E=mc^2 using three unit systems

[69dodge]

A couple of people have missed my point that there should be no numerical conversion constants within a system.

You would expect that in a scientific/engineering system of units the force required to accelerate 1 <primary mass unit> by 1 <primary length unit> per <primary time unit>2 would be 1 <primary force unit>. Unfortunately that is not the case in FPS; instead, 1 <primary force unit> is the force required to accelerate 1 <primary mass unit> by g, the acceleration due to gravity (and the same word is used for the primary force unit and the primary mass unit).

Numerical conversion constants within a system is a completely different issue from whether some natural constant or quantity has a value of 1 in the system. That's of negligible benefit (in fact it's better if there aren't any such quantities, because it encourages both sloppy thinking and the practice of using different systems for different purposes; and in any case the value won't stay at exactly 1 as measurements or definitions of units are refined).

I don't understand the distinction you're making here between a "numerical conversion constant" and a "natural constant or quantity".

Let me ask you this: Is there any fundamental reason to pick a system of units in which Newton's second law of motion contains no extra constant (F = ma, instead of F = kma) but Newton's law of gravitation does contain one (F = G m₁m₂ / r², instead of F = m₁m₂ / r²)? Inertia and gravitation are essentially the same thing, according to our current best theory (general relativity).

You say there's no reason for natural constants or quantities to have the value 1, but when you're used to a system of units in which some quantity does have the value 1, it's easy to overlook the fact that any constant is there at all which could conceivably have a different value. 'Of course, F should equal ma,' your embedded quote above basically says, 'F = kma would be silly.' But is the k there any sillier than the G in F = G m₁m₂ / r² ?

And see, for example, this discussion of general relativity, where the author finds it convenient to use units in which c = 8 pi G = 1, because then he can write Einstein's equation without any constants. (That's the same G as in Newton's law of gravitation.)

 

[Lucky]

I hope someone's still interested in this topic (I'm a slow poster).

Standardization is helpful, but what sorts of problems are you going to make the forced standard unit suited for? And what do you do when you are going things that the units are not well suited for?

That depends on whether we're talking about everyday weights and measures or science and engineering. Specialised systems claim to be 'convenient' in two ways: (1) they are designed to give some physical quantities or constants a numerical value of 1, and (2) units are of the right order of size, so numbers are not too large or small.

(1) is obviously irrelevant for everyday purposes, and it shouldn't trouble scientists and engineers to include an extra number or two in their calculations. (2) is important for everyday purposes – we don't want to talk about giga or nano units of potatoes, beer etc. – but again shouldn't matter to scientists or engineers. My view is that the advantages of using a single system for all purposes greatly outweigh any small gain in convenience to scientists from number size and a few constants of 1. Choice of base unit should be to suit commerce rather than science - scientists can happily cope with powers of 10. Obviously there has to be compromise for different purposes, and power-of-10 prefixes will be required (the gram is a poor choice of base unit because it's too small for almost all everyday measures).

Most physics textbooks (outside freshman texts) are in the Gaussian system. Some of them are 'bilingual' or only recently starting to adopt the SI. The CGS system is just as sensible as the MKS (on which the SI is based). Most fundamental physicists use CGS (with electrostatic or electromagnetic units, depending on the situation), because it is much more convenient.

I have no means of knowing how true this is (it obviously depends on what country you're talking about), but it's mostly a matter of cultural inertia. There's no absolute sense in which cgs is more convenient than SI, unless you're specifically talking about electromagnetism. But here several rival 'convenient' systems (and hybrid ones) are commonly used, even for the same purpose.

I don't understand the distinction you're making here between a "numerical conversion constant" and a "natural constant or quantity".

Let me ask you this: Is there any fundamental reason to pick a system of units in which Newton's second law of motion contains no extra constant (F = ma, instead of F = kma) but Newton's law of gravitation does contain one (F = G m₁m₂ / r², instead of F = m₁m₂ / r²)? Inertia and gravitation are essentially the same thing, according to our current best theory (general relativity)

You say there's no reason for natural constants or quantities to have the value 1, but when you're used to a system of units in which some quantity does have the value 1, it's easy to overlook the fact that any constant is there at all which could conceivably have a different value.

But F = kma would have no additional meaning. Because of the definitions of mass and force, k is dimensionless, so it would mean exactly the same as F = ma in any well-behaved system of units. (By 'well-behaved' I mean that there are no dimensionless conversion constants between any base or derived units.) All physical quantities and constants necessarily have dimensions, so cannot equal (the number) 1. They can be hidden in 'natural' systems only by disregarding the units, and therefore the quantities, you are working with (that's what I meant by encouraging sloppy thinking).

