The Common (Non-Radar) Method

There is, of course, the more common and familiar method. It does not use radar. Instead, it uses two distinct standards, namely, identically constructed clocks and standard rods Taylor and Wheeler (1992). The measuring procedure itself, we recall, consists of (i) locating the event by counting standard rods, and (ii) determining its time by counting at that location the ticks of the clock, which is synchronized to the standard clock.

One is now confronted with a question of consistency: Is this common non-radar method compatible with the radar method, even if the spacetime framework is based on inertially expanding or accelerated clocks?

Consider the common method of measuring an event. It consists of starting with a geometrical clock having a spacetime history as depicted in Figure 2 or 3. Such a clock is a standard of time and of length. Thus a physicist forms a spatial array of adjacent clocks AB, BC,$\cdots$, EF,$\cdots$ which are identically constructed and synchronized. The definition is as follows:

 

Clocks AB, XY, $\cdots$ are said to be identically constructed if their frequency shift factors are all the same:

$$k_{AB}=k_{XY}=\cdots\equiv e^{\Delta\tau}~.$$ (14)

This definition is illustrated in Figures 7 and 8.

The ticking of adjacent clocks is synchronized by synchronizing the pulses impinging on their shared radar unit. This procedure guarantees synchronization of all clocks. It is exemplified in Figure 7. There the three clocks AB, BC, and CD have the phases of their internal pulses adjusted to tick in synchrony.

Suppose standard clock AB has its $n$th (resp. $(n+1)$st) ticking event at its left (resp. right) radar unit A (resp. B). These events are exhibited by Eq.(2) or (3). Then, by induction, the left radar unit of the $M$th identically constructed clock has its $N$th ticking event at

Having formed a linear array of such clock, the physicist uses the lattice of events generated by their tick-tock actions as a standard to measure an arbitrary event. The common method of measuring an event consists of counting (i) how many clocks separate it from the standard clock ($M=0$), and (ii) how many clock ticks elapse before this event happens. The result of these two counts is the pair of integers

$$\begin{array}{cc} M=m& \left( \begin{array}{c} \textrm{res... ...\textrm{temporal measurement} \end{array} \right). \end{array}$$ (15)

They comprise the measurement of the given event in units of time and spatial extent as furnished by the standard geometrical clock.