A When and Why Was the Definition of the Meter Changed?

[pervect]

The wording of the definition of the meter has apparently changed recently. I'm wondering about the motivation for the change. The current definition is:

The meter, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s-1, where the second is defined in terms of ΔνCs.

The older definition (1983, I think) was

The metre is defined as the length of the path traveled by light in a vacuum in 1299 792 458 of a second. .

Does anyone know when and why the change was made?

 

[Dale]

I'm wondering about the motivation for the change.

The motivation was to standardize all of the definitions. Now, they all have the same structure.

 

[PeterDonis]

Does anyone know when and why the change was made?

The "when" is simple: it was made as part of a major change in the definitions of a number of SI base units, the overall purpose being to fix the value of Planck's constant in order to remove the need to define the kilogram in terms of a physical artifact:

https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units#Impetus_for_change
The "why" for the particular change in the wording of the meter definition is less clear, but it looks like it was probably to make the wording clearer and more rigorous, and to make the wording of all the unit definitions more standardized. The wording of the SI definition of the second was also changed, even though the physical meaning of that definition, like that of the meter, remained the same.

 

[Orodruin]

the overall purpose being to fix the value of Planck's constant in order to remove the need to define the kilogram in terms of a physical artifact:

This was not the only problematic definition. There was also the definition of the temperature unit (remedied by fixing the Boltzmann constant) and the archaic definition of the ampere (remedied by defining the elementary charge). We also stopped using carbon to define amount of substance and fixed the Avogadro number. (Although personally I think amount of substance is a silly dimension to introduce ...)

 

[PeterDonis]

personally I think amount of substance is a silly dimension to introduce

I think this unit is mainly for chemists, since it's a lot more convenient for them to give things like reaction enthalpies per mole instead of per molecule.

 

[Dale]

I think this unit is mainly for chemists, since it's a lot more convenient for them to give things like reaction enthalpies per mole instead of per molecule.

Yes, but why make it dimensionful? They could have made it dimensionless, like the radian. To me it would have made more sense to make radians dimensionful and moles dimensionless.

 

[PeterDonis]

why make it dimensionful?

Because "number of elementary entities" is a dimension. It's not a dimensionless number like the fine structure constant.

To me it would have made more sense to make radians dimensionful

I agree that having radians be dimensionless is confusing, particularly when you start talking about angular frequencies: does a inverse second mean a radian per second or a cycle per second?

 

[Orodruin]

Because "number of elementary entities" is a dimension.

I agree with Dale. It is just a number and as such should be dimensionless. That does not mean that you could not define a unit for it. It is not a number per volume or anything else similar, it is just a number, like $10^5$ is a number. To me it would make more sense to not give numbers physical dimension.

 

[Dale]

I am guessing that this was the decision of a committee and, given the variety of opinion between us 3, it was probably not a unanimous decision.

 

[Orodruin]

I am willing to listen to counter arguments, bit I think I will be hard to convince ...

 

[PeterDonis]

it is just a number, like $10^5$ is a number

Not quite, because $10^5$, as a number, is not a number of anything. "Number of elementary entities" is a number of a specific kind of thing. Just as "number of meters", "number of seconds", etc. are numbers of specific kinds of things.

 

[Orodruin]

Not quite, because $10^5$, as a number, is not a number of anything. "Number of elementary entities" is a number of a specific kind of thing. Just as "number of meters", "number of seconds", etc. are numbers of specific kinds of things.

Still not convinced. Do you also want a separate physical dimension for oranges?

 

[PeterDonis]

Do you also want a separate physical dimension for oranges?

If such a thing were useful enough to qualify for SI units, sure. Systems of units are chosen for human convenience, not because they're built into the laws of physics. Even if one uses "natural" units that set as many physical constants as possible equal to $1$, one still has to choose at least one unit based on the convenience of the humans using the system of units.

Consider radians, which @Dale brought up. Should radians be dimensionless? @Dale thinks not, and gave a good reason for why not (which I agreed with). But the SI committee says they are. Who is "right"? There is no unique answer. It's a choice for human convenience.

