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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.

Community
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Yanis Mekrebi
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    Related: https://physics.stackexchange.com/q/327246/2451 , https://physics.stackexchange.com/q/9314/2451 , https://physics.stackexchange.com/q/406454/2451 , https://physics.stackexchange.com/q/505360/2451 – Qmechanic Jul 04 '22 at 12:40
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    The speed of light is a constant, not defined. How we express it need not include the meter. –  Jul 04 '22 at 22:02
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    "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 – Filip Milovanović Jul 05 '22 at 16:10
  • @21380 as a constant we can obviously define a value for it. In fact is commonplace to define $c=1$. – J. Manuel Jul 06 '22 at 11:26
  • And it's even useful in compound units. The mass (energy equivalent) of an electron is sometimes given as $0.511\text{MeV}/c^2$. – A. R. Jul 06 '22 at 12:54

5 Answers5

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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.

Steeven
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    Imagine the issues of retooling everything using a third system of units, when some countries haven’t even converted to metric. – ZeroTheHero Jul 04 '22 at 12:23
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    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". – SoronelHaetir Jul 04 '22 at 23:25
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    @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. – Turksarama Jul 05 '22 at 00:30
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    @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 ( https://www.nytimes.com/2020/08/18/science/foot-surveying-metrology-dennis.html?nl=todaysheadlines&emc=edit_th_20200819 ) 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. – Peter Cordes Jul 05 '22 at 06:13
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    @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. – LorenzoDonati4Ukraine-OnStrike Jul 06 '22 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.

Sandejo
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    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. –  Jul 04 '22 at 22:04
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    @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. – leftaroundabout Jul 05 '22 at 12:45
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    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. – Henry Jul 05 '22 at 15:40
  • “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. – J. Manuel Jul 06 '22 at 11:36
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    @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. – TonyK Jul 06 '22 at 14:55
  • @J.Manuel Not only is it possible to define the 3 quantities independently, it’s been done. The speed of light must be defined based only on the physical behaviour of light, else it would not be the speed of light. While the metre and second are currently defined to depend on one another and the speed of light, that was not always the case. For instance, historically, the metre and second have respectively been defined in terms of things the distance between two marks on a rod in a vault and a certain fraction of a day—definitions independent of each other and the speed of light. – Sandejo Jul 06 '22 at 15:03
  • @TonyK We certainly can define all physical constants (except I believe the fine structure constant). You end up with a different unit system though, but you can do it. In fact it is common to define the speed of light to be $c=1$. – J. Manuel Jul 07 '22 at 09:35
  • @Sandejo You cannot define a meter that has a length of $1 \mathrm{m}≡1 , \mathrm{heightOfTheStatueOfLiberty}$, a second that has a span of $1 \mathrm{s}≡ \mathrm{0.5 days}$ and a speed of light $c=299792458 \mathrm{m/s}$. In your example, once they independently defined the meter and the second, the speed of light was doomed to be that many meters sticks that light travels in a second. There were no freedom to define the speed of light. – J. Manuel Jul 07 '22 at 10:22
  • @J.Manuel Setting $c=1$ does not define the speed of light. In saying that, we first define $c$ to be the speed at which light propagates in vacuum. Then, we define units of length and time such that, in those units, $c=1$, i.e. length and time have the same unit. Also, it is entirely possible to define a metre and second as you suggested; they just wouldn’t be consistent with the current definitions. However, while you could then define some constant $c$ by $c=299792458\ \mathrm{m/s}$, this would not be the speed of light, as it is not the speed at which light propagates in vacuum. – Sandejo Jul 07 '22 at 13:51
  • @Sandejo. Maybe you don't like $c=1$. Perhaps just define $c=1 , \mathrm{UnitOfVelocity}$. In unit systems one could start by define the “lengths” or spans of fundamental units and then get into specific numerical values for the constants in that system. This is the SI case. However, one could definitively start by defining the constants and then get the specific spans of the units of that system. An example of this kind of unit system is Planck units. Your system just need to be consistent within itself. – J. Manuel Jul 07 '22 at 15:43
  • @Sandejo. “The four universal constants that, by definition, have a numeric value 1 when expressed in these units are”. Quote from Plank units page in Wikipeadia. Now, maybe we disagree on what it means defining something. I saw that used setting instead of defining. Since this is out of my expertise I’ll rest my case. – J. Manuel Jul 07 '22 at 15:50
  • @J.Manuel I have nothing against natural units. I'm just saying that I think you're misunderstanding what is being defined when we say $c=1$. When the Wikipedia page on Planck units says "[t]he four universal constants that, by definition…" it is referring to the definition of the units, not the constants. – Sandejo Jul 07 '22 at 16:16
  • @leftaroundabout You are using two contexts of the word "define" interchangeably, which is producing confusion. To define, as in give a definition, and to define, as in declare a value for. We can indeed give a definition for the "speed of light" and a meter, but we do not declare the value of the speed of light as we do for the meter. To suggest we can "declare the value" of the speed of light is clearly nonsense. That value is a constant, a given. How we typically express it is the thrust of this question. –  Jul 07 '22 at 18:06
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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.

