Ideal local phase reception by stationary Rindler observers

In the flat affine Minkowski model with signature $(-,+,+,+)$ and $x^0=ct$, use the smooth right-wedge chart and stationary congruence of The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance. A stationary observer has fixed $\rho >0,y,z$, worldline $z(\tau )$ on an open proper-time interval in seconds, and four-velocity $U=dz\mathrm{/}d\tau$ with $\eta (U,U)=-c^2$. These are accelerated timelike worldlines, not global inertial frames.

Let $\theta$ be the dimensionless affine phase with constant nonzero future null covector $k=d\theta$ and ideal vacuum rays of Adopted ideal plane-wave and frequency-reading model (i). As an additional local reception assumption, adopt that an ideal point receiver comoving with a stationary Rindler observer assigns, at the event $z(\tau )$, the instantaneous received angular frequency

$\omega (\tau ):=-\frac{d(\theta \circ z)}{d\tau }=-k(U(\tau ))>0,$

and linear frequency $\nu (\tau )=\omega (\tau )\mathrm{/}(2\pi )$ in the conventions of Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave and SI units and coordinate conventions of the special-relativity framework. The phase derivative and positivity are mathematical facts of The phase rate along a worldline is the wave covector evaluated on the four-velocity; their identification with this ideal receiver's reading is the assumption adopted here. This is the pointwise equivalence to the momentarily comoving inertial phase receiver, together with the proper-time clock convention of Adopted ideal-clock and ideal-vacuum-light interpretation. Reception comparisons use one and the same affine wave and events joined by one of its radial rays. This assumption does not postulate an emitter mechanism.

The instantaneous quantity may vary along an accelerated worldline. It is not an assertion that an observer's entire proper-time signal is monochromatic or that an arbitrary finite-duration Fourier measurement equals this value. For possible finite-apparatus use, define the local wavelength ${\lambda }_{\mathrm{l}\mathrm{o}\mathrm{c}}:=2\pi c\mathrm{/}\omega$ and the translational acceleration length $L_{\alpha }:=c^2\mathrm{/}\alpha$, where $\alpha$ is the observer's proper-acceleration magnitude; for this congruence $L_{\alpha }=\rho$. The intended local ray regime has ${\lambda }_{\mathrm{l}\mathrm{o}\mathrm{c}}\mathrm{/}{L}_{\alpha }\ll 1$ throughout the observation segment, receiver size small relative to $L_{\alpha }$, and an observation duration long enough for several cycles but short relative to $L_{\alpha }\mathrm{/}c$. These qualitative scale separations give no numerical tolerance or measured uncertainty. The equality above is exact only inside the adopted ideal model; no accuracy claim for a real receiver follows from it.

This supplements, without changing, the accelerated-reception exclusion in Adopted ideal plane-wave and frequency-reading model (iv). It includes no detector dynamics, intensity, polarization, finite-bandwidth pulse or envelope, medium, dispersion, absorption, quantum or gravity model, no energy identification with $\mathrm{\hslash }\omega$, and no empirical outcome or proof of the model's applicability. It is a sourced adopted idealization, not a deductive or experimentally established law.