 

[Orodruin]

If such a thing were useful enough to qualify for SI units, sure. Systems of units are chosen for human convenience, not because they're built into the laws of physics.

I am not arguing that mol should not exist. I am arguing that I think it should not have physical dimension.

 

[PeterDonis]

I am not arguing that mol should not exist. I am arguing that I think it should not have physical dimension.

How would you change the SI definition of the mol to accomplish this?

 

[Orodruin]

How would you change the SI definition of the mol to accomplish this?

I would just introduce it as a unit for dimensionless numbers. There is nothing wrong with having different units for the same type of physical quantity. For example, we do not measure atomic radii in meters. The mol would just be an auxiliary unit for dimensionless numbers, meaning $N_A$ would be a conversion constant (like the speed of light in natural units) $N_A = 1 = 6.02214076\cdot 10^{23}/{\rm mol}$ (so essentially the unit mol would represent the number $6.02214076\cdot 10^{23}$).

 

[PeterDonis]

The mol would just be an auxiliary unit for dimensionless numbers, meaning $N_A$ would be a conversion constant (like the speed of light in natural units) $N_A = 1 = 6.02214076\cdot 10^{23}/{\rm mol}$ (so essentially the unit mol would represent the number $6.02214076\cdot 10^{23}$).

Isn't this equivalent to what the SI definition of the mol says? It refers to "elementary entities", but that just means "things that are being counted". Counting numbers are dimensionless, aren't they?

 

[Orodruin]

Isn't this equivalent to what the SI definition of the mol says? It refers to "elementary entities", but that just means "things that are being counted". Counting numbers are dimensionless, aren't they?

In the SI system, the unit mol has physical dimension different from 1 (typically denoted $\mathsf N$). This is what I am arguing against.

 

[PeterDonis]

In the SI system, the unit mol has physical dimension different from 1

How is that reflected in the SI definition?

Previously, I was interpreting "number of elementary entities" as denoting a dimension. Is that what you are referring to?

 

[Dale]

The mol would just be an auxiliary unit for dimensionless numbers, meaning NA would be a conversion constant (like the speed of light in natural units)

Or treated like % which is a symbol officially accepted for use with the SI but is not a unit and is simply defined as a number.

 

[Orodruin]

How is that reflected in the SI definition?

"By convention physical quantities are organized in a system of dimensions. Each of the seven base quantities used in the SI is regarded as having its own dimension, which is symbolically represented by a single sans serif roman capital letter."

 

[PeterDonis]

Each of the seven base quantities used in the SI is regarded as having its own dimension

Ah, ok. So mol would have to no longer be an SI base unit if it were to be considered as a label for a dimensionless number.

 

[Orodruin]

Ah, ok. So mol would have to no longer be an SI base unit if it were to be considered as a label for a dimensionless number.

Indeed. What also bugs me is this text about "dimensionless quantities"

Another class of dimensionless quantities are numbers that represent a count, such as a number of molecules, degeneracy (number of energy levels), and partition function in statistical thermodynamics (number of thermally accessible states).

So, "number of molecules" is a dimensionless number, but "amount of substance" is not.

 

[pervect]

Thanks, everyone. I found a wiki discussion of some of the issues at <<link>>. I'd known that people were working on redefining the kilogram to get rid of the artifact kilogram, but I didn't know that they'd finally done it. I was surprised that they revisited the definition of the meter as a consequence, but I can see some of the logic, all of the defintions are very similar now.

I think the new defintions may be a bit more confusing to students at the introductory level, eing a bit more abstract. I suppose we'll see.

 

[vanhees71]

At the introductory level it's indeed very hard if not impossible to introduce students to the new SI. The SI's purpose is not to provide didactically feasible and simple definitions of the units but to provide as accurate standards as possible given the contemporary technology of metrology.

