# What is the meaning of a fundamental constant? Dimensioned or dimenionless

• nonequilibrium
In summary, the significance of constants is that they define how humans use units. The two dimensionless constants have universal relevance, while the other constants only have meaning in relation to humans.
nonequilibrium
So I was thinking about the question "What if Boltzmann's constant $$k$$ were different?" but then I got thinking about the nature of the question.

What is the significance of a constant? Can you say one dimensioned constant is more fundamental than the other one? For example, I can imagine someone saying "$$c$$ is more fundamental than the gas constant $$R$$" but does that have any meaning other than a personal liking for the speed of light?

"What if I change $$k$$?" Is that a well-defined question? Should I specify "If everything else stays the same"? And is that certainly non-contradictory? (For example how do I know if other constants are maybe defined using $$k$$?)

And it is often said that the real fundamental constants are the dimensionless ones. Why is this? I remember reading a quote that: any change in a dimensioned quantity is unnoticeable if it not accompanied with a change in a dimensionless quantity. If a dimensionless quantity changes, then certainly something measurable changes. Now why is this? And does this make it a more fundamental constant?

Also they say one can use natural units in which, for example, $$c = 1$$. I really CAN'T understand this: How can it NOT have a unit? If I then say $$v = 1/2$$, surely that is not correct? Is it not like saying "My bag weighs 2" and then assuming somebody will add the unit kg? For example if $$c = 1$$, it would be allowed to use the mathematical statement $$e^c = 1$$, but if I were to use a non-natural unit system, I would not be allowed to do that, because then I'd have a dimensioned exponent...

Any other comments about constants are welcome.

One last one: is there a fundamental difference in asking "What if $$k$$ had a different value?" and "What if $$k$$ could differ in different parts of our universe?" ?

In my view constants, fundamental or otherwise, are neither more nor less fundamental than dimensionless numbers - they are different.

Dimensionless numbers allow something not available with constants - they allow comparison of different systems.

The dimensionful constants are simply artifacts of our chice of units. The dimensionless constants are the ones that define physics. They are the ones which are independent of our choice of units.

Hm, I don't seem to get the difference. Aren't dimensionless constants artifacts of our choice of numeric system?

$$e^{i \pi} + 1 = 0$$

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mr. vodka said:
Hm, I don't seem to get the difference. Aren't dimensionless constants artifacts of our choice of numeric system?

No, but their numeric values are. That's why they often retain symbol form.

the ratio of a circle's circumference to it's diameter is pi. It's not absolutely 3.14(etc...) that's just how we represent it in the base 10 number system.

As far as I understand, natural units were created to removed constants. For example, Einstein's E=mc^2 is simplified to E=m when using natural units (i.e Planck units).

mr. vodka said:
So I was thinking about the question "What if Boltzmann's constant $$k$$ were different?" but then I got thinking about the nature of the question.
The Boltzmann k has no real physical meaning. It's only used to defined the temperature unit. If you would define a temperature unit different from Kelvin, then Boltzmann k would change.

mr. vodka said:
What is the significance of a constant? Can you say one dimensioned constant is more fundamental than the other one? For example, I can imagine someone saying "$$c$$ is more fundamental than the gas constant $$R$$" but does that have any meaning other than a personal liking for the speed of light?
c doesn't have any real meaning either. It basically defines the unit of length.

Only dimensionless constants have a real meaning. All others only define what humans use for units.
In fact there are only two dimensionless number in classical physics which have some univeral relevance. One is the fine structure constant and the other a ratio for the gravitational constant.
But even those dimensionless constants might not be univeral. Recently there have been reports (again) that these dimensionless constants vary in the universe. But still there are the only ones with meaning, rather than being an arbitrary definition of mankind physical units.

mr. vodka said:
Also they say one can use natural units in which, for example, $$c = 1$$. I really CAN'T understand this: How can it NOT have a unit?
You use different unit for length so that c comes out the numerical value 1. Then you drop the units which are the same in all equations anyway. Strictly speaking there still should be a unit. Especially if you don't want to lose the ability to make a check for unit consistency.

mr. vodka said:
One last one: is there a fundamental difference in asking "What if $$k$$ had a different value?" and "What if $$k$$ could differ in different parts of our universe?" ?
Surely Aliens *don't* use the same idea of assign temperature a scale such that water has the special temperatures 273 Kelvin and 373 Kelvin for melting and boiling. In such a case they'd get a different k.

