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Its Equivalence With The Radar Method

Now compare Eq.(16) with Eqs.(18) and (13) or Eq.(17) with Eqs.(18) and (14). Observe that for both cases [*]

$\displaystyle m$ $\textstyle =$ $\displaystyle \frac{n_2 -n_1}{2}$ (16)
$\displaystyle n$ $\textstyle =$ $\displaystyle \frac{n_2 +n_1}{2}~.$ (17)

One sees that the radar method is equivalent to the common method provided one identifies the radar pulse data $(n_2 -n_1)/2$ with the $m$th distant clock, and $(n_2 +n_1)/2$ with its $n$th ticking event. This equivalence is new. It extends the fundamental and familiar result based on a lattice array of free-float clocks to (i) the case of an array of inertially expanding clocks and to (ii) the case of a array of accelerated clocks. Put differently, it gives physical validity to the concepts ``inertially expanding frame'' and ``accelerated frame''.

Figure 9: Lattice of spacetime graduation events (heavy dots) determined and calibrated by a single geometrical clock AB which is expanding inertially. The spacetime history of the e.m. pulse bouncing inside this clock is the heavy zig zag line left of the middle. The clock is bounded by two straight lines A and B, the histories of the receding reflectors which keep the e.m. pulse trapped inside the clock. The other straight lines indicate the receding reflector histories of identically constructed clocks, if they were to form an array of adjacent geometrical clocks. The hyperbolas (dashed lines) are the times simultaneous with the tickings of the standard clock AB. The 45$^0$ lines emanating to the left from A and to the right from B are the histories of the two trains of pulses escaping from A and B. The fact that AB is a standard clock implies that all graduation events of the calibrated lattice lie on these histories. Based on the method of pulsed radar, each graduation event (e.g. the encircled dot) is labelled by two unique integers, namely two numbered ticks (the dots in the square and in the diamond) of the clock.They are AB's ``radar coordinates'' of that graduation event.
\includegraphics[scale=.6]{array_of_expanding_clocks}
Figure 10: Lattice of spacetime graduation events (heavy dots) as determined and calibrated by a single geometrical clock CD which is accelerating. The spacetime history of the e.m. pulse bouncing inside this clock is the heavy zig zag line between the two hyperbolas C and D. The clock is bounded by these two hyperbolas, the histories of the two accelerating reflectors which keep the e.m. pulse trapped inside the clock. The other hyperbolas indicate the accelerated reflector histories of identically constructed clocks, if they were to form an array of adjacent geometrical clocks. The straight lines (lightly dotted) are the times simultaneous with the tickings of the standard clock CD. The 45$^0$ lines emanating to the left from C and to the right from D are the histories of the two trains of pulses escaping from C and D, with those escaping from C ultimately crossing the event horizon of clock CD. The fact that CD is a standard clock implies that all graduation events of the calibrated lattice lie on these histories. Based on the method of pulsed radar, each graduation event (e.g. the encircled dot) is labelled by two unique integers, namely two numbered ticks (the dots in the square and in the diamond) of the clock. They are CD's ``radar coordinates'' of that graduation event.
\includegraphics[scale=.765]{array_of_accelerated_clocks}


next up previous contents
Next: IDENTICALLY CONSTRUCTED CLOCKS AS Up: MEASURING EVENTS VIA RADAR Previous: The Common (Non-Radar) Method   Contents
Ulrich Gerlach 2003-02-25