To achieve this accurateness, however, in as a technology-independent way as possible, one uses what's to the best of our knowledge of today are fundamental constants to define system of units. These constants are Plancks constant ("action quantum") $h$ and the speed of light in vacuo, $c$, and the charge of an electron, $-e$.

Now one needs one more constant to build up the system of units. The natural choice would be the Newtonian gravity constant $G$, but that's the bete noire among the natural constants that cannot be accurately measured today. That's why there's still one material-dependent constant left, and that's $\Delta \nu_{\text{Cs}}$, i.e., the frequency of the groundstate hyperfine transition of Cs-133, defining the base unit second since 1967 by setting its value to 9 192 631 770 Hz, where Hz=1/s is the unit of frequency. Based on this everything else follows with the constants stated above: The speed of light is fixed to 299 792 458 m/s defining the base unit m based on the base unit of time, s. The kg then is defined via Planck's constant which since 2019 set to $6.626 070 15  \cdot 10^{–34} \text{J} \cdot \text{s}$ via the use of the already defined units m and s given that $1 \text{J}=1 \text{kg} \cdot \text{m}^2/\text{s}^2$. Setting the elementary charge to $1.602 176 634  \cdot 10^{–19} \text{C}$ defines, again under reference to the above defined s, to the base unit Ampere for the electric current given that 1 C=1 As. For the temperature unit, K, one needs to fix another constant, the Boltzmann constant $k_{\text{B}}=1.380 649 ⋅ 10^{-23} \text{J}/\text{K}$. Finally, now also the Avogadro number, defining the unit 1 mol of a substance as the number $N_{\text{A}}=6,022 140 76 \cdot 10^{23}/\text{mol}$.

 

[Herman Trivilino]

I think the new defintions may be a bit more confusing to students at the introductory level, eing a bit more abstract. I suppose we'll see.

Yes. It'll be interesting to see how it's presented in the newer introductory college-level texts. And the response in the literature.

 

[Orodruin]

Yes. It'll be interesting to see how it's presented in the newer introductory college-level texts. And the response in the literature.

The old definition is still equivalent so I see no reason why you could not first present the new definition first and then give something like the old definition as clarification:

"The meter, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s^-1, where the second is defined in terms of Δν_Cs. This means that the metre is the length of the path traveled by light in a vacuum in 299 792 458 of a second."

 

[HallsofIvy]

The "radian measure" of an angle is defined as the length of the arc cut off by that angle with vertex at the center of a circle of radius r, divided by r. The numerator and denominator are both linear measurements with the same linear units. The ratio is dimensionless. That is why radians are (and should be) dimensionless.

 

[PAllen]

Still not convinced. Do you also want a separate physical dimension for oranges?

Silly fact: at current worldwide annual production, it would take a bit more than a million times the age of the universe to produce a mole of oranges.

 

[DrDu]

(Although personally I think amount of substance is a silly dimension to introduce ...)

As a chemist, I always wonder why we need the Coulomb. You can express it easily via the Faraday constant in terms of the mole :-)

 

[Dale]

The "radian measure" of an angle is defined as the length of the arc cut off by that angle with vertex at the center of a circle of radius r, divided by r. The numerator and denominator are both linear measurements with the same linear units. The ratio is dimensionless. That is why radians are (and should be) dimensionless.

This is the SI convention, but it is not the only convention possible. The angle in some system of units is not necessarily equal to the ratio, but in general it is proportional to the ratio: $\theta = k \frac{s}{r}$. For radians k is a dimensionless 1 making radians dimensionless, but for degrees $k=180^{\circ}/\pi$ which could be considered dimensionful if degrees were given a dimension. Hence the dimensionality of angles is a convention. SI could have chosen k to be a dimensionful 1 rather than a dimensionless 1.

 

[Vanadium 50]

, it would take a bit more than a million times the age of the universe to produce a mole of oranges.

But in 2008-2009 Zimbabwe was able to produce a mole of Zimbabwe dollars overnight.

 

[HallsofIvy]

"Degrees"? What in the world are "degrees"? Every one knows that radians are the only way to measure angles!