For instance, when dealing with gravitation it aids understanding to be aware of G, and that it appears to be a physical property of spacetime, that we don't know for sure how 'constant' it actually is, and whether it would have different values in different universes. Physical properties of the universe should be illuminated, not obscured, by the system of units (I concede that SI is not entirely successful here).

I can see why theoretical physicists find Planck units useful for some purposes, but there are several problems:
(1) They can't be defined exactly (I don't know what this would mean e.g. for instrument calibration).
(2) They couldn't be used to define a practical system, as they are the wrong dimensions and the wrong size (as well as being imprecise).
(3) They obviously preclude any theory in which one or more of them isn't constant.

Actually it might be interesting to investigate a physics in which, say, velocity, energy, gravitation (the dimensions of G) and permittivity are fundamental dimensions (or perhaps even one in which some fundamental constants are dimensionless), but that's not the one we have.

In my web trawling to remind myself what Gaussian units are (I have never used them) I found someone who agrees with me.

What is more important to us: that our equations should look pretty, or that the our fundamental units should be sensible? I think that sensible units are of vital importance, especially if we are going to make quantitative calculations (we are!), whereas the prettiness or otherwise of our equations is of marginal concern.

Also this one who discusses the difficulties in an amusing way:

philosophically minded physicists want only the three base quantities of the Gaussian system to better reflect the underlying science. They dream of a visit to Woolworths where they enquire of the sales assistant
"Excuse me, have you any 600,000,000 ergs per second light bulbs?"
"Why, certainly, sir. We sell all our electrical goods by Gaussian units because in advanced texts on electrodynamics we find that the tensor calculus equations which unify the electric and magnetic fields are just so much clearer!"
"Great, I'll take 59,958,491,600 / c of them, please."
"While you're here, may I interest you in any five millistatvolt AA cells?"

 

[Art Vandelay]

(1) is obviously irrelevant for everyday purposes,

That's not always true. For instance, since most things have a specific gravity that is close to that of water, it is useful to set the specific gravity of water to one (specific gravity unit).

(2) is important for everyday purposes – we don't want to talk about giga or nano units of potatoes, beer etc.

Why not? People deal quite well with kilograms, kilometers, and milliliters.

All physical quantities and constants necessarily have dimensions, so cannot equal (the number) 1.

Is pi a physical constant?

(1) They can't be defined exactly (I don't know what this would mean e.g. for instrument calibration).

Neither can any other unit. They are defined more precisely than metric units traditionally have been, and there is little practical use to further precision.

(2) They couldn't be used to define a practical system, as they are the wrong dimensions and the wrong size (as well as being imprecise).

Prefixes can take care of the size. And what do mean "wrong dimensions"?

(3) They obviously preclude any theory in which one or more of them isn't constant.

They are, by definition, constant.

Also this one who discusses the difficulties in an amusing way:

Why would anyone say "600,000,000 ergs" rather than "600 megaergs"? How can one possibly buy 2 s/m of light bulbs?

 

[69dodge]

Actually it might be interesting to investigate a physics in which, say, velocity, energy, gravitation (the dimensions of G) and permittivity are fundamental dimensions (or perhaps even one in which some fundamental constants are dimensionless), but that's not the one we have.

By "a physics", do you mean a world or a description of a world?

Similarly, when you say "that's not the one we have", are you talking about our world or about some particular description of our world, e.g. our current theories of physics expressed in SI units?

The same world can be described in different ways.

I can see why theoretical physicists find Planck units useful for some purposes, but there are several problems:
[...]
(3) They obviously preclude any theory in which one or more of them isn't constant.

Did you know that the meter is currently defined as the distance that light travels in a certain amount of time (1 / 299 792 458 s)?

You could say that this definition precludes any theory in which the speed of light isn't constant. But as far as we can tell, the speed of light is constant, and so this definition is fine, and for a variety of reasons it has been determined to be the most useful one.

Any system of units and dimensions has some assumptions built in. If the assumptions match the way the world really works, then everything's ok. If we discover that they don't, we can always pick a new system.

 

[Yllanes]

(1) is obviously irrelevant for everyday purposes, and it shouldn't trouble scientists and engineers to include an extra number or two in their calculations.

