Why is length considered a fundamental (base) quantity?

[Ahsan Khan]

Hello all,

Textbooks define fundamental or base quantities as those quantities which are not expressed in terms of other quantities and they define derived quantities as those quantities which are expressed in terms of other quantities. I have the basic understanding that the choice of a set of fundamental quantities is quite arbitrary, means it will not be wrong for example to consider electric charge as fundamental quantity and then rather define current in terms of electric charge and time to be a derived quantity however in the SI system, scientists agree to take those 7 quantities as fundamental ones.

The new definition of one meter has now been defined in terms of a fundamental constant (called speed of light in vacuum c) and time. As per this new definition one meter is defined as the distance that light travels in vacuum in 1/299792458 of a second. Here, isn't one meter (length), not expressed in terms of time and speed of light c? I am not sure if scientists view speed of light c as physical quantity (speed) or, not as physical quantity, but just as a constant in nature. In case c is a physical quantity, then length is expressed in terms of two physical quantities: speed and time and if c is not taken as physical quantity even then length is expressed in terms of atleast one another quantity namely time; so shouldn't length be rather a derived quantity in this system, how can we say those 7 quantities are fundamental ones when two or three of them are expressed in terms of another?

Thanks a bunch!

Regards!

 

[PeroK]

how can we say those 7 quantities are fundamental ones when two or three of them are expressed in terms of another?

The Wikipedia page, for example, explains the Defining Constants (like $c$, $h$, $e$ etc.) and the Base Units (for time, length, mass etc.).

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

 

[Ibix]

Originally we thought of distance and time as completely distinct concepts. Einstein's relativity blurred that somewhat and also showed us that $c$ was a natural conversion factor between uniys of time and units of distance, and is also the speed at which light travels. The new unit system reflects this insight, defining as much as possible in terms of an arbitrary unit of time multiplied by a natural constant, with the value of the natural constant defined so that the other units match their pre-2018 values to a very high precision.

When doing relativity it's very common to use seconds for the unit of time and define $c=1$ so our unit of distance is the light second. That's a great idea and simplifies a lot of maths, but light seconds are a bit large for every day use - so in the SI units we define $c$ to be a rather clumsy number so that our unit of length is more hunan-sized and match pre-redefinition sizes. It would be mathematically easier to define $c=3\times 10^8\mathrm{ms^{-1}}$, but then we'd all have to buy new rulers and we'd probably lose a few airliners to old-metric/new-metric conversion errors, which is too high a price to pay for physical constants being round numbers.

 

[apostolosdt]

Hello all,
[...] I am not sure if scientists view speed of light c as physical quantity (speed) or, not as physical quantity, but just as a constant in nature. In case c is a physical quantity, then length is expressed in terms of two physical quantities: speed and time and if c is not taken as physical quantity even then length is expressed in terms of atleast one another quantity namely time; so shouldn't length be rather a derived quantity in this system, how can we say those 7 quantities are fundamental ones when two or three of them are expressed in terms of another?

Thanks a bunch!

Regards!

Interesting remark. However, defining one physical quantity as fundamental and others as derived is sort of imposing tags within a selected unit system. What really counts is the economy of one choice of fundamental units' set and the convenience of employing the latter to define the rest of the quantities. For instance, in the so-called natural system of units, we choose three physical constants as being the fundamental units, but we do not hesitate to define the two of them in terms of the other. So, by setting $c=1$ and $\hbar=1$, time is defined as distance and length as inverse mass, and believe me, that is a HUGE help in advanced physics.

Units, whatever system one chooses to work with, are kind of "rulers" for allowing us to assign numerical values to measured quantities; otherwise, what does a length of just 11.12 might mean, right? Zee goes even further (I think in his "Gravity" book) to suggest that we should replace all the important physical constants with 1! (Although, Zee's style is well known to be a bit too bizarre occasionally.)

Perhaps if you tried to think in terms of dimensions things might be clearer. Hope that helps.

 

[Dale]

I have the basic understanding that the choice of a set of fundamental quantities is quite arbitrary, means it will not be wrong for example to consider electric charge as fundamental quantity and then rather define current in terms of electric charge and time to be a derived quantity however in the SI system, scientists agree to take those 7 quantities as fundamental ones.

This really is the answer to your own question. It is simply an arbitrary convention. The fundamental quantities and their fundamental dimensions are an arbitrary convention, regardless of the definitions of the unit sizes or their practical realizations.