 

[pervect]

"Degrees"? What in the world are "degrees"? Every one knows that radians are the only way to measure angles!

That's somewhat amusing - I am assuming that it was intended to be, of course.

But, being serious, I do have to say that people do use degrees. We can generalize this by saying that a statement that starts with "Every one knows" is generally false, usually there is someone that doesn't know something, and/or is willing to argue some particular point.

Exactly what to replace a statement about "what everyone knows" with is somewhat unclear.

 

[Orodruin]

Exactly what to replace a statement about "what everyone knows" with is somewhat unclear.

Oh come on! Everybody knows that!

 

[maline]

Once we are listing our pet peeves with the SI, here is mine: I think it's awful that Avogadro's number is now an arbitrarily chosen constant. This number, or rather its reciprocal, once represented an important physical quantity: the mass of a baryon in grams. Of course both "baryon" and "gram" require further specification, but the choice of Carbon-12 selects the baryons in a satisfactory way, and we had just gotten around to a solid definition of the gram. So Avogadro's number should be an experimental fact, not open to definition by fiat.
If they would have gone the other way and set a number for $N_A$ while keeping the Carbon-12 standard, thereby defining the gram and kilogram, I would be happy with that too. But fixing both the kilogram and the mole numerically removes the physical meaning of the mole/amu/Avogadro number, and so makes the system more arbitrary rather than less.

Of course, I am also offended that they used $h$ rather than $\hbar$ for the kilogram definition. Can you imagine, $\hbar$ is now an irrational number!

 

[Dale]

This number, or rather its reciprocal, once represented an important physical quantity: the mass of a baryon in grams.

We can now write the mass of a baryon directly in the new kilogram standard. Why does Avogadro’s number need to be tied to the mass of a baryon?

 

[maline]

We can now write the mass of a baryon directly in the new kilogram standard. Why does Avogadro’s number need to be tied to the mass of a baryon?

Of course it doesn't need to be. But it pains me that something that once had physical meaning has been redefined as a mere convention.
The mole is not just "some big number that we divide our quantities by to make them manageable". It is the conversion factor between amu and grams, and amu is/was (a more rigorous form of) "number of baryons". The new definitions lose contact with that structure.

 

[Orodruin]

Of course, I am also offended that they used $h$ rather than $\hbar$ for the kilogram definition. Can you imagine, $\hbar$ is now an irrational number!

In SI base units, yes. In reasonable units $\hbar = 1$.

 

[DrDu]

Once we are listing our pet peeves with the SI, here is mine: I think it's awful that Avogadro's number is now an arbitrarily chosen constant. This number, or rather its reciprocal, once represented an important physical quantity: the mass of a baryon in grams. Of course both "baryon" and "gram" require further specification, but the choice of Carbon-12 selects the baryons in a satisfactory way, and we had just gotten around to a solid definition of the gram. So Avogadro's number should be an experimental fact, not open to definition by fiat.

The mass difference between different nucleons (not to speak of baryons in general) and even between the mass for the same nucleon in different nuclei is far greater than the deviations of the true mass of N_A C-12 atoms from 12g. Hence, I don't see a problem here.

 

[maline]

The SI didn't knowingly change the values of any of the units, so I expect that the mass of 1 mol of Carbon-12 is still exactly 12g, to within current measurement accuracy. What bothers me is that this fact no longer play any definitional role.
I think that as much as possible, units should be values with specific physical relevance. Of course we are limited by the need to keep fixed the values currently in use, so we are forced to use large, ugly multiples of the physical values. The mole was the one case where the old value actually did have significance, and they went and stuck in a big ugly number anyway!

 

[Dale]

I think that as much as possible, units should be values with specific physical relevance.

As far as I know none of the SI units satisfy that criterion. I think only natural units would.

 

[maline]

As far as I know none of the SI units satisfy that criterion. I think only natural units would.