An extra number may be the difference between a correct calculation and one were rounding errors destroy everything. It is madness to do a computer simulation of an atom or molecule and use 4.8·10^-10 esu or 1.6·10^-19 C as the charge of an electron. Or 1.6·10^-24 g as the mass of a proton. Starting with those numbers, it is quite unlikely that a numerical calculation is going to end up OK. The sensible way to go is to use atomic units, were the potential energy is just 1/r and we measure energies in hartrees (in terms of the ionisation energy of the hydrogen atom, 13.6 eV). In his everyday life, an atomic physicist will also use atomic units. When we are dealing with atoms and molecules, using grams or kilograms doesn't make sense and the same goes with ergs or joules. You seem to think that we should use the same system for all situations, but that's clearly not a good idea. As I said, numerical calculations force us to tailor the units to fit the problem and everyday usage makes us choose convenient units.

For non-scientific purposes, of course we should use only one metric system.

(2) is important for everyday purposes – we don't want to talk about giga or nano units of potatoes, beer etc. – but again shouldn't matter to scientists or engineers. My view is that the advantages of using a single system for all purposes greatly outweigh any small gain in convenience to scientists from number size and a few constants of 1. Choice of base unit should be to suit commerce rather than science - scientists can happily cope with powers of 10. Obviously there has to be compromise for different purposes, and power-of-10 prefixes will be required (the gram is a poor choice of base unit because it's too small for almost all everyday measures)

There is no need, nor any gain, to use always the same system. Why do you say his has advantages? Scientists are quite capable to use many different systems. A physics student will use a different system in each subject and there is nothing wrong with that.

There's no absolute sense in which cgs is more convenient than SI, unless you're specifically talking about electromagnetism.

Electromagnetism is a broad subject. In solid state physics you study the magnetic and electric properties of matter, superconductivity, semiconductors, etc. Light is an electromagnetic phenomenon, so there goes optics. And anyway, as I've been saying, you can use different systems for different situations.

But F = kma would have no additional meaning. Because of the definitions of mass and force, k is dimensionless, so it would mean exactly the same as F = ma in any well-behaved system of units. (By 'well-behaved' I mean that there are no dimensionless conversion constants between any base or derived units.)

(Emphasis mine). In the atomic units I mentioned before, energy has the dimensions of length^-1. This doesn't mean that every well-behaved system has to make this choice.

All physical quantities and constants necessarily have dimensions, so cannot equal (the number) 1. They can be hidden in 'natural' systems only by disregarding the units, and therefore the quantities, you are working with (that's what I meant by encouraging sloppy thinking).

This is false. To continue with atomic units, we measure energy in terms of the ionisation energy of the hydrogen atom. This has the value of 1 rydberg or 0.5 hartrees (hartrees are more common). You say this is sloppy, because we are disregarding the actual value of this energy. But that's not the case. What's 1 joule? It is the work of a force of 1 N in a displacement of 1 m. Are we disregarding this quantity by giving it the value 1? The bottom line is that you always need to define something as 1 or any other 'pure number'. The sensible thing to do is to choose a quantity of the order of magnitude of the rest of quantities that you are going to manipulate.

For instance, when dealing with gravitation it aids understanding to be aware of G, and that it appears to be a physical property of spacetime, that we don't know for sure how 'constant' it actually is, and whether it would have different values in different universes.

Do you think a relativity worker doesn't know this?

Physical properties of the universe should be illuminated, not obscured, by the system of units (I concede that SI is not entirely successful here).

A proliferation of ugly constants is what obscures the formulae and makes laws seem complicated. A joule is a completely artificial unit (convenient for human scales). A hartree is a natural unit, in a similar way as a radian is a natural unit for angles.

I can see why theoretical physicists find Planck units useful for some purposes, but there are several problems:
(1) They can't be defined exactly (I don't know what this would mean e.g. for instrument calibration).

Of course they can be defined exactly. They cannot be defined exactly in terms of metric units. I mean, G = 6.6742(01)·10^-11 N·m²·kg^-2, there's an uncertainty there. But it goes both ways: 1 joule has an uncertainty measured in atomic untis. But they can be defined exactly within their system. Saying G cannot be equal to 1 is like saying that 1 N·m cannot be equal to 1 joule.

(2) They couldn't be used to define a practical system, as they are the wrong dimensions and the wrong size (as well as being imprecise).

I have answered this already. They are the right size for their respective uses. nobody is saying we should use geometrised units to weigh our potatoes.

(3) They obviously preclude any theory in which one or more of them isn't constant.

