The relativity of measurements

In physics, as well as in everyday life, “small” and “large” have no absolute meaning. Their meaning always correspond to a comparison, often kept implicit.

Quantitatively, we translate this by using units. Units are arbitrary quantities used as references to specify quantities of the same nature. The result of any physical measurement is then specified at least by a number and a unit. The number indicates how many times the unit “fits” in the measured quantity (see Fig. 1).

Figure 1: The measurement of quantities is never absolute, but always defined in comparison to units. For example, when you go to the market and buy 5kg of oranges, it means that the mass of your oranges is five times the standardized and agreed upon mass of 1kg. You can see this clearly if your seller uses a good old mechanical balance, where your fruits are compared to a standard masses.

Any quantity can be used as a reference unit to express other quantities of the same nature. However, to facilitate communication and economical exchanges, we try to all use the same units, most commonly the International System of Units. While this is convenient, it often makes us lose sight of the fact that measurements are relative to the choice of a specific reference unit.

The situation is similar to the way we deal with the positions or speeds of objects. We specify positions and speeds by numbers associated with appropriate units, and a reference point from which the positions or speeds are measured (or specified). See Figure 2 for an example where we tend to keep the reference frame implicit. The important point is that the position, velocity and orientation of an object are relative to the choice of a reference frame.

Fig 2. Speed is always relative to a reference frame. Imagine that you walk in a fast train. You are walking at 300 km/h relative to the ground, but only at 5 km/h relative to the train. This means that a change in the position reference implies a change in your speed. Image source: gizmodo.

Imposing that the general form of the equations of physics must be independent of the chosen reference frame is extremely constraining in the development of physics theories. This exercise results in the so called relativity theories. This method is amazingly far reaching. The whole of fundamental classical physics, including mechanics, electrodynamics and of course the general relativistic description of gravitation can be formulated by applying the relativity principle to position, orientation and motion! However, relativity has no reason to be restricted to position, orientation and velocities. We will discuss how scale relativity extends relativity theories to include considerations for the effect of a change in the choice of the reference unit — or the scale at which the quantity is considered. The pertinence of this consideration lies in an often neglected aspect of units. A unit is more than just a reference quantity. It also carries information about the resolution.

The micrometer, the meter, the kilometer, the Astronomical Unit, the parsec are all reference units of length or distance. In principle, we can measure any length by using any one of these units. However it would be quite cumbersome and even impractical to discuss the size of micro-organisms using the astronomical unit or the distance between stars in terms of micrometers. This is not only a question of suitability, it has to do with the meaning of the considered quantities.

First, measurements depend on the measuring instruments. A measuring instrument always has a limited range of operation and a limited accuracy. For example, a meter-stick is a convenient tool for measuring the size of a book-shelf, but it becomes very impractical to measure the size of a parmecium or the size of a continent.

Second, and more fundamentally, the use of an inappropriate unit questions the very definition of both the quantity being measured and the unit used to report the measurement. For example, measuring the distance between the Earth and the Sun with a micrometer precision is more of a conceptual rather than a technological challenge. We first need to define the distance between the Earth and the Sun in a way that remains meaningful when considered with a micrometric precision.

Consequently, the result of a measurement includes a typical error, most often corresponding to the precision of the measuring apparatus. This precision can be thought of as the resolution or scale of inspection achieved during the measurement. We can also regard it as the contribution of the smallest significant detail that can be taken into account with a given measurement method. Usually, we also express physical quantities using a unit comparable to the precision or measurement error. For example, if we measure the length of a pencil using a ruler graduated at a millimetric scale or a caliper, we get different results (see Fig. 3). Measuring the length of the pencil with an even higher accuracy requires to reflect about the actual meaning of “length of the pencil”. Such a reflection can be guided by the specific motivations for making the effort to achieve such a great precision.

Figure 3: Two ways to measure a pencil, with different resolutions. At the bottom is a ruler graduated at the 1 millimeter scale. We may write the result as 13.2 ± 0.1 cm. The error (± 0.1) indicates the resolution or precision of the measurement. On top is a caliper, graduated at the 0.1 millimeter scale (100 micrometer). The measurement of the length of the same pencil may then be 132.40± 0.05 mm. With the graduated ruler, the length of the pencil can be well defined and measured with the graduated ruler. However, with the caliper, one may have to be careful and take into account the fact that the eraser is squishy.

What aspects of the pencil would we have to consider to define and measure its length with micrometric precision? What about nanometric precision? Can we still talk about the length or any other aspects of the pencil when considering picometric or even femtometric precisions? Inversely, one may consider the length of the pencil at very coarse resolutions: meters, kilometers, astronomical units or even parsecs. It does not seem very interesting, because the measurement of the length of the pencil is then zero. Note that it is zero, and not “almost zero”. Indeed, assume that the only available resolution is the Astronomical Unit (AU). The pen then measures 0 ± 1 AU. Saying that it’s “almost zero” would imply we have some knowledge of the pencil at resolutions finer than 1 AU.

We see that at both ends of the range of resolutions, the length of the pencil loses its significance. The notion of “length of the pencil” may have significance only over a relatively narrow interval of resolutions.

Other systems, in quantum, chaotic or complex domains may display structures proper to their own nature over a broader range of scales. Physics runs in all sorts of difficulties when approaching this type of objects. Scale relativity proposes to approach these difficulties by including relative resolutions, alongside with relative positions, relative orientations and relative motions, as additional parameters specifying reference frames from which observations are made. As the name suggests, scale relativity provides an extension to relativity theories such as Galilean relativity, special relativity and general relativity. As such, scale relativity includes earlier relativity theories and brings consideration for the resolution scale at a fundamental level. Soon we will be discussing how physics usually deals with scales in nature.

Animation about the metric system

TED-Ed. 2016. Why the Metric System Matters — Matt Anticole.