 

[Mister T]

The new definition of one meter has now been defined in terms of a fundamental constant (called speed of light in vacuum c) and time.

That happened in 1983.

I am not sure if scientists view speed of light c as physical quantity (speed) or, not as physical quantity, but just as a constant in nature.

It's both.

In case c is a physical quantity, then length is expressed in terms of two physical quantities: speed and time and if c is not taken as physical quantity even then length is expressed in terms of at least one another quantity namely time; so shouldn't length be rather a derived quantity in this system, how can we say those 7 quantities are fundamental ones when two or three of them are expressed in terms of another?

You are confusing the definition of quantities with the definitions of the units used to measure those quantities. I know what is meant by a fundamental unit, but I don't know what is meant by a fundamental quantity.

The meter is defined in terms of the second (a unit) and the speed of light (a quantity). There is no reason to conclude from that that the meter can't be taken as a fundamental unit.

 

[Ahsan Khan]

Originally we thought of distance and time as completely distinct concepts. Einstein's relativity blurred that somewhat and also showed us that $c$ was a natural conversion factor between uniys of time and units of distance, and is also the speed at which light travels.

Ok so as of now distance and time are not completely distinct concepts. For a given value of time there is definite distance that light travels means if we just assign a unit for time, the unit of distance pops up itself and c is the conversation factor. I think it's like a deeper understanding that distances are travelled in time, so the former is inseparable with the later, correct me if I am wrong. Somewhat that we arrived about heat and work that a work of about 4.18 joule causes a transfer of 1 calorie heat. We see work and heat as related from then onwards and consider them as one physical quantity merging the two; so physics textbooks should now start taking distance and time as one physical quantity relating them only a by conversion factor c. Why are they keeping them as separate quantities, adding an unnecessary quantity to the list. They should list the 6 quantities as fundamental and should remove either distance or time from the list of 7 fundamental quantities. Seeing both distance and time in the list of fundamental quantities violates the very definition of fundamental quantities anyway. The definition of fundamental quantities says quantities which cannot be expressed in terms of other quantities.
Regards!

 

[PeroK]

They should list the 6 quantities as fundamental and should remove either distance or time from the list of 7 fundamental quantities.

I doubt they'll do as you demand. How are you going to start teaching physics without distance or time? It's absurd.

 

[Ahsan Khan]

For instance, in the so-called natural system of units, we choose three physical constants as being the fundamental units, but we do not hesitate to define the two of them in terms of the other.

This is too shocking for me. What about definition of fundamental quantities then? What is then the meaning of classifying physical quantities into two categories of fundamental and derived when just like a derived quantity and fundamental quantity can also be expressed in terms of other physical quantities. We should in such case discard the idea of fundamental quantities and abolish the term fundamental quantity from textbooks and physics curriculum altogether and announce that there are no such things as fundamental quantities, every quantity can be expressed in terms of other hence all quantities are derived only. Help me where I am missing!

Regards

 

[Ibix]

Help me where I am missing!

Fundamental physics appears to lie in dimensionless quantities like the fine structure constant. Not having any units, these are the same whatever unit system you use.

All dimensionful quantities have values that depend on some part of your choice of unit system. These quantities can be fundamental in the sense of appearing in equations however you construct them, although their actual values depend on the unit convention.

 

[Ahsan Khan]

I doubt they'll do as you demand. How are you going to start teaching physics without distance or time? It's absurd.

I am not demanding it but to me it seems more absurd to first define something and then go against your own definition. For example if I say only those 7 colours are to be put in a box A which are found in a rainbow and any colour which is formed by mixing any combination of more than one colours of rainbow are to be put in box B and then I formed a cyan color by combination of blue and green and ordered it to be put in box A. Won't that be a violation of my own rule?

 

[anorlunda]

Help me where I am missing!

I think you're placing too much significance to the word fundamental.

 

[Vanadium 50]

You are confusing the definition of quantities with the definitions of the units used to measure those quantities.

This.

 

[Ahsan Khan]

Fundamental physics appears to lie in dimensionless quantities like the fine structure constant.

Can you tell me some examples of what you call fine structure constant (dimensionless quantities).

 

[apostolosdt]

There are no fundamental (physical) quantities. I'm afraid that's a misconception echoing the naming of some base units as fundamental, and the rest as derived. That distinction cannot be extended to the physical quantities measured by those units. Everyone of us occasionally loosely refers to "fundamental quantities" when one actually means "fundamental units" in some chosen system.