Not really, 69dodge already answered this.

Actually it might be interesting to investigate a physics in which, say, velocity, energy, gravitation (the dimensions of G) and permittivity are fundamental dimensions (or perhaps even one in which some fundamental constants are dimensionless), but that's not the one we have.

I don't understand this paragraph.

In my web trawling to remind myself what Gaussian units are (I have never used them) I found someone who agrees with me.

I disagree with that page. He seems to think that students are not capable of using more than one system of units in their life.

I am now going to give a brief technical note, following Jackson't appendix. This will illuminate the idea that you are always setting something equal to one.

A system of nis in the context of electromagnetism has to set values for three constants, k1 to k3. The first comes from Coulomb's force F = k1 q q' / r². The second one comes from Ampère's force between two wires and k3 from Faraday's law of induction. Maxwell's equations are

[latex]
\begin{align}
\vec\nabla\cdot\vec E & = 4\pi k_1 \rho\\
\vec\nabla\times\vec B &= 4\pi k_2 \alpha \vec J + \frac{k_2\alpha}{k_1}\frac{\partial \vec E}{\partial t}\\
\vec \nabla\times \vec E + k_3 \frac{\partial \vec B}{\partial t}&=0\\
\vec\nabla\cdot\vec B &=0
\end{align}
[/latex]

And we have two ligatures, k1/k2 = c^2 and alpha = 1/k3 (alpha is the electromagnetic constant), so only k1 and k3 are independent. Some systems:

[latex]
\footnotesize
\begin{array}{ccccc}
\text{System} & k_1 & k_2 & \alpha & k_3\\
\hline
\text{esu} & 1 & c^{-2} (t^2\ell^{-2}) & 1 & 1\\
\text{emu} & c^2 (\ell^2t^{-2}) & 1 & 1 & 1\\
\text{Gaussian} & 1 & c^{-2} (t^2l^{-2}) & c (\ell/t) & c^{-1} (t/\ell)\\
\text{Heaviside-Lorentz} & \frac{1}{4\pi} & \frac{1}{4\pi c^{2}} (t^2l^{-2}) & c (\ell/t) & c^{-1} (t/\ell)\\
\text{SI} & \frac{1}{4\pi\epsilon_0} (ml^3t^{-2}q^{-2}) & \frac{\mu_0}{4\pi}=10^{-7} (mlq^{-2}) & 1 & 1\\
\hline
\end{array}
[/latex]
And we are only in free space. We need to choose the macroscopic fields D, H. P and M. Let us only say that they have all the same dimensions in the Gaussian system, but different dimensions in the SI. This comes for the choice of k3. The SI sets k3 = 1, so that E and cB have he same dimensions (as can be sen in Maxwell's equations). Once this choice is made, the choices for the other 4 magnitudes are made so that the macroscopic Maxwell's equations are simple. I wanted to write this so that you can see how every system sets several things equal to 1.

 

[Art Vandelay]

(Emphasis mine). In the atomic units I mentioned before, energy has the dimensions of length^-1.

No, it doesn't. That's what he's talking about with "sloppy thinking". Just because you give the pressure in terms of inches of mercury doesn't mean that pressure actually has dimension of inches.

This is false. To continue with atomic units, we measure energy in terms of the ionisation energy of the hydrogen atom. This has the value of 1 rydberg or 0.5 hartrees (hartrees are more common). You say this is sloppy, because we are disregarding the actual value of this energy.

I believe that what he's saying is that referring to it as "1" is sloppy, not that referring to it as "1 rydberg" is sloppy.

 

[Yllanes]

No, it doesn't. That's what he's talking about with "sloppy thinking". Just because you give the pressure in terms of inches of mercury doesn't mean that pressure actually has dimension of inches.

It's not the same. You say 'mm of Hg', not just 'mm'. I mean, you could call 1 mmHg = 1 vandelay, which is not the same as saying 1 mm = 1 vandelay. You are not changing any formulae is what I mean. In atomic units, I say energy is measured in the same units as length^-1 because the potential energy of an electron is -1/r, the expressions do reflect this assertion.