In fact there are only two dimensionless number in classical physics which have some univeral relevance.

Perhaps you'd care to expand on this statement?

I have a wall chart stuffed full of dimensionless numbers.

Which ones do you mean? Most of them are either arbitrary and related to the choice of human physical units, or they are related to a process which is derivable from more fundamental concepts.

For example, if I decide to have a scale different from Kelvin, then the numerical value of the Boltzman constant would change. It's really just arbitrary.

In classical physics there are only two constant. Namely the fine structure constants and the gravitational constant.

If you add particle physics you get bunch of constant more, but maybe one day these will be derived too:
http://math.ucr.edu/home/baez/constants.html

There are 154 dimensionless numbers on this chart - mainly related to fluid mechanics.

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OK, but if it's related to fluid mechanics then is goes into the category "constants derivable from fundamental principles". Probably if you ever manage to solve Navier-Stokes, then you can find these constants. So it's rather a mathematical constant, like Feigenbaum's constant. It's not something that is "encoded into physical reality independently".

Perhaps you'd like to produce something more concrete than your say-so?

Do you actually read my posts? You should at least make an attempt to understand them.
Or if you want to read some authority's opinion then read John Baez.

I also recommend the Baez FAQ entry.

You can of course have an infinite number of dimensionless constants (α, α/2, α², sin(α), etc.). There are a much smaller number that are considered "fundamental". I think it would be difficult to make a case for any of the fluid-mechanics dimensionless numbers being "fundamental". Generally that label is reserved for the dimensionless constants that govern the simplest interactions we can study.

Studiot said:
Dimensionless numbers allow something not available with constants - they allow comparison of different systems.
That is a good point: if I say "say c has a different value", it is not as clear as "say $$\frac{c}{\overline{T_E}}$$ changes" with the numerator being the average period of the Earth around the sun: not coincidentally, this is a dimensionless "constant". Okay the numerator is not an actual constant, but it's the best I could find for a conceptual example.

Gerenuk said:
The Boltzmann k has no real physical meaning. It's only used to defined the temperature unit. If you would define a temperature unit different from Kelvin, then Boltzmann k would change.

c doesn't have any real meaning either. It basically defines the unit of length.

Only dimensionless constants have a real meaning. All others only define what humans use for units.
In fact there are only two dimensionless number in classical physics which have some univeral relevance. One is the fine structure constant and the other a ratio for the gravitational constant.
But even those dimensionless constants might not be univeral. Recently there have been reports (again) that these dimensionless constants vary in the universe. But still there are the only ones with meaning, rather than being an arbitrary definition of mankind physical units.You use different unit for length so that c comes out the numerical value 1. Then you drop the units which are the same in all equations anyway. Strictly speaking there still should be a unit. Especially if you don't want to lose the ability to make a check for unit consistency.Surely Aliens *don't* use the same idea of assign temperature a scale such that water has the special temperatures 273 Kelvin and 373 Kelvin for melting and boiling. In such a case they'd get a different k.
Hm, of course the numerical value for our dimensioned constants depend on our choice of units, but when I mean "different k" surely I don't simply mean a different set of units. I've heard "k doesn't have a physical meaning, it's due to the definition of temperature" a lot of times, but I don't get its point. For example, as you say, the speed of light is also due to our defnition of meter and second (okay, in the new system it's the other way around, but that's not really important for this discussion), but the fact that in a certain system c has THAT value means something? Hypothetically, if I just look at a beam of light passing, it has THAT exact speed, its speed doesn't become something trivial, but is predetermined by nature, somehow. Why can't k be the same way? If k has to do with how we define temperature, namely using the trippelpoint of water, then k has a physical meaning having to do with the physical system of water in such an environment? I wonder if this in essence means one could derive k theoretically by modelling a system of water.

Two other remarks:
- Oh okay, so the units in a natural unit system are simply implicit? That solves that problem for me

- Which two dimensionless ones do you mean? I don't recognize the second one with a ratio of G with something else.

