Is the definition of a meter tautological?

Question

The speed of light is defined as $c=299{,}792{,}458\,\mathrm{m/s}$, and a meter is defined as the distance that light travels in a $1/299{,}792{,}458=1/c$ of a second, but then we would have defined a meter in terms of the speed of light, but we also defined the speed of light in terms of a meter, seems a bit circular for me.

My guess is that we defined a meter as the distance that light travels in a $1/299{,}792{,}458$ of a second so that the speed of light would be exactly $299{,}792{,}458\,\mathrm{m/s}$, but then why didn't we define it as the distance light travels in a $1/100$ of a second, that would make $c=100\,\mathrm{m/s}$, which is much more easy to remember and manage.

Please tell me if there are any ambiguities in my question, I'll do my best to fix them, thanks.

Answer 1

Theoretically, we have not defined the speed of light in terms of the metre. We have defined it as a specific distance (that light can cover in one second).

Now take that distance and divide it with $299792458$, and then you have a smaller portion of a distance. That portion is defined as a metre.

So, there's no circular metre definition here.

Why this number? you may reasonably ask. The answer is that while we can change the definitions of fundamental units such as the metre so that they become more future-safe and universally accessible and thus scrap an old definition, we can't just change their values to something entirely different. Because those fundamental units have already been in use in everything from research to daily life through centuries.

If we suddenly redefined the metre to be just $1/100$ of the distance covered by light in a second (which is an enormously long distance, by the way), then we would have to alter every ruler, every length scale, every textbook in the world, not to speak of altering people's uses, mindsets, traditions and so on. (Also, making the metre so enormously long as you suggest, might cause the use of the metre-unit to die out from every-day life and other units better fitting to the human-scale might become more used.)

Such a value-redefinition would be an enormously impractical task to implement - to get this through, you might want a better reason than just that the definition becomes easier to remember. Nevertheless, it is an interesting question that goes to the historical roots of how standardisation is done.

Answer 2

Your definition of the speed of light is wrong. The speed of light is a physical constant, defined independently of the metre or the second. Instead, it is defined by saying that there’s this physical phenomenon called “light,” and we let $c$ denote the speed at which it propagates in vacuum, which turns about to be Lorentz invariant. Having established that, we can see that the definition of the metre is not circular.

Answer 3

This answer addresses the original title of this question: "Is the definition of a meter circular?".

No it’s not . The key idea here is the use of extremely precise “ticks” to define the second, and consequently, by virtue of the universal constancy of the speed of light define the meter.

“The meter is $\frac{1}{299,792,458}$ part of the path length traveled by light after $9,192,631,770$ periods of the radiation corresponding to the transition between the hyperfine levels of the unperturbed ground state of the Cs-133 atom”.

In this definition there is no explicit use of the speed of light, only fractions and clock ticks. However, technically we are resetting (redefining) the speed of light to be exactly $c=299,792,458 \, \mathrm{m/s}$, because $9,192,631,770$ ticks of this specific Cs-133 radiation is exactly $1 \mathrm{s}$ since 1962. By the way, this is a definition similar to the one using the meridian, where the defining “big-length” moved from 1 quarter of a meridian to 1 light-second, but makes the new meter a more reproducible, time stable, universally available standard.

Relatively to your next concern, the reason for choosing those exact fractions is to avoid rock the boat by destroying the metric system.

Answer 4

The current definition of the metre in the distance that light travels in $1/299,792,458$ of one second. This implies a value of the speed of light. Based on the definition and that of the second ($9,192,631,770$ cycles of radiation from the transition between hyperfine levels of the ground state of $^{133}\text{Cs}$), it is possible to realise the metre.

The original (1791) definition of the meter was chosen to be equal to $1 / 10,000,000$ of the distance between the North Pole and equator through Paris. Subsequent redefinitions were chosen such that changes in the length of the metre were minimal in practice. Values implied by prior definitions remained approximately correct.

Answer 5

No. Please refer to:

https://en.wikipedia.org/wiki/History_of_the_metre

A metre (as spelled EU-style) was standardized in 1889 as the length of a bar of platinum-iridium held at melting point of ice in Paris and other locations around the globe.

Later new standardizations were adopted based on optical measurements - firstly in 1960 one based on the wavelength of a specific krypton-86 transition; then that one you refer to in your post based on the speed of light was adopted in 1983 and this was updated lately in 2019.

Now that highly precise measurements of the speed of light are available - and this, as others here point out, is constant - we can use this to provide a preciser measure of the metre.

So it's not so much a circular definition as a sort of bootstrapped definition procedure.