Lets do it one step at a time. The speed of light is something around 3·10⁸ m/s. But that is a definition, not a measured value. Nowadays, we define the second and then we define the metre as the distance light travels in 3·10^-8 s. So time and length are the same, really. Measuring them with different units is like measuring the distance in the North-South direction in miles and the distance in the East-West direction in kilometres. If anyone is not convinced of this, I recommend the parable of the surveyors, the first chapter of Taylor & Wheeler's Spacetime Physics. So, I claim that distance and time have the same dimensions, provided we make the natural choice and set c = 1. If you disagree at this point, of course you are going to deny that length and 1/energy have the same dimensions in atomic units. If you, however, agree at this point, then you will have to accept what follows:

We have c = 1, so now E = m in Einstein's celebrated formula. We can measure masses in energy units. For example, the mass of an electron is 511 MeV.

Now we go to hbar, another fundamental constant of nature. Its value is 6.6·10^-16 eV·s which, using c as a conversion factor, is like saying hbar = 2.0 · 10^-7 eV·m, if we measure time as well as distance in metres (which is perfectly reasonable). In atomic units we go even further and set hbar = 1. This means that 1 m = 2·10⁷ eV^-1. In other words, if hbar = c = 1, then energy and 1/length have the same dimensions.

As I said before, you are always setting something = 1 (see the table for example). In Gaussian units, the electric and magnetic fields (all four of them), have the same dimensions. In the SI they do not. This is the same situation.

Another example. When an electron experiments a transition from one state to another with lower energy, it emits EM radiation of a wavelength lamdba = hc / (E1-E2). This factor is annoying if you are working all the time with atomic spectra, because you measure transition wavelengths but want to tabulate energy levels. So you get rid of it by measuring the energy in cm^-1. Then you only have to subtract two energies and take the inverse to get the wavelength. It may seem a small improvement, but it does matter if you aer doing it all the time. Experimental energy levels are usually tabulated in cm^-1 (the values in hartrees are usually theoretical values, obtained through Hartree-Fock or Density Functional theories). The standard reference is NIST. For example, energy levels of sodium (big file, 288 levels).

The same thing is done in gravitation. I am quoting from Misner, Thorne & Wheeler: (and keeping their horrible notation 'sec' for 's'):

Throughout this book, we use "geometrized units," in which the speed of light c, Newton's gravitational constant G, and Boltzman's constant k are all equal to unity. The following alternative ways to express the number 1.0 are of great value:

1.0 = с = 2.997930... · 10¹⁰ cm/sec
1.0 = G/c² = 0.7425 · 10²⁸ cm/g;
1.0 = G/c⁴ = 0.826 · 10^-49 cm/erg;
1.0 = G·k/c⁴ = 1.140 · 10^-65 cm/K;
1.0 = c²/G^1/2 = 3.48 · 10²⁴ cm/gauss^-1.

One can multiply a factor of unity, expressed in any one of these ways, into any term in any equation without affecting the validity of the equation. Thereby one can convert one's units of measure from grams to centimeters to seconds to ergs to.... For example:

Mass of sun = M = 1.989 · 10³³ g
= (1.989 · 10³³ g) · (G/c²)
= 1.477 · 10⁵ cm
= (1.989 · 10³³ g) ·(c²)
= 1.788 · 10⁵⁴erg.
The standard unit, in terms of which everything is measured in this book, is centimeters. However, occasionally conventional units are used; in such cases a subscript "conv" is sometimes, but not always, appended to the quantity measured:
M_conv= 1.989 · 10³³ g.

 

[Art Vandelay]

It's not the same. You say 'mm of Hg', not just 'mm'.

Actually, people often speak of the pressue being equal to a certain number of inches, without mentioning Hg. And even with mentioning Hg, it's not quite correct. "Hg" is not a unit. What really happening is that you're basically multiplying by the density (except that it's force per volume, rather than mass per volume) of Hg, so you have inches * (pounds/inches^3)=pounds/inches^2=pressure.

In atomic units, I say energy is measured in the same units as length^-1 because the potential energy of an electron is -1/r, the expressions do reflect this assertion.

Just because you use the same untis doesn't mean that they have the same dimensions.

So time and length are the same, really.

Not quite.

Measuring them with different units is like measuring the distance in the North-South direction in miles and the distance in the East-West direction in kilometres.

No, because there is a real metric defined in 3D. In 4D, there is a psuedo metric. If they're the same, how can "measuring" c have any meaning?

[qyote]In atomic units we go even further and set hbar = 1.[/quote]You can set it equal to three elephants and a tutu. That doesn't make it so. You can't just remove the units.

In other words, if hbar = c = 1, then energy and 1/length have the same dimensions.

Except that hbar isn't equal to 1.

When an electron experiments a transition from one state to another with lower energy,

Experiences, you mean?