EDIT: I suppose a clearer example is changing the fundamental constant $$\frac{c_{''here''}}{c_{''there''}} = 1$$. This seems like a non-arbitrary well-defined question, once "here" and "there" are specified.

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mr. vodka said:
Hm, of course the numerical value for our dimensioned constants depend on our choice of units, but when I mean "different k" surely I don't simply mean a different set of units. I've heard "k doesn't have a physical meaning, it's due to the definition of temperature" a lot of times, but I don't get its point.
Imagine you are the first human and you haven't defined temperature units or measured Boltzmann constant yet. Now you can arbitrarily define one of them and the other will follow.
Just historically Kelvin was defined first as to make the water transition points 273 and 373. The Boltzmann constant follows from using thermometers defined this way.

mr. vodka said:
For example, as you say, the speed of light is also due to our defnition of meter and second (okay, in the new system it's the other way around, but that's not really important for this discussion), but the fact that in a certain system c has THAT value means something?
Imagine you have no rulers. How would you proceed to gauge one? The only way is to
*define* the speed of light and use a fixed reference for seconds to create a unit for length.
Or what else?
Again historically the meter was defined to be a special length which was passed from generation to generation. At one point they noticed that this is arbitrary and instead they fixed the value of the speed of light determined with the current meter rule.

mr. vodka said:
- Which two dimensionless ones do you mean? I don't recognize the second one with a ratio of G with something else.
I once thought a lot about all kinds of natural units. Basically you can set many physical constants to have a numerical value one, just by picking the right units. However there is one constant that you cannot circumvent: all dimensionless combinations cannot be changed. One is the fine structure constant. The other is the gravitational constant divided by some other constants as to give a dimensionless number. I forgot which units you need for it, but maybe you can find out yourself :) [haven't got my notes here]
Candidates are the dielectric constant, speed of light, charge of electron, mass of electron and so on (maybe that's even enough).

Hm, the main thing I'm having trouble with is that it seems that you're saying that dimensioned constants are meaningless... Well okay they're meaningless on their own, if you don't know what the units are, but the process of defining units is choosing things in nature as to give meaning to the later derived constants. For example, go back to the old definition of meter and second, the ones not using c; I'll call them symbolically the "stick" and "drop" respectively. Okay once you had that, you could 'compare' the velocity of light with these units. You saw that in one drop, light passed about 300 million sticks. Okay if you had chosen another stick, you would've gotten another number, but it's precisely because the amount of sticks light passes depends on the definition of a stick, that light gets a physical significance: the fact the numerical value changes can be seen as nature nullifying your arbitrary choice of a stick. So I can't really follow when you say "c doesn't have any real meaning". Are there then parts of what I've said in this post that you don't agree with?

Maybe there is nothing I don't agree with.
I'll put in other words: There is no way an alien civilization could deduce our numeric value for the speed of light, because it depends on an arbitrary choice of meter.
However they could determine the fine structure constant, which is supposed to be universal (apart from recent claims)

And I fully agree with your last post. But then saying "k does not have a physical significance" seems unjustified, doesn't it? I think you'd agree the speed of light (not the numerical value) has a physica significance, so why shouldn't k? After all the reason you're using to call k that is also applicable to c, so if k does not have a physical significance, that can't be the reason.

It has the same significance as a cup for a volume measure. It's arbitrary, but you do need at least some reference, so you use the cup.
It's a matter of view I guess.

Do you actually read my posts? You should at least make an attempt to understand them.
Or if you want to read some authority's opinion then read John Baez.

Yes I did indeed read your posts - and responded politely.

Furthermore when asked for further information I supplied it.

In turn I asked you for further information and received what appears to be a rude response.

I did not seek an opinion I asked for a chain of sold reasoning to back up statements, apparently plucked out of the air.

In particular you have stated that there are only 2 fundamental dimensionless numbers (I do not call them constants, which have a different definition in my dictionary).

Therefore I ask for a derivation of the

Knudsen Number from these two numbers.

I hold that this cannot be done, but will applaud if you demonstrate.

Just out of curiousity: what would you call a constant?