They don't now, but they were originally intended to. The metre was $10^{-7}$ times the length of a curve from the Earth's equator to its north pole. The gram was the mass of a cubic centimeter of water at standard atmospheric pressure and freezing temperature. And the (older) second, of course, was $\frac 1{24\times 60\times 60}$ of the Earth's mean solar day.
The ideal of choosing units based on Nature is what gave us the SI in the first place. Unfortunately the old definitions failed, due to the values involved not being truly fixed nor easy to measure, and the newer definitions were constrained to be equal to the old ones in value. if we were creating new units today, we would probably use natural units times powers of ten, and perhaps the Cesium hyperfine transition frequency times a power of ten. We certainly would not use numbers like 299,792,458!

 

[Orodruin]

They don't now, but they were originally intended to. The metre was $10^{-7}$ times the length of a curve from the Earth's equator to its north pole. The gram was the mass of a cubic centimeter of water at standard atmospheric pressure and freezing temperature. And the (older) second, of course, was $\frac 1{24\times 60\times 60}$ of the Earth's mean solar day.

Honestly, those were horrible definitions as they relied on arbitrary artefacts and resulted in units that were not very well defined.

 

[maline]

Honestly, those were horrible definitions as they relied on arbitrary artefacts and resulted in units that were not very well defined.

Of course we know they didn't work well, and perhaps people should have foreseen that. But the motive was to make the units as non-arbitrary as possible, and I think that's still an admirable ideal.
And yes, nowadays our perspective is so broad that we think of the planet Earth as an "arbitrary artifact". So much the better!

 

[cmb]

To argue that the 'Mole' is not dimensionless is like arguing that the number 1 is not dimensionless, because you have to have one of something?

Errr... no, not really.

A Mole is dimensionless, whereas a mole of [something] has the dimension [something].

 

[Herman Trivilino]

The mole is not just "some big number that we divide our quantities by to make them manageable". It is the conversion factor between amu and grams, and amu is/was (a more rigorous form of) "number of baryons". The new definitions lose contact with that structure.

No, they don't. All they do is make the conversion factor exact.

If you had an apparatus that you used to measure the conversion factor you would continue to use the same apparatus in the same way. It's just that the apparatus now calibrates rather than measures. There's nothing less physical about that.

 

[Dale]

They don't now, but they were originally intended to. ...

You and I have very different opinions on what constitutes a physically meaningful quantity. To me all of those quantities you have identified as being physically meaningful are not, while the fundamental constants of nature are physically meaningful.

I mean, the mass of a cubic centimeter of water is only physically meaningful to me if I am weighing a volume of water. Planck’s constant is physically meaningful then, but it is also physically meaningful if I am measuring other things besides a volume of water.

 

[maline]

I mean, the mass of a cubic centimeter of water is only physically meaningful to me if I am weighing a volume of water.

I think the idea was to define the gram in terms of the centimeter, with the conversion factor being the most "natural" density available. Pure water was seen as the archetypical 'measurable substance".

To me all of those quantities you have identified as being physically meaningful are not, while the fundamental constants of nature are physically meaningful.

I probably don't disagree with you on most of those judgements. The difference is the difference in perspective between the eighteenth and twenty-first centuries. Things like the details of our planet, or the freezing point of water, were once seen as primal and indispensable elements of Reality. Nowadays we know a bit more about with things are truly fundamental, so the old Tremendously Important Facts have become contingent bits of trivia.
My point is the ideal that I think they were aiming for with these definitions: to describe our quantities relative to fundamental aspects of Nature, with a minimum of arbitrary choice. I admire that goal, and I think that the old definition of Avogadro's number was the last piece of the SI to still exemplify that, without the ugly numbers.

 

[Orodruin]

without the ugly numbers.

The entire point of the "ugly" numbers is to ensure that all of the archaic definitions hold to measurement accuracy (or at least very close to it). As such, those "ugly" numbers appear as a relic of the old definitions.

The main point of the definitions is to make the units as well defined as possible, thus referring to measurements with as little measurement uncertainty as possible (and also not subject to changes over time as artefacts are prone to).