 

[Yllanes]

No, because there is a real metric defined in 3D. In 4D, there is a psuedo metric. If they're the same, how can "measuring" c have any meaning?

The time coordinate doesn't behave in the same way as the space coordinates. So what? We can measure all of them in seconds, or in metres, we can add them: s^2 = x^2 - t^2, etc. The only reason why our conventional units for time and length are different is that there is no way to choose one that would be convenient for both at human levels. A length of one second is too long, a time of one metre is too short... So we use different units for the two of them. But even then I can talk of a length of one second, or one light-year, and there is no misunderstanding, the conversion factor c is used. If you always measured time in metres, there would be no need at all for c. Speeds would have no dimensions and range from 0 to 1.

Measuring c has meaning if you define your unit of length and your unit of time independently, as we did until 1984. But now we define c as a conversion factor between metres and seconds. So nowadays, changing from s to m is like changing from miles to km. You cannot measure c any more than you can refine the factor in 1 mi = 1.609344 km. You can, however, measure the speed of a light ray and decide whether it is equal to c.

You can set it equal to three elephants and a tutu. That doesn't make it so. You can't just remove the units.

Except that hbar isn't equal to 1.

In conventional units it isn't. But we can choose the units so it is equal to 1...
I don't know why this is so hard to accept, I thought the table I gave a couple of posts back illustrated how different choices of units make new constants appear and disappear.

Let me give yet another example. There are several concepts of mass: we have inertial mass (the one in F = ma) and active and pasive gravitational masses (the ones in F = G m' m'' /r^2). These are different magnitudes. The first tells us a body's resistance to changing its state of motion when a force is applied, the second how strong the gravitational field it creates is and the third how it acts under the effect of a gravitational field. There is no a priori reason why they should be related at all, but they seem to be. The equivalence principles say that all these masses are proportional. This means that we can choose our units so the constant relating them is exactly 1 and measure all of them in the same units (grams, kg, whatever). According to your logic, they have different dimensions, even though they are measured in the same units. If you don't think the different masses have different dimensions, then please explain how this is different from distance/time, E/B, or the other examples.

Experiences, you mean?

Yes, thanks. I was trying to write a long post in a broken keyboard, the frustation makes the Spanish come out sometimes...

 

[69dodge]

Just because you use the same untis doesn't mean that they have the same dimensions.

Can you give some idea of what you mean by "dimensions"? Because to my way of thinking, same units implies same dimensions. For example, inches are a unit of length, so anything that has units of inches necessarily has dimensions of length.

[qyote]In atomic units we go even further and set hbar = 1.[/quote]You can set it equal to three elephants and a tutu. That doesn't make it so. You can't just remove the units.

Except that hbar isn't equal to 1.

You seem to be ascribing more reality to the assignment of units than I think is warranted. What hbar "is" depends on the system of units one uses to describe it.

I think the problem is that you expect the dimensions that we assign to a quantity to tell you everything there is to know about the quantity. They don't, and they aren't meant to. Energy and torque have the same dimensions (length times the dimensions of force), but they aren't the same sort of thing. Radians and degrees are both dimensionless, but if I want to describe an angle, I can't just give you a pure number. I still have to tell you whether it's in radians or degrees; otherwise, you won't know how big of an angle I'm talking about.

 

[ponderingturtle]

I hope someone's still interested in this topic (I'm a slow poster).

That depends on whether we're talking about everyday weights and measures or science and engineering. Specialised systems claim to be 'convenient' in two ways: (1) they are designed to give some physical quantities or constants a numerical value of 1, and (2) units are of the right order of size, so numbers are not too large or small.

(1) is obviously irrelevant for everyday purposes, and it shouldn't trouble scientists and engineers to include an extra number or two in their calculations. (2) is important for everyday purposes – we don't want to talk about giga or nano units of potatoes, beer etc. – but again shouldn't matter to scientists or engineers.
[snip]

But F = kma would have no additional meaning. Because of the definitions of mass and force, k is dimensionless, so it would mean exactly the same as F = ma in any well-behaved system of units.

Here you are violateing rule 1, as it shouldn't bother people to have an extra number in their calculations right?


So when do those rules matter and when are they a failure of well behaved unit systems?

 

[Lucky]

Too much to answer in one go!

we don't want to talk about giga or nano units of potatoes, beer etc.

Why not? People deal quite well with kilograms, kilometers, and milliliters.
...
Prefixes can take care of the size.
...
Why would anyone say "600,000,000 ergs" rather than "600 megaergs"?

