Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave

Adopt the SI coordinate and unit conventions of SI units and coordinate conventions of the special-relativity framework: in every inertial chart $x^0=ct$ with $c>0$ in metres per second and spatial coordinates in metres, so $c$ is the model's conversion constant. Let $k$ be a future null wave covector and $\theta$ its affine phase (Affine phase functions and wave covectors on affine Minkowski space). Take $\theta$ dimensionless; its covector components then have unit ${\mathrm{m}}^{-1}$.

Angular convention. Here angles use radian arc length on the unit circle. An explicit construction, also fixing $\pi$, is $a(t):=2{\int }_0^t(1+r^2{)}^{-1}\mathrm{d}r$ for $0\le t\le 1$, and $\pi :=2a(1)$. These are finite Riemann integrals (The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$): the integrand is continuous with positive denominator by 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 and integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. By The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive, $a^{\mathrm{\prime }}(t)=2\mathrm{/}(1+t^2)>0$; The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$ makes $a$ strictly increasing, and Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$ makes it a bijection onto $[0,\pi \mathrm{/}2]$. This is the arc-length parameter of the quarter-circle $q(t)=((1-t^2)\mathrm{/}(1+t^2),2t\mathrm{/}(1+t^2))$: the sum, product and quotient rules of Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$ give $q^{\mathrm{\prime }}(t)=(-4t,2(1-t^2))\mathrm{/}(1+t^2{)}^2$, hence $\mathrm{\mid }{q}^{\mathrm{\prime }}(t)\mathrm{\mid }=2\mathrm{/}(1+t^2)$, and $\mathrm{\mid }q(t)\mathrm{\mid }=1$.

For $\alpha \in [0,\pi \mathrm{/}2]$, let $t$ be the unique value with $a(t)=\alpha$ and define $\cos \alpha :=(1-t^2)\mathrm{/}(1+t^2)$ and $\sin \alpha :=2t\mathrm{/}(1+t^2)$; for $\alpha \in [\pi \mathrm{/}2,\pi ]$ use the unique $t$ with $a(t)=\pi -\alpha$ and define $\cos \alpha :=-(1-t^2)\mathrm{/}(1+t^2)$ and $\sin \alpha :=2t\mathrm{/}(1+t^2)$. The definitions agree at $\pi \mathrm{/}2$. The rational function $(1-t^2)\mathrm{/}(1+t^2)$ decreases continuously from $1$ to $0$, since its difference at $v>u\ge 0$ is $-2(v^2-u^2)\mathrm{/}[(1+v^2)(1+u^2)]<0$. Thus cosine is a strictly decreasing bijection $[0,\pi ]\to [-1,1]$; write $\arccos$ for its inverse. For spatial coordinate vectors $v,w\in {\mathbb{R}}^3$ in one named inertial frame, write $v\cdot w:={\sum }_{i=1}^3{v}_i{w}_i$ and $\mathrm{\mid }v\mathrm{\mid }:=\sqrt{{\sum }_{i=1}^3{v}_i^2}$. If $v,w$ are nonzero, their norms are positive and their angle is $\arccos ((v\cdot w)\mathrm{/}(\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }))$: The Cauchy-Schwarz inequality for finite sums, applied to the three real coordinate entries, gives $\mathrm{\mid }v\cdot w\mathrm{\mid }\le \mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$, so division by $\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }>0$ puts the argument in $[-1,1]$. For unit vectors it is $\arccos (v\cdot w)$. These formulas give $\sin \alpha \ge 0$ and ${\sin }^2\alpha +{\cos }^2\alpha =1$ on $[0,\pi ]$ and fix a full turn as $2\pi$.

(a) Frame data as wave data. In the inertial chart of a frame $(o;e_0,e_1,e_2,e_3)$ (Inertial frames and inertial charts of the affine Minkowski model) let $(\varpi ,s)$ be the frame data of $k$ given by Frame data of a future null wave covector: the unique pair $(\varpi ,s)$ with $\mathrm{\mid }s\mathrm{\mid }=1$, so $\varpi =-k(e_0)>0$ and $\mathrm{\mid }s\mathrm{\mid }=1$. Define the angular frequency $\omega :=c\varpi$ (unit ${\mathrm{s}}^{-1}$), the linear frequency $\nu :=\omega \mathrm{/}(2\pi )$ (unit ${\mathrm{s}}^{-1}$, hertz), the wave vector $\kappa :=(\omega \mathrm{/}c)s$ (unit ${\mathrm{m}}^{-1}$, with $\mathrm{\mid }\kappa \mathrm{\mid }=\omega \mathrm{/}c$), the wavelength $\lambda :=2\pi \mathrm{/}\mathrm{\mid }\kappa \mathrm{\mid }=2\pi c\mathrm{/}\omega =c\mathrm{/}\nu$ (unit $\mathrm{m}$), and the propagation direction $s$ (dimensionless). In these conventions the phase is $\theta (p)=-\omega t+\kappa \cdot x_{\mathrm{v}\mathrm{e}\mathrm{c}}(p)+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ with $t:=x^0(p)\mathrm{/}c$ the coordinate time in seconds, so the components of $k$ in the chart are

$k_a=(-\omega \mathrm{/}c,(\omega \mathrm{/}c)s_1,(\omega \mathrm{/}c)s_2,(\omega \mathrm{/}c)s_3)\mathrm{.}$

(b) Observer frequency. For an ideal observer whose four-velocity is $U_o=cn$ with $n\in V$ future unit timelike, $\eta (n,n)=-1$ (Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition), define the observer angular frequency of the wave as

${\omega }_n:=-ck(n)=-k(U_o)(\text{unit }{\mathrm{s}}^{-1}),$

and the observer linear frequency as ${\omega }_n\mathrm{/}(2\pi )$. By The phase rate along a worldline is the wave covector evaluated on the four-velocity this is minus the rate of change of the phase with respect to the observer's own proper time, and the sign convention makes it positive.

(c) Well-definedness. $\varpi =-k(e_0)>0$ and $\mathrm{\mid }s\mathrm{\mid }=1$ in every inertial frame, hence $\omega >0$ and $\lambda >0$ are well defined and independent of the choices made in their computation (frame-data lemma); ${\omega }_n>0$ for every future unit timelike $n$ and every future null wave covector $k$ (A future timelike vector pairs negatively with every nonzero future causal vector, The metric pairing is a linear bijection to the dual, and the raised covector in an orthonormal frame), so every ideal observer reads a positive angular frequency, and ${\omega }_n$ depends only on $k$ and the observer vector $n$. The classification of $k$ as future null is the one of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones.

(d) Scope. This item fixes definitions, units and sign conventions only. It does not assert that any physical light is described by such a phase; that identification is adopted in post-sr-ideal-plane-wave-model. It contains no quantum identification of energy with $\mathrm{\hslash }\omega$, no amplitude or intensity content, no medium or dispersion model, no source or detector model, and no measurement or empirical claim. No choice principle is used.