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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.

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    $\begingroup$ The speed of light is a constant, not defined. How we express it need not include the meter. $\endgroup$
    – 21380
    Jul 4 at 22:02
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    $\begingroup$ "but then why didn't we define it as the distance light travels in a 1/100 of a second, that would make c=100m/s, which is much more easy to remember and manage" - to borrow programmer jargon: because of backward compatibility; we didn't want to break all that legacy physics $\endgroup$ Jul 5 at 16:10
  • $\begingroup$ @21380 as a constant we can obviously define a value for it. In fact is commonplace to define $c=1$. $\endgroup$
    – J. Manuel
    Jul 6 at 11:26
  • $\begingroup$ And it's even useful in compound units. The mass (energy equivalent) of an electron is sometimes given as $0.511\text{MeV}/c^2$. $\endgroup$
    – Andrew Ray
    Jul 6 at 12:54

5 Answers 5

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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.

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    $\begingroup$ Imagine the issues of retooling everything using a third system of units, when some countries haven’t even converted to metric. $\endgroup$ Jul 4 at 12:23
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    $\begingroup$ Even changing it to 300000000m/s would be enormously annoying for all the stated reasons, and that would be a very minor change in the length of the meter, at least in terms of human perception. Close enough that incredible care would be required to tell if someone were talking about "old meter" or "new meter". $\endgroup$ Jul 4 at 23:25
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    $\begingroup$ @SoronelHaetir how annoying, I wonder? I'm not sure in how many circumstances a change of 0.07% would be noticeable. GPS probably, and atomic clocks, and maybe not anything else. $\endgroup$
    – Turksarama
    Jul 5 at 0:30
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    $\begingroup$ @Turksarama: Surveying of land, probably; US customary units apparently have 2 versions of the foot, and NIST is planning to drop one of them in 2023. An article about it ( nytimes.com/2020/08/18/science/… ) mentions geological surveying as one case that's affected by the difference of about 1/100th of a foot per mile, or 2 ppm. This hypothetical change to the metre (to make c=3e6 m/s exactly) is ~692 parts per million (0.07 percent). Probably just barely small enough for precision manufacturing to ignore. $\endgroup$ Jul 5 at 6:13
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    $\begingroup$ @Turksarama 0.07%, i.e. 700ppm, may not sound a big deal when measuring distances in everyday life. However that definition may influence other units, for example, where that kind of accuracy is not unheard of. For example, common quartz crystals used for quartz watches have a frequency tolerance of around 20-30ppm (and their resonant frequency depends on the geometrical dimensions of the actual crystal inside the component). Other electronic equipment needs accuracy of that order or better in component manufacturing and characteristics. $\endgroup$ Jul 6 at 14:51
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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.

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    $\begingroup$ This is the correct answer, though for a constant, I'd not say we "define" it, rather, we "express" it. We define the meter because we invented it, but we express the constant c in whatever terms we choose, but those various terms are all exactly the same thing. $\endgroup$
    – 21380
    Jul 4 at 22:04
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    $\begingroup$ @21380 no, "define" is the right word – but as the answer states, the definition is not given by any number+unit, but instead by an experiment that can be performed to measure the speed. $\endgroup$ Jul 5 at 12:45
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    $\begingroup$ We have defined the metre in terms of the second and a stated number, which we have asserted to be constant since experimental evidence suggests the speed of light is constant. Suppose that turned out to be false in some sense (due to an accelerating expansion of the universe or some other as yet unknown reason and our experiments were not yet accurate enough to spot it) - so things we measured and then measured again would appear to be changing length. Then we could then redefine distance or have some other measure of what we cared about. $\endgroup$
    – Henry
    Jul 5 at 15:40
  • $\begingroup$ The speed of light is a physical constant, defined independently of the metre or the second”. It is impossible to define this 3 quantities independently from each other. Define 2 get the third. $\endgroup$
    – J. Manuel
    Jul 6 at 11:36
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    $\begingroup$ @J.Manuel: We don't define the speed of light; it is a given. We define the second (in terms of various vibrational frequencies of Caesium-133), and then we define the metre in terms of the speed of light and the second. $\endgroup$
    – TonyK
    Jul 6 at 14:55
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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.

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  • $\begingroup$ I wanted to write "you cannot define a speed in m/s before you know what a meter is", but I think I understand what you mean: (1) Define a second. (2) Take the physical distance light travels in a second. (3) Divide that into 299,792,458 equal fractions. That is the definition of meter. For some reason, you mention second ... only second ;-) even though it should be first. The number 299,792,458 comes first, even though it's the last step. $\endgroup$ Jul 4 at 23:36
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    $\begingroup$ @Peter: IMHO definitions don't have to happen in a fixed order. Instead, definitions form a DAG, where each edge points from the more fundamental value to the thing that is defined in terms of it. If you want, you could topologically sort that DAG to get an ordered list of definitions, but you don't actually need to do this, because the choice of which topological ordering to use makes no difference. $\endgroup$
    – Kevin
    Jul 4 at 23:58
  • $\begingroup$ @Kevin Well, I take issue with you using the meter before you defined it, and the second, too. $\endgroup$ Jul 5 at 0:00
  • $\begingroup$ @Peter: I didn't write this answer, but my point is that it's fine, because you can use a topological sorting algorithm to see that (3) obviously has to "come before" (1), and the same algorithm can also verify that you haven't made a circular definition. Presenting (3) before (1) is neater, I suppose, but it's unnecessary, because you can work that out on your own anyway. $\endgroup$
    – Kevin
    Jul 5 at 0:02
  • $\begingroup$ When it says "Define the speed of light to be exactly..." (note bolded 'exactly'), it's just a (posh?) way of saying "define things so that [the measured outcome of the physical phenomenon of] the speed of light is exactly...". Where "things" is means "metre". $\endgroup$
    – Pablo H
    Jul 5 at 6:34
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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.

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    $\begingroup$ "Approximately" in this answer means "indistinguishable by any metrology equipment available at the time". $\endgroup$
    – The Photon
    Jul 4 at 15:00
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    $\begingroup$ To make the point even clearer, we could say the metre is defined as 30.66331898849837 times the wavelength of the radiation of the Cs hyperfine transition in vacuum. $\endgroup$ Jul 5 at 12:51
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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.

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