Since this thread is about both quantities and their units, perhaps it's better to limit the term "fundamental" to the units to avoid confusion and wrong conclusions.

Also, the equivalence of distance and time is a relativistic concept; it does not apply to Newtonian mechanics, but that's well-known to the OP I guess.

 

[Ahsan Khan]

This.

Do you mean that we definite fundamental quantities completely arbitrary but we still have the authority to connect the unit of our one of the defined fundamental quantities with our another defined fundamental quantity?

 

[Dale]

Why are they keeping them as separate quantities, adding an unnecessary quantity to the list.

You can read the BIPM minutes about why they made that choice. It is an arbitrary choice so they discussed it and as a committee came to a decision. They wrote their minutes so that interested people could see why.

They should list the 6 quantities as fundamental and should remove either distance or time from the list of 7 fundamental quantities.

In principle you could have just 1 base dimension as is done with Geometrized Units. In that system the only base dimension is length and all other dimensions are derived from length. By the way, I would not use the term "fundamental", I would just call them "base" dimensions.

What about definition of fundamental quantities then?

I get the impression that you may be overemphasizing the word "fundamental" here. In the SI brochure the BIPM never refers to fundamental dimensions or fundamental units. It refers only to the constants as fundamental. The units and the dimensions are referred to as base units and base dimensions.

You may want to read the actual text directly, it seems like you may be overreacting to a slightly misworded presentation of the concepts: https://www.bipm.org/en/publications/si-brochure

 

[PeroK]

Also, the equivalence of distance and time is a relativistic concept; it does not apply to Newtonian mechanics,

Nor to non-relativistic QM. What's to become of the Schrodinger equation?

 

[PeroK]

In principle you could have just 1 base dimension as is done with Geometrized Units. In that system the only base dimension is length and all other dimensions are derived from length.

That's no use for anything but GR. That wouldn't do for the entirety of physics that the SI units must support.

 

[Ahsan Khan]

There are no fundamental (physical) quantities. I'm afraid that's a misconception echoing the naming of some base units as fundamental, and the rest as derived. That distinction cannot be extended to the physical quantities measured by those units. Everyone of us occasionally loosely refers to "fundamental quantities" when one actually means "fundamental units" in some chosen system.

Since this thread is about both quantities and their units, perhaps it's better to limit the term "fundamental" to the units to avoid confusion and wrong conclusions.

Also, the equivalence of distance and time is a relativistic concept; it does not apply to Newtonian mechanics, but that's well-known to the OP I guess.

I like (digest) what you said in the later part of this post. Also your saying there are no fundamental quantities is now making some sense to me but I have in one the textbooks which is considered quite authentic they use terms like fundamental quantities and fundamental units in similar sense, I am sharing the screenshot of the page of the book.

 

[Dale]

That's no use for anything but GR. That wouldn't do for the entirety of physics that the SI units must support.

Agreed. It is a reductum ad absurdum argument. If you insist on removing any redundant dimension then you will wind up with only one dimension. So insisting on removing any redundant dimension is a bad strategy to begin with.

 

[Dale]

There are no fundamental (physical) quantities. ...

Since this thread is about both quantities and their units, perhaps it's better to limit the term "fundamental" to the units to avoid confusion and wrong conclusions.

That is exactly the opposite of the BIPM's usage, which I would consider authoritative.

 

[Ahsan Khan]

You can read the BIPM minutes about why they made that choice. It is an arbitrary choice so they discussed it and as a committee came to a decision. They wrote their minutes so that interested people could see why.

Ok it would be interesting, I hope it will give a good insight to my quest.

In principle you could have just 1 base dimension as is done with Geometrized Units. In that system the only base dimension is length and all other dimensions are derived from length. By the way, I would not use the term "fundamental", I would just call them "base" dimensions.

With just length as dimension(physical quantity) I can form dimensions of mass, electric charge, temperature?

I get the impression that you may be overemphasizing the word "fundamental" here. In the SI brochure the BIPM never refers to fundamental dimensions or fundamental units. It refers only to the constants as fundamental. The units and the dimensions are referred to as base units and base dimensions.

You are right but many of textbooks made me think so.

 

[Dale]

With just length as dimension(physical quantity) I can form dimensions of mass, electric charge, temperature?

In standard Geometrized Units as linked above mass has dimensions of L and electric charge also has dimensions of L. The standard Geometrized Units doesn't have temperature, but you could extend it to include temperature by setting $k_B=1$.

