the special-relativity framework specialized to the wave covector.

(c) Natural units and restoration. In natural units $c=1$ with $x^0=t$ one has $k_a=(-\omega ,\omega s)$, $\lambda =2\pi \mathrm{/}\omega$, and the Doppler and aberration factors $\gamma (1-\beta \cdot s)$ and $\sqrt{(1\mp \mathrm{\mid }\beta \mathrm{\mid })\mathrm{/}(1\pm \mathrm{\mid }\beta \mathrm{\mid })}$ are unchanged; restoring dimensions means replacing $k_0$ by $-\omega \mathrm{/}c$, $k_i$ by $(\omega \mathrm{/}c)s_i$, writing every velocity as $v=c\beta$ with the dimensionless $\beta$, and replacing $\lambda$ by $2\pi c\mathrm{/}\omega =c\mathrm{/}\nu$. The dimensionless factors $\gamma (1-\beta \cdot s)$ and the longitudinal square roots need no $c$ restoration because they contain only $\beta$, $s$ and $\gamma$; the $c$ restoration affects the map between $\omega$, $\nu$, $\lambda$ and the components of $k$, not the shift factors themselves.

(d) Scope. This remark introduces no quantum identification of energy with $\mathrm{\hslash }\omega$, no amplitude or intensity content and no measurement claim; it is a units and convention statement carrying the scope of the items it cites. No choice principle is used.