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 axis. The
common radar unit B has electromagnetic pulses bouncing off it.
There are those from A and those from C. Consider
consecutive pulses from A and 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

if and only if they have - within a prespecified accuracy - coincident starting ( and ) and coincident termination ( and ) 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 measured at B there are corresponding sequences and 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

then

and

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 and , 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 -sequences of pulses departing from C and
arriving and counted at B, there is a matched -sequence generated
by clock AB also at B. The ratio

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 and
-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,
, 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''.