My point was that scientists and engineers can handle prefixes and explicit powers of ten without a problem, but for ordinary use people don't like very large or small prefixes any more than large or small numbers. It's interesting that prefixes beyond m and K are very uncommon in everyday measures.

Is pi a physical constant?

pi belongs in the logical (mathematical) universe, not the physical universe.

By "a physics", do you mean a world or a description of a world?

Similarly, when you say "that's not the one we have", are you talking about our world or about some particular description of our world, e.g. our current theories of physics expressed in SI units?

The same world can be described in different ways.

I meant physics as a human construct. We don't know whether our choice of mass, length, time and charge as fundamental dimensions is a matter of history or convention, an artifact of our psychology, or the reflection of a deep truth about the universe. Obviously we could choose any other four quantities (that we currently regard as derived), provided we were careful that our combination didn't exclude anything (if it turns out that some dimensions are equivalent then we won't need four). And perhaps we could have a physics in which all the fundamental constants really are dimensionless (and can be derived mathematically), but I put 'investigate' because we probably can't.

Did you know that the meter is currently defined as the distance that light travels in a certain amount of time (1 / 299 792 458 s)?

Yes, that's the current way the standard is defined, but that's a matter of convenience, and says nothing about whether c truly is constant (or whether length and time are equivalent). If it turns out that the assumption is incorrect then we will simply define a different standard, perhaps by specifying some narrow condition for the measurement, or by abandoning c.

I believe that what he's saying is that referring to it as "1" is sloppy, not that referring to it as "1 rydberg" is sloppy.

Yes, except that in the case of the fundamental constants they aren't even referred to as 1; they simply disappear. The history of physics should teach us to be wary of disregarding hidden quantities.

In the atomic units I mentioned before, energy has the dimensions of length-1.

I can talk of a length of one second, or one light-year, and there is no misunderstanding, the conversion factor c is used. If you always measured time in metres, there would be no need at all for c. Speeds would have no dimensions and range from 0 to 1.

Right, that's what I'm getting at. What precisely do you mean by these statements? Because they could mean very different things, depending on how literally you're talking. You could mean that a coherent system of dimensionally unbalanced equations will always give the correct results, because it's mathematically equivalent to having hidden constants of 1 with the appropriate units. I called this 'sloppy', but OK I can see that physicists aren't going to bother with dimensionally balanced equations all the time when there's no obvious gain.

Or you could be saying that dimensional analysis is meaningless, or that the system corresponds to a physics in which we actually need fewer fundamental dimensions, because some of them have been found to be equivalent for all purposes. If so, I disagree. True, in relativity we can regard two of the fundamental dimensions as equivalent (in a sense), but this seems not to work in all areas of physics. For instance, if ε₀ and G are dimensionless then mass and charge have the same dimensions, but we don't know what this means. It may be that we really can dispense with a dimension or so (consistently throughout a unified physics), but as I said, the system of units shouldn't constrain the physical theories.

I believe that physicists, and certainly physics teachers, sometimes use statements about the equivalence of incommensurate quantities without really considering what they mean by this. And even if you think that physicists are always crystal-clear in their minds about their meaning, you surely can't think that students are (the point about suitable units for teaching purposes has got sidelined, though I would like to discuss it when I have time).

It is madness to do a computer simulation of an atom or molecule and use 4.8·10-10 esu or 1.6·10-19 C as the charge of an electron. Or 1.6·10-24 g as the mass of a proton. Starting with those numbers, it is quite unlikely that a numerical calculation is going to end up OK. The sensible way to go is to use atomic units, were the potential energy is just 1/r and we measure energies in hartrees (in terms of the ionisation energy of the hydrogen atom, 13.6 eV).

I take your point, but even so you can give any quantity you like the numerical value 1 without saying that (for example) the dimensions of energy are length^-1.

For instance, when dealing with gravitation it aids understanding to be aware of G, and that it appears to be a physical property of spacetime, that we don't know for sure how 'constant' it actually is, and whether it would have different values in different universes.

Do you think a relativity worker doesn't know this?

Of course they 'know' it, but suppose natural units were used for all purposes, and G, c etc. never appeared as quantities in any theory or calculation, then there's a lower probability that any physicist would think about their physical significance.

Yllanes, have you considered that convenience may be largely a matter of what you're used to? You have said several times that SI isn't widely used in physics, and it's common for students to use different systems of units for different purposes. But I never used anything other than SI, including for electromagnetism (admittedly undergraduate only), and it seemed perfectly convenient to me. It never once occurred to me that physics would be easier without the 'ugly' constants.

