Rindler radar distances worked: reflectors at twice and at half the observer's radius

Worked computation in the Rindler chart of The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance. Let $c>0$, fix ${\rho }_0>0$ and take $y_0=z_0=0$; let $O$ be the congruence worldline $\rho ={\rho }_0$ with Rindler time $\theta =c\tau \mathrm{/}{\rho }_0$, and take the emission event $e_{-}$ at ${\theta }_e=0$, so that $e_{-}=x^{-1}(0,{\rho }_0,0,0)$ in the inertial chart $x$.

(a) Reflector at ${\rho }_m=2{\rho }_0$. The future null line from $e_{-}$ of constant $V$ has $V=-{\rho }_0$, and it meets $\rho =2{\rho }_0$ at the Rindler time ${\theta }_m=\log 2$; using $\sinh (\log 2)=3\mathrm{/}4$ and $\cosh (\log 2)=5\mathrm{/}4$ (Addition formulas, identities, parity, and derivatives of the hyperbolic functions, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm) the reflection event is $q_m=x^{-1}(3{\rho }_0\mathrm{/}2,5{\rho }_0\mathrm{/}2,0,0)$, and the return event is $e_{+}=x^{-1}({\rho }_0\sinh (2\log 2),{\rho }_0\cosh (2\log 2),0,0)=x^{-1}(15{\rho }_0\mathrm{/}8,17{\rho }_0\mathrm{/}8,0,0)$ at Rindler time $\theta (e_{+})=2\log 2$. Thus $({\theta }_e+\theta (e_{+}))\mathrm{/}2=\log 2={\theta }_m$, the radar time assigned to $q_m$ is $({\rho }_0\mathrm{/}c)\log 2$ plus the base constant of $O$'s proper time, and the radar distance is ${\rho }_0\log 2$. The two worldlines $\rho ={\rho }_0$ and $\rho =2{\rho }_0$ are at rest distance ${\rho }_0$ in the slice $\theta =\log 2$; since $e>2$ gives $\log 2<1$ (The elementary numerical bound $2<e<3$, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm), the radar distance is strictly smaller than the rest distance.

(b) Reflector at ${\rho }_m={\rho }_0\mathrm{/}2$. The future null line from $e_{-}$ of constant $U$ has $U={\rho }_0$, and it meets $\rho ={\rho }_0\mathrm{/}2$ at the Rindler time ${\theta }_m=\log 2$; using $\sinh (\log 2)=3\mathrm{/}4$ and $\cosh (\log 2)=5\mathrm{/}4$ the reflection event is $q_m^{\mathrm{\prime }}=x^{-1}(3{\rho }_0\mathrm{/}8,5{\rho }_0\mathrm{/}8,0,0)$ and the return event is again $e_{+}=x^{-1}(15{\rho }_0\mathrm{/}8,17{\rho }_0\mathrm{/}8,0,0)$ at Rindler time $2\log 2$. The radar time is again $({\rho }_0\mathrm{/}c)\log 2$ plus the base constant and the radar distance is again ${\rho }_0\log 2$, while the rest distance in the slice $\theta =\log 2$ is now ${\rho }_0\mathrm{/}2$; since $e<3<4$ gives $\log 2>1\mathrm{/}2$ (The elementary numerical bound $2<e<3$, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm), the radar distance is now strictly larger than the rest distance.

(c) The two cases (a)-(b) give the same radar distance ${\rho }_0\log 2$ at distinct rest distances ${\rho }_0$ and ${\rho }_0\mathrm{/}2$; this demonstrates noninjectivity of the assignment as the reflector radius varies. To prove that radar distance is not determined by the rest distance alone, also take ${\rho }_m=3{\rho }_0\mathrm{/}2$. With the same emission at ${\theta }_e=0$, the reflection occurs at ${\theta }_m=\log (3\mathrm{/}2)$, at $q_m^{\mathrm{\prime }\mathrm{\prime }}=x^{-1}(5{\rho }_0\mathrm{/}8,13{\rho }_0\mathrm{/}8,0,0)$, and the return occurs at $2\log (3\mathrm{/}2)$, at $e_{+}^{\mathrm{\prime }\mathrm{\prime }}=x^{-1}(65{\rho }_0\mathrm{/}72,97{\rho }_0\mathrm{/}72,0,0)$. This reflector has rest distance ${\rho }_0\mathrm{/}2$, equal to case (b), but radar distance ${\rho }_0\log (3\mathrm{/}2)<{\rho }_0\log 2$, because $3\mathrm{/}2<2$ and $\log$ is strictly increasing. These equal inputs and unequal outputs prove the stated failure of dependence on rest distance alone. The original two cases both satisfy ${\theta }_m=({\theta }_e+\theta (e_{+}))\mathrm{/}2$ and have round-trip proper time $2{\rho }_0\log 2\mathrm{/}c$; the third case satisfies the same midpoint identity and has round-trip proper time $2{\rho }_0\log (3\mathrm{/}2)\mathrm{/}c$.

(d) Scope. Conditional model computation with ideal clocks, ideal light rays and ideal reflectors in the flat classical model; no measurement, apparatus, source or statistical claim, and no empirical inference, is made, and the idealizations of The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (v) are retained.