The operational radar protocol of an ideal observer

Adopt the ideal-clock and ideal-light assumptions of Adopted ideal-clock and ideal-vacuum-light interpretation and the SI conventions of SI units and coordinate conventions of the special-relativity framework. Let $x:I\to A$ be a future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space) with proper-time function $\tau$ (Proper time along a future timelike worldline), and let $s_{-}<s_{+}$ be parameter values in $I$. Suppose an event $q$ and the events $e_{-}:=x(s_{-})$, $e_{+}:=x(s_{+})$ satisfy: $q-e_{-}$ is future null and $e_{+}-q$ is future null (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones, Oriented time-oriented affine Minkowski space (the spacetime primitives)). Under the adopted ideal-light assumption an ideal ray is emitted at $e_{-}$, reaches $q$, and an ideal ray emitted at $q$ reaches $e_{+}$; the event $q$ is idealized as an instantaneous reflection that changes nothing else. The radar protocol of the observer assigns to $q$

(i) the radar time ${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q):=\frac{1}{2}(\tau (s_{-})+\tau (s_{+}))$, the ideal clock's reading midway between emission and reception, and

(ii) the radar distance ${\rho }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q):=\frac{c}{2}(\tau (s_{+})-\tau (s_{-}))$, half the round-trip light time multiplied by $c$.

Two events measured by the same observer are radar-simultaneous when their radar times agree. The assignment does not change under increasing $C^1$ reparametrizations with everywhere positive derivative and the same clock origin of the observer's worldline, because the proper-time function does not (Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps).

Conventions and scope. (a) The protocol is a definition of an operational assignment conditional on the existence of the emission, reflection and reception events, and it assumes an available ideal reflector and ideal apparatus. (b) The split of the round trip into a "time" and a "distance" uses the symmetric convention in which half of the round-trip time is assigned to the outgoing and half to the returning segment; the protocol determines the round-trip quantity $\tau (s_{+})-\tau (s_{-})$ operationally, while the derived one-way assignment is a convention and does not measure a one-way light speed independently of a synchronization convention. (c) The conditional assignment is defined for the stated future timelike worldline, including an accelerated one, but no general existence, uniqueness or synchronization theorem for accelerated observers is asserted. Nothing is claimed about dispersive or absorbing media, finite-resolution apparatus, global single-valuedness or detector models, and no measurement outcome, statistical inference or empirical comparison is asserted.

• Well-definedness. The data $\tau (s_{-})$ and $\tau (s_{+})$ are real numbers determined by the worldline and the two parameters (Proper time along a future timelike worldline, Curves, regular causal curves and future timelike worldlines in affine Minkowski space), so ${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q)$ and ${\rho }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q)$ are well defined; the existence hypotheses on $q$ are exactly the condition that $q$ is connected to the observer's history by a future null outgoing segment from $e_{-}$ and a future null returning segment to $e_{+}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones). Since $\tau$ is invariant under increasing $C^1$ reparametrizations with everywhere positive derivative and the same clock origin of the worldline (Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps), so are the radar time and radar distance.

• The two null segments are the operational content of the adopted assumption. The existence of ideal rays along the null displacements $q-e_{-}$ and $e_{+}-q$ is supplied by the ideal-light clause of Adopted ideal-clock and ideal-vacuum-light interpretation, which identifies ideal light rays in vacuum with future null straight lines; the protocol therefore describes an assignment conditional on that idealization and does not describe a physical mirror, a finite apparatus, or a medium.

• The half-split is a convention, not a derived fact. The protocol measures the round-trip time $\tau (s_{+})-\tau (s_{-})$ operationally; writing half of it as a one-way time and $c\mathrm{/}2$ times it as a distance is the symmetric convention adopted here. For inertial observers this convention is shown to agree with the simultaneity slices of the observer in Radar synchronization of an inertial observer reproduces its inertial chart, while the rotating-ring example below concerns synchronization by the congruence's local rest spaces, not a failure of a single rotating observer's radar assignment. Neither statement is part of the definition.