# A SI meter definition changed?

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#### pervect

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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 travelled by light in a vacuum in 1299 792 458 of a second. .
Does anyone know when and why the change was made?

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#### Dale

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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

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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:

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

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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

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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

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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

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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

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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

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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

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

#### PeterDonis

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 $N_{\text{A}}=6,022 140 76 \cdot 10^{23}/\text{mol}$.

"SI meter definition changed?"

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