DOPPLER MEASUREMENT AND PLAIN-NUMBER SPEED HAVE BECOME PARADIGMS

Doppler speed measurement inherently treats speed as a fraction of the speed of the signal--in the generic case as a fraction the speed of light. This plain unit-free number--this fraction which physicists often denote by a Greek beta--has become paradigmatic. The formulas in relativity don't work unless the speed is represented by a plain dimensionless number (v/c, or beta)--almost as if nature is suggesting that we think of speed that way instead of in terms like "meters per second." Moreover, Doppler speed measurement has become an increasingly common part of life.

I am trying to think of some other examples besides airport control towers and highway speed traps. How about measuring the approach or recession of a star? A fractional increase of a ten-to-seventh part in the frequency of its starlight signifies it is approaching at a ten-to-seventh of the standard speed. A ten-to-seventh part of the speed of light is, as I mentioned earlier, is freeway speed, around sixty-seven miles per hour.

In practice, speeds are actually read directly from the fractional increase or decrease in frequency. The underlying relativistic formulas, which involve a slight correction of the practical, are even more explicit in requiring the unit-free or dimensionless version of speed s=v/c. I'm writing s because I don't have a beta. The relativistic Doppler frequency-ratio formula (notice it is for a dimensionless ratio ƒ of frequencies, not for a difference expressed in terms of some unit) is ƒ=square root[(1+s)/(1-s)]. The speed has to be a pure unit-free number or else it wouldn't work: could not be added to one.

The inverse formula tells the dimensionless speed of approach or recession from the (also dimensionless) frequency ratio:

s=(1-ƒ²)/(1+ƒ²)

Here as well, the dependent variable must be unit-free for things to work, and what the formula provides is also a pure number. As mentioned earlier, it's a general rule that relativistic formulas, which are the more correct, depend on the dimensionless speed and not on "meter per second" speeds.

In case anyone was wondering, meters are officially defined nowadays by reference to the speed of light--so at bottom all speed measurements, not merely those explicitly Doppler, are technically anchored to the speed of light. The speed of light can no longer be measured in modern-day metric units because it has been assigned an exact value by the definition of the meter. Since 1983 the meter has been defined as the distance light in vacuo travels in 1/299792458 of an atomic clock second. You might say that speeds can now, in effect, ONLY be reckoned as fractions of the in vacuo speed of light, because computationally that underlies what is required to express them in meters per second! This is only a further confirmation of the way in which Doppler speed measurement (where a percentage frequency shift is interpreted as a percentage of the speed of light) and the underlying unit-free form of speed have become paradigmatic.