PRINCIPLES OF MEASUREMENT

Geometrical clocks play a fundamental role in the development of the measurement of space and time. However, in order not to appear arbitrary, following Rand and Peikoff, we shall remind ourselves telegraphically of the nature of measurement from a perspective which requires no specialized knowledge and no specialized training.

A process of measurement involves two concretes: the thing being measured and the thing that is the standard of measurement. The relationship between the two is reciprocal: either one may serve as a standard. Measurements pertain to the attributes of these concretes. The choice of one of them as a standard is based on having its attribute serve as a unit of measurement. The process of measurement consists of establishing a relationship to this unit which serves as a standard of measurement.

Within certain limits the choice of a standard is optional. However, the primary standard must be in a form (e.g. platinum meter rod in Paris, or Cesium clock at N.I.S.T. in Boulder, Colorado, etc.) easily accessible to a physicist and it must represent the specific attribute which serves as a unit of measurement (e.g. 1 meter of length, or 1 second = 9,192,631,770 Cesium cycles of time, etc.). Moreover, once a standard has been chosen, it becomes immutable for all subsequent measurements. Any chosen standard satisfies this principle. A standard gets copied in the form of secondary standards. Their purpose is to establish - usually by a process of counting - a quantitative relationship between the standard and any other instance of the attribute of the thing to be measured.

Whenever certain concretes have attributes which can be related to the same standard of measurement, one says that these concretes are commensurable. The importance of commensurability lies in the fact that it is an equivalence relation: If concrete $\mathcal A$ is commensurable with concrete $\mathcal B$, then $\mathcal B$ is commensurable with $\mathcal A$; if $\mathcal A$ is commensurable with $\mathcal B$, and $\mathcal B$ is commensurable with $\mathcal C$, then $\mathcal A$ is commensurable with $\mathcal C$. Using this fact, and omitting explicit reference to the specific measurement of their attributes, but retaining their existence, a physicist integrates these concretes into an equivalence class.

Thus, based on commensurability with a standard rod, one forms an equivalence class, the concept length. Or, based on commensurability with an entity undergoing a periodic process, one forms another equivalence class, the concept time.

A century ago physicists thought that the concept of length and of time required two independent standards, one for each. But in 1905 it was realized that these two standards are not independent. In fact, they are related by a universal conversion factor, the speed of light in vacuum. Thus starting in 1983 both length and time have been defined by referring to a single standard, a unit of time as determined by the tickings of a Cesium atomic clock.

By having such a clock control the pulse repetition rate of a mode-locked femtosecond-laser, one generates a phase-coherent train of pulses Udem et al. (2002). Introduce this train into the one-dimensional resonance cavity of a geometrical clock with ends at relative rest as shown in Figure 4. A resonance condition is obtained when (twice) the length of that cavity is adjusted to equal the spacing in that train of pulses. This resonance condition accomplishes two things:

1. It establishes the relationship between a Cesium-controlled standard of time, i.e. the duration between successive femtosecond pulses, and the corresponding standard of length, i.e. the size of the resonance cavity.
2. It makes the geometrical clock into a single representative of a standard of time and of the space measurements. The periodic tickings of the pulse bouncing back and forth inside provides copies of that standard of time, while the adjusted cavity size furnishes that standard of length.

Figure 4: Free-float geometrical clock driven at resonance by a periodic train of pulses. The partial transmissivity of partial reflector A admits part of their amplitude into the interior of geometric clock AB. When their period matches the reflection time inside the clock, resonance prevails, and they form a single pulse which bounces back and forth (heavy zig-zag line) inside AB. This bouncing is the ticking of the geometrical clock. The ticking interval is the standard of time, while the spatial extent of AB is the associated standard of length. Note that the picture omits, among others, the laser pulses reflected from A and the partially transmitted pulses emerging from B.