 

[Ahsan Khan]

In standard Geometrized Units as linked above mass has dimensions of L and electric charge also has dimensions of L. The standard Geometrized Units doesn't have temperature, but you could extend it to include temperature by setting $k_B=1$.

Ok so it is only the units which had been arbitrary categorized, the quantities (dimensions) are actually all not so fundamental; there can be too much dependence of certain dimensions on another depending on the system of unit chosen the way time can be written in terms of length(with c as conversion factor) that is time can have dimension of L and as you told in Geometrized units mass and electric charge can have dimension of L as well.

It means I was in a false belief that the seven base quantities should in no way be inter-related and hence their(those 7 base quantities) units should never be written in terms of other units. However it turns out the dimensions of even base quantities are related and can be written in terms of other dimensions. So a classification of base quantity is just a way of constructing a particular system of units, they are chosen arbitrary and they (the base quantities)are not supposed to be unrelated to be a base quantity. As many people here pointed out I am confusing fundamental units for fundamental quantities. Though there are fundamental units there is no such thing is fundamental quantities.

It means the rule that one can't be expressed in terms of another is true for units (like meter etc) which are fundamental and not for the quantities.

Just because length and time are base quantities it doesn't mean they can't be expressed in terms of one another, right?

Having said that, I further want to clarify how would you define fundamental units as they say.

Can a fundamental unit(not talking about fundamental quantity this time) be also allowed to express in terms of other fundamental unit just like a base quantity.

Meter is a fundamental unit and second is also a fundamental unit, one meter has been defined in terms of seconds.

This now making me think that even fundamental units can be expressed in terms of other fundamental units. So I am again confused that why then they say it in the definition that fundamental units are those which can't be expressed in terms of other units? Till now I was thinking that the rule that one can't be expressed in terms of other has failed for quantities and I was ok with it because I am being told quantities are not fundamental at first place but now I see even units which were fundamental can be expressed in terms of other units, so it appears even units are not fundamental.

I want to know what is your stand on this statements:

(1) Fundamental units are those which can't be expressed in terms of other units.

(2) Base quantities are those which can't be expressed in terms of other quantities.

Thanks a lot for your continues help!

 

[apostolosdt]

Nor to non-relativistic QM. What's to become of the Schrodinger equation?

Correct! Thanks!

 

[Mister T]

Ok so it is only the units which had been arbitrary categorized, the quantities (dimensions) are actually all not so fundamental; there can be too much dependence of certain dimensions on another depending on the system of unit chosen the way time can be written in terms of length(with c as conversion factor) that is time can have dimension of L and as you told in Geometrized units mass and electric charge can have dimension of L as well.

You are still mixing up quantities and the units used to measure them. The units we've been talking about in this thread are also dimensions. (Most of the units we use in physics are dimensions, an exception being the radian, which is a unit but not a dimension.)

It means I was in a false belief that the seven base quantities should in no way be inter-related and hence their(those 7 base quantities) units should never be written in terms of other units.

They are seven base dimensions. They are NOT quantities.

 

[vanhees71]

Metrology is a really interesting topic. Of course for science and technology it's mandatory to have a compatible and complete system of units, and to a certain extent it is an arbitrary choice, how to set up such a system of units.

First of all, it is clear that the choice of a system of units depends on our best current knowledge of the physical laws, and this is today the general theory of relativity as far as space and time measurements are concerned and quantum theory. So the choice of a set of "base units" for a sufficient set of "base quantities" is based on the presently most comprehensive theories about the fundamental laws of Nature.

Second, however, one also should have as precise "a realization" of the units, i.e., concrete measurement procedures to measure the corresponding quantities in the theoretically defined units.

Since 2019 we are very close to the ideal definition of a set of base units in terms of fundamental constants of nature. Fundamental constants are those constants that are considered to take everywhere and at every time fixed values as soon as the base units are chosen, and thus in this way you can everywhere and at any time precisely "realize" these base units and therefore also any "derived units" you need.

 

[apostolosdt]

Metrology is a really interesting topic. [...]

Onnes liked to emphasize that through his motto "Door meten tot weten" which means sth. like "By measuring to knowing". (Dutch members can correct me, of course.)

 

[Dale]

I want to know what is your stand on this statements:

(1) Fundamental units are those which can't be expressed in terms of other units.

(2) Base quantities are those which can't be expressed in terms of other quantities.