And a bit more on topic: I was wondering, I think everybody agrees that if you change a dimensionless constant that there would be a noticeable difference (at least in principle), but why must this be a physical constant? For example, if the ratio $$\frac{\textrm{distance that light travels in one second}}{\textrm{length of my forearm}}$$ changed, surely that would be observable too?

Is it because the physics that makes up the length of my forearm can be traced down to physical constants, making it more a practical clarity issue to demand the ratio be one of physical constants. Or is it more fundamental why I can't use my forearm?

@Studiot: I wrote down well-defined reasoning. What's really rude is when someone is too lazy to spend a minute thinking about what people write and instead continuously nags it's just an opinion.
My first statement was "If you would define a temperature unit different from Kelvin, then Boltzmann k would change." and you'd do well searching for the next.
As I don't get the impression that you even try to understand what I wrote about derived constants, I feel no need to explain the difference to you again. The Knudsen Number is derived.

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@Vodka: True, in the relation between speeds and lengths doesn't change. If there was something like a fixed length of universal forearms, then the ratio between lengths could be called a constant. I think there isn't such a length. But you may call the ratios of particle masses a constant maybe. So you are right. If you manage to find to related measures, then they are some kind of constant.
But are there any others than masses?

You clearly failed at your own hurdle not reading another's posts.

1) Nowhere in this thread have I discussed Kelvin and Boltzman.

2) Mr Vodka appears the only one who has noticed a distinction I drew about dimensionless numbers (such as Reynolds No) which is that they allows comparison of different systems not different systems of measurement as for instance setting c = 1 does. This importance of this facet is huge.

3) I have not seen, or do not know how, to derive Knudsens number from your two 'fundamental numbers - nor its seems can you.

I apologise to Mr Vodka that his very excellent question is being so debased when I tried to contribute to the discussion, and turned the other cheek several times to your jibes.

Studiot said:
1) Nowhere in this thread have I discussed Kelvin and Boltzman.
That's exactly your problem! I mention Kelvin and Boltzman and you have go by skipping this fact and calling out I haven't explained any facts.

Studiot said:
3) I have not seen, or do not know how, to derive Knudsens number from your two 'fundamental numbers - nor its seems can you.
Then think again. People can't even derive the most basic quantum mechanical problems, yet quantum mechanics is known to perfectly explain these problems.
Just because no-one has a computer powerful enough to derive all your number, doesn't mean they are not derivable.

Studiot said:
I apologise to Mr Vodka that his very excellent question is being so debased when I tried to contribute to the discussion, and turned the other cheek several times to your jibes.
Maybe you call "Perhaps you'd like to produce something more concrete than your say-so?" one of your cognitive highlights, where you contribute to the discussion. You admitted yourself that you didn't even think about Boltzmann when I wrote about it.

Gerenuk, enough of this nonsense. I will play no longer.

Mr Vodka.

Consider the ratio of two lengths. inches/millimetres. This is not a dimensionless number and can be rewritten with a true constant in the form

millimetres = 25.4 x inches

Here 25.4 is a true constant.

This constant allows us to compare measurement systems of length.

Other more exotic constants are always the same (non variable) when put into an equation. But that is the nature of a constant.

Now consider the ratio of inertial forces/viscous forces in a fluid. Measuring both forces in the same units results in the dimensionless Reynolds Number.

This is not a constant but can vary over a very large range.

However certain specific values of this number are critical in that they denote the change from laminar to turbulent flow.

Thus at a Re of 1400 an airflow will change from laminar to turbulent, as will a water flow, as will a flow of granular material such as sand or salt.

This variable dimensionless number allows us to compare (aspects of) different physical systems and make predictions about their behaviour.

Before this bickering gets any worse I recommend a brief visit to the following pages:

http://en.wikipedia.org/wiki/Dimensionless_physical_constant
http://math.ucr.edu/home/baez/constants.html

From there, go where you will.

PS: I guess what is implicit in this is that constants relevant to many-body phenomena can be calculated (given "enough" computational power and time) from more fundamental things. There is, of course, the debate of reductionism versus emergence buried in some of that, but for the purpose of this thread, I think one can ignore that.

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Thank you for the links, Gokul. I did google J Baez but did not find these.

I also looked again at the title of this thread and realize that I was too hasty when reading it.

