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

1. Sep 19, 2010

### 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...

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?" ?

2. Sep 19, 2010

### Studiot

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.

3. Sep 19, 2010

### Staff: Mentor

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.

4. Sep 19, 2010

### nonequilibrium

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

5. Sep 20, 2010

### Pythagorean

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

Last edited: Sep 20, 2010
6. Sep 20, 2010

### Pythagorean

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.

7. Sep 20, 2010

### Sakha

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).

8. Sep 20, 2010

### Gerenuk

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.

9. Sep 20, 2010

### Studiot

Perhaps you'd care to expand on this statement?

I have a wall chart stuffed full of dimensionless numbers.

10. Sep 20, 2010

### Gerenuk

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

11. Sep 20, 2010

### Studiot

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

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12. Sep 20, 2010

### Gerenuk

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".

13. Sep 20, 2010

### Studiot

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

14. Sep 20, 2010

### Gerenuk

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.

15. Sep 20, 2010

### Staff: Mentor

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.

16. Sep 20, 2010

### nonequilibrium

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.

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.

Last edited: Sep 20, 2010
17. Sep 20, 2010

### Gerenuk

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.

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.

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).

18. Sep 20, 2010

### nonequilibrium

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?

19. Sep 20, 2010

### Gerenuk

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)

20. Sep 20, 2010

### nonequilibrium

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.