J. Manuel
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  • 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. – Peter - Reinstate Monica Jul 04 '22 at 23:36
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    @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. – Kevin Jul 04 '22 at 23:58
  • @Kevin Well, I take issue with you using the meter before you defined it, and the second, too. – Peter - Reinstate Monica Jul 05 '22 at 00:00
  • @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. – Kevin Jul 05 '22 at 00:02
  • 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". – Pablo H Jul 05 '22 at 06:34
  • I think it would be better to make point 1 be your point 2, and then a new point 2 "Define the metre as one 299,792,458'th part of the distance travelled by light in one second." (Alternatively, "define the metre as that distance such that light travels 299,792,458 metres in one second") – Martin Bonner supports Monica Jul 05 '22 at 10:14
  • I completely revamped this answer to address the concerns mention in the comments above keeping the main point that was to show the standard is not circular. Therefore, if you feel some comments a little out of scope, don't!. They have been properly posed before and were valids at that time. – J. Manuel Jul 05 '22 at 21:19
  • ”In this definition there is no explicit use of the speed of light…” If that were the case, then what would determine “the path length traveled by light after [1 second],” if not the light’s speed? Also, you seem to define the speed of light independently of the behaviour of light, which makes one wonder how this actually defines the speed of light, as opposed to just some arbitrary speed. As I understand it, you seem to be defining $c$ in terms of the metre and the second, without reference to the speed of light, and then you define the metre in terms of the speed of light and the second. – Sandejo Jul 06 '22 at 14:55
  • @Sandejo. There is no explicit use of the speed of light because you don’t need to know its value when setting up the experiment to get a meter stick. If you lose your meter stick somewhere in alpha centaury a new meter stick could be easily remade by a simple computer routine that runs the following algorithm: “Turn on the laser and start the clock; wait until $9,192,631,770$ clock ticks; measure the distance reached by the laser light; divide that distance by $299,792,458$ and you get a meter”. All this routine includes are 2 big integers and the discription of a device. – J. Manuel Jul 07 '22 at 11:08
  • When you “measure the distance reached by the laser light,” you are assuming that the laser beam travels at a specific speed, commonly called the speed of light. Without knowing that the laser beam always travels at a specific speed (i.e. without knowing the invariance of the speed of light), this method would not work. This is where you use the speed of light to define the metre. – Sandejo Jul 07 '22 at 14:40
  • @Sandejo. I said, it doesn’t use the speed of light explicitly. By no means I said it doesn’t do it implicitly. In the second paragraph I wrote: “… by virtue of the universal constancy of the speed of light define the meter…”, therefore, all you’re saying is already there. I don’t understand what your point is. – J. Manuel Jul 07 '22 at 15:18
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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.

FTT
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    "Approximately" in this answer means "indistinguishable by any metrology equipment available at the time". – The Photon Jul 04 '22 at 15:00
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    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. – leftaroundabout Jul 05 '22 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.

Trunk
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