But F = kma would have no additional meaning. Because of the definitions of mass and force, k is dimensionless, so it would mean exactly the same as F = ma in any well-behaved system of units.

Here you are violateing rule 1, as it shouldn't bother people to have an extra number in their calculations right?

So when do those rules matter and when are they a failure of well behaved unit systems?

I didn't mean them as rules. I was trying to illustrate the difference between dimensionless constants (like the 32 that was overlooked in the original problem) and physical quantities whose value can be set to 1 but should still have units. It's the dimensionless constants that a well-behaved system (by my definition) doesn't have.

 

[Yllanes]

True, in relativity we can regard two of the fundamental dimensions as equivalent (in a sense), but this seems not to work in all areas of physics. For instance, if ε₀ and G are dimensionless then mass and charge have the same dimensions, but we don't know what this means.

I know begin to understand what you mean. You have to realise that what you call 'hidden constants' are really dimensionless, we are not cheating. They are not equal to 1 N/m*C or whatever, they are equal to the pure number 1. But you cannot make everything equal to 1. I gave before the example of electromagnetism. There you have to choose three constants, satisfying some ligatures. We can make some of them equal to 1, but not all of them. For example, k1/k2 =c^2, so if we make k2 = 1, we must make k1 = c^2. Here k2 is dimensionless and k1 has dimensions of a speed squared. The same thing happens in other contexts. There is no way to make all of the constants dimensionless.

Of course they 'know' it, but suppose natural units were used for all purposes, and G, c etc. never appeared as quantities in any theory or calculation, then there's a lower probability that any physicist would think about their physical significance.

They determine the value of the natural units. The Bohr radius, a hartree, etc. are a combination of several physical quantities.

Yllanes, have you considered that convenience may be largely a matter of what you're used to? You have said several times that SI isn't widely used in physics, and it's common for students to use different systems of units for different purposes. But I never used anything other than SI, including for electromagnetism (admittedly undergraduate only), and it seemed perfectly convenient to me. It never once occurred to me that physics would be easier without the 'ugly' constants.

Fair enough. For basic electromagnetism there is no advantage to the Gaussian system. Its advantages come when studying electrodynamics, optics, or the electromagnetism of matter, for example. And in the end, Gaussian vs. SI is largely a matter of personal preference. (Think of it as using a programming language or another). But most of the advanced textbooks, papers, handbooks, etc. are in the Gaussian system so you have to know it. And this inertia is the main reason why it is still being used and will be for some time. So I was exaggerating when I said the Gaussian system is better. I prefer it, and so do a lot of people, but of course you can do everything in electrodynamics with the SI.

What really is more convenient is the use of natural units. There is one (very important) area where you must use them: computer calculations. But even in theoretical work they are the logical way to go. They are not arbitrary, but chosen according to sensible arguments. Think of it this way: there's no way a possible extraterrestrial intelligence is going to use m, kg, etc. but they are probably going to use the Bohr radius, the hartree, or some simple multiples of them because they are defined by physics, not by humanity. They express simple relations between the relevant physical constants. But for things occuring at human scales (like engineering), the natural units make no sense: they are tiny. So you use different systems of units for different situations.

I didn't mean them as rules. I was trying to illustrate the difference between dimensionless constants (like the 32 that was overlooked in the original problem) and physical quantities whose value can be set to 1 but should still have units. It's the dimensionless constants that a well-behaved system (by my definition) doesn't have.

Again, these constants are dimensionless, they shouldn't have units. If you measure time and distance in metres, the speed of light is 1 m/m = 1.

 

[ponderingturtle]

I didn't mean them as rules. I was trying to illustrate the difference between dimensionless constants (like the 32 that was overlooked in the original problem) and physical quantities whose value can be set to 1 but should still have units. It's the dimensionless constants that a well-behaved system (by my definition) doesn't have.

The 32 in the orrigional post was not a dimensionless constant, it was g(or mabey G) the acceleration do to gravity. That was because of a mistake in what the actual units are and how they are defined.

And as you did not even follow the sugestions you set up, why sould anyone? Who in every day life calculated accelerations who is not an engeneer?

The point is that sceintists do not subscribe to an absolute universal set of units for a number of very good reasons. So all you ever need to use is the units that are conveniet to the problem at hand and if you need to introduce new units, well that is fine as well.