Mr Vodka only referred to constants, with or without dimensions, so my introducing variable dimensionless numbers was a bit off thread. It's just that the context I normally use dimensionless quantities is as I described.

Sorry for any confusion this has caused.

mr. vodka said:
And I fully agree with your last post. But then saying "k does not have a physical significance" seems unjustified, doesn't it? I think you'd agree the speed of light (not the numerical value) has a physica significance, so why shouldn't k? After all the reason you're using to call k that is also applicable to c, so if k does not have a physical significance, that can't be the reason.
k and c are "in the same boat". They both have the same physical significance, i.e. they tell you something about your choice of units. Because of the fact that through judicious choice of units you can make a dimensionful constant have any value there is simply no additional information in the constant. It is only when you make dimensionless comparisons that you start to gain any information about physics besides the choice of units.

For example, we might say that c is very fast. That is certainly the case if we are using anthropomorphic units like a second which is about the duration of a heartbeat and a meter which is about the height of a child. By c being large in such units we mean that in one heartbeat a pulse of light would go past many children laid head to foot. On the other hand we might use stellar units and say that c is very slow, e.g. if our unit of distance is some average distance between galactic groups and our unit of time is some average lifetime of a star. It all depends on our choice of units.

Here are a couple of links from a while back where I explored the idea of the physical significance of dimensionful vs. dimensionless numbers:
https://www.physicsforums.com/showpost.php?p=2011753&postcount=55
https://www.physicsforums.com/showpost.php?p=2015734&postcount=68

Thank you for the replies.

Hm DaleSpam interesting two posts!

A few questions: (if you don't have time for them all, then I'm also happy if you could just look at question 3

1) "So, basically we have just c doubling and the permittivity and permeability halving and no other changes." Well I suppose there could be more changes, right? Would it in principle suffice to check each of the dimensionless fundamental constants for a change?

2) So let's for a moment define meter as the length of a certain rod and a second in a similar way (earth's orbit or something). Now let $$L_c$$ be the distance light travels in one second, expressed in meter. Now define $$\beta = \frac{L_c}{[L_c]}$$, then beta is a dimensionless number! It's equal to the amount of rods light passes in a certain fraction of Earth's orbit. What if say $$\beta' = 2 \beta$$? Surely this is an observational difference. So does that imply there must be a change in one of the 26(I think) fundamental dimensionless constants? I suppose it does, right? Because if it doesn't, then there wouldn't be an observational difference, although there certainly is. How do you know what dimensionless constants changed?

3) If I want to focus on changing k, does it mean that I have to find a dimensionless constant with k in it, so I know what has to change for it to be observable? The weird thing is, k has Kelvin as units (I mean, it has that unit in its unit), so there probably won't be any dimensionless constant for it! Unless N_A (Avog. number) = R/k can be seen as a dimensionless constant, but Baez didn't count it as one, and it might just be a definition? And if one argues "well then maybe k can't be involved in any observational difference", but surely that is wrong if we see PV = kNT => k = PV/NT. If I don't change the definitions involving the right hand side, then a change in k most definitely will be observational?

mr. vodka said:
2) So let's for a moment define meter as the length of a certain rod and a second in a similar way (earth's orbit or something). Now let $$L_c$$ be the distance light travels in one second, expressed in meter. Now define $$\beta = \frac{L_c}{[L_c]}$$, then beta is a dimensionless number! It's equal to the amount of rods light passes in a certain fraction of Earth's orbit. What if say $$\beta' = 2 \beta$$? Surely this is an observational difference. So does that imply there must be a change in one of the 26(I think) fundamental dimensionless constants? I suppose it does, right? Because if it doesn't, then there wouldn't be an observational difference, although there certainly is. How do you know what dimensionless constants changed?
A similar reasoning works for particle masses. However I think you cannot find a universal length to compare. Because there is no distinct special rod that you might describe to an alien "over the telephone". And also the orbit of a planet is derivable from just knowing particle positions and their arbitrary velocities.
So you're right for masses, but I guess one can't find an equivalent example for lengths.

I don't see why this is an issue of universality... If I keep my rod, go to bed and overnight beta changes, then the next morning I can see the change by comparing my rod with L_c. I'm not interested in knowing if beta changed for aliens too.

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