Adjacent Clocks

To compare the operation of two adjacent geometrical clocks, AB and BC one notes that they have radar unit B in common. Assume that all three radar units A, B, and C move collinearly along the $z$ axis. The common radar unit B has electromagnetic pulses bouncing off it. There are those from A and those from C. Consider $a$ consecutive pulses from A and $c$ consecutive pulses from C:

These sequences are depicted in Figures 5 and also in 6. We say that these two sequences are matched relative to B, and we write

$$\{p_i,p_{i+1},\cdots ,p_{i+a}\}_B \sim \{q_j,q_{j+1},\cdots , q_{j+c}\}_B~,$$

if and only if they have - within a prespecified accuracy - coincident starting ($p_{i}$ and $q_{j}$) and coincident termination ($p_{i+a}$ and $q_{j+c}$) pulses. The subscript B on these sequences serves as a reminder that the pulses are being counted at radar unit B.

The electromagnetic pulses impinging on B get partially reflected and partially transmitted. Thus for every pulse sequence $\{p_i,p_{i+1},\cdots ,p_{i+a}\}_B$ measured at B there are corresponding sequences $\{p_i,p_{i+1},\cdots ,p_{i+a}\}_A$ and $\{p_i,p_{i+1},\cdots ,p_{i+a}\}_C$ measured at A and C respectively. Thus one has the following proposition (``Invariance of matched sequences''):

The property of being matched is invariant as each sequence of pulses travels from one radar unit to another, i.e. if

$$\{p_i,p_{i+1},\cdots ,p_{i+a}\}_B \sim \{q_j,q_{j+1},\cdots , q_{j+c}\}_B~,$$

then

$$\{p_i,p_{i+1},\cdots ,p_{i+a}\}_A \sim \{q_j,q_{j+1},\cdots , q_{j+c}\}_A$$

and

$$\{p_i,p_{i+1},\cdots ,p_{i+a}\}_C \sim \{q_j,q_{j+1},\cdots , q_{j+c}\}_C~.$$

The validity of this proposition is an expression of the principle of the constancy of the speed of light, that is, of the fact that light pulses cannot overtake each other. If two pulses, say $p_i$ and $q_j$, are coincident on the world line of radar unit B, then they are still coincident after they have travelled to the world line of any other radar unit, regardless of its motion.

As measured by atomic clock B, the ticking rates of geometrical clocks AB and BC need not be uniform, and in general they are not. This is evident from Figure 5. This deficiency is remedied by calibrating the rate of pulses coming from C in terms of AB. Thus for every $c$-sequences of pulses departing from C and arriving and counted at B, there is a matched $a$-sequence generated by clock AB also at B. The ratio

$$\frac{c}{a}= \frac{\textrm{(\char93 of ticks of clock BC)}}{\textrm{(\char93 of ticks of clock AB)}}$$ (3)

is the normalized ticking rate of clock BC. The normalization is relative to clock AB. Conversely, the inverse of Eq.(4),

$$\frac{a}{c}= \frac{\textrm{(\char93 of ticks of clock AB)}}{\textrm{(\char93 of ticks of clock BC)}}$$ (4)

is the ticking rate of AB normalized relative to BC. Because of the invariance of matched sequences, it does not matter whether the ratios, Eqs.(4)-(5), were measured at radar unit A, B, or C.

We say that the adjacent clocks AB and BC are normalizable if both ratio (4) and ratio (5) are non-zero for every matched pair of $a$ and $c$-sequences along the world lines of the two adjacent clocks. A basic and obvious aspect of normalizability for adjacent clocks is its reciprocal property: If AB is normalizable relative to BC, then BC is normalizable relative to AB. Thus all collinear clocks AB, BC, BD, BE, $\cdots$, which share radar unit B, are mutually normalizable.

Of particular utility are clocks which are commensurable. Their distinguishing property is obtained by subdividing the set of normalizable geometrical clocks further and selecting those whose normalized ticking rates, Eq.(4) or (5), are constant for all matched starting and termination pulses. Such clocks allow one to view the boost-invariant accelerated and the boost-invariant expanding inertial frames from a single perspective, which is developed in Section VII.

Before giving the precise general definition of commensurability (Section IV+.1667emB, we interrupt the developement by illustrating the above constellation of definitions, applying them to various combinations of inertial and accelerated clocks.

Nota bene: For the purpose of verbal shorthand we shall allow ourselves to refer to ``geometrical clocks'' simply as ``clocks''. However, for atomic clocks we shall use no such shorthand. Thus clocks without the adjective ``atomic'' are understood to be geometrical clocks, while atomic clocks are always referred to by means of the modifier ``atomic''.