Again, for the official reference on the SI see the brochure pdf here:
https://www.bipm.org/en/publications/si-brochure

or the nice html version here:
https://www.nist.gov/pml/special-publication-330

On p 129 of the .pdf they state:

Prior to the definitions adopted in 2018, the SI was defined through seven base units from
which the derived units were constructed as products of powers of the base units. Defining
the SI by fixing the numerical values of seven defining constants has the effect that this
distinction is, in principle, not needed, since all units, base as well as derived units, may be
constructed directly from the defining constants. Nevertheless, the concept of base and
derived units is maintained because it is useful and historically well established

So basically they indicate that the whole concept is outdated and is essentially just kept for historical consistency and usefulness.

So, for example, on p 131 the meter which is a base unit is essentially defined as $$1 \mathrm{\ m} = \frac{9192631770}{299792458} \frac{c}{\Delta \nu_{Cs}}$$
So its status as a base unit is nothing more than a convention with historical roots and has nothing whatsoever to do with the idea of some inability to define or express the meter in terms of other quantities.

In other words, the base units is simply a defined set of units that are called "base units".

 

[f95toli]

Note also that length is not unique in any way; ALL the base units (except the second) depends on time.
E.g. the Ampere can be realized by counting the number of charges that passes per second.

The reason for this reliance on time (=the realization of the second) is that this is by far the easiest quantity to measure; a regular atomic clock is accurate to 1 part on 10^15 or so; and the new optical clocks that are likley to replace the Cs fountains in a few years are at least 3 orders of magnitude better than that.
For the other units the realisations are typically at best accurate at a level of 1 part in 10^8. Hence,, time is much, much easier to measure than any other quantity and the fact that you need an accurate clock to measure e.g. electrical current is not going to be a significant source of errors.

Also, it is important to keep in mind that the SI is a practical system; having a "philosophically" satisfying system is useful unless you can also use it to calibrate instrumentation. The BIMP is ultimately controlled by a collection of governments and they are more interested in having a system that works in the real work than one that is "tidy".

Lastly, metrologists do NOT talk about "fundamental units"; it should be "base units" which is NOT the same thing. I don't think anyone would argue that there is anything "fundamental" about say luminous intensity, but it is a base unit

 

[Ahsan Khan]

They are seven base dimensions. They are NOT quantities.

Ok so these seven base dimensions can be inter-related (without violating the definition of base dimension)?

 

[Ahsan Khan]

Again, for the official reference on the SI see the brochure pdf here:
https://www.bipm.org/en/publications/si-brochure

or the nice html version here:
https://www.nist.gov/pml/special-publication-330

On p 129 of the .pdf they state:
So basically they indicate that the whole concept is outdated and is essentially just kept for historical consistency and usefulness.

So, for example, on p 131 the meter which is a base unit is essentially defined as $$1 \mathrm{\ m} = \frac{9192631770}{299792458} \frac{c}{\Delta \nu_{Cs}}$$
So its status as a base unit is nothing more than a convention with historical roots and has nothing whatsoever to do with the idea of some inability to define or express the meter in terms of other quantities.

In other words, the base units is simply a defined set of units that are called "base units".

They say prior to 2018 the seven(classified) base units were defined and the rest derived units were constructed using powers(combinations) of these base units. But with time we are now after 2018 are able to construct units of ANY quantity using the fundamental constants of nature and hence as of now those seven base units have remain in principle, no more different from the other derived units but we now are still giving them tag of base units for historical reason and their usefulness. I understand this. But let's say we are in year 2006. In 2006 that seven base units were not outdated rather they were of special significance. If we were to discuss this topic in 2006, how can any justify meter and second as base units(units which can't be expressed in terms of other) while also expressing meter in terms of second? The relationship : One meter is distance that light travels in 1/299792458 second!

Regards!

 

[Mister T]

But let's say we are in year 2006. In 2006 that seven base units were not outdated rather they were of special significance. If we were to discuss this topic in 2006, how can any justify meter and second as base units(units which can't be expressed in terms of other) while also expressing meter in terms of second? The relationship : One meter is distance that light travels in 1/299792458 second!

Derived units were constructed from combining the base units. In other words, the derived units depended on the base units for their definition.

The process of changing this arrangement so that the distinction between base units and derived units is a convention actually began in 1983 when they defined the meter in terms of the speed of light.

 

[Dale]

But let's say we are in year 2006. ...

I don't know, but you can look for historical versions of the SI document. They publish everything they do very publicly. You should be able to find the documents that were extant in 2006.

I don't see the point, so I will leave that as an exercise for you.