## September 21, 2006

### Dimensional Analysis

#### Posted by John Baez

With your help, I would like to start amassing a collection of wisdom on gnarly issues in physics. Let’s start with dimensional analysis. I thought I had this pretty much figured out, until Kehrli pointed out a couple of things that surprised me:

• Dimensionless constants can depend on our choice of units.
• Dimensionful constants often don’t depend on our choice of units.

With your help, I would like to start amassing a collection of wisdom on gnarly issues in physics. They’re “gnarly” not because they’re technical, but because they involve slippery concepts. Their clarification may require not so much hard calculations as patient, careful thought. So, I’m not talking about something like whether N = 8 supergravity is four-loop renormalizable. I’m talking about something like why the future is different than the past - why there’s an arrow of time.

Gnarly issues often evoke passionate arguments. I hope we can discuss them in a friendly and calm manner - perhaps agonistically, but never antagonistically.

Gnarly issues also attract digressions and crackpots. If anyone posts any comments that seem too aggressive, digressive, or nutty I’ll just delete them. What I really want are insightful comments that include references and links to relevant literature.

It’s common in physics to assign quantities “dimensions” built by multiplying powers of mass ($M$), length ($L$) and time ($T$). For example, force has dimensions $M L T^{-2}$. Keeping track of these dimensions can be a powerful tool for avoiding mistakes and even solving problems.

This raises some questions:

• What’s so special about mass, length and time? Do we have to use three dimensions? No - we often use fewer, and sometimes it’s good to use more. But is there something inherent in physics that makes this choice useful?
• What’s the special role of dimensionless quantities - those with dimensions $M^0L^0T^0$? In what sense is a dimensionless quantity like the fine structure constant more fundamental than a dimensionful one like the speed of light?

I thought I had these pretty much figured out, until Vera Kehrli pointed out two things that surprised me:

• Dimensionless constants often depend on our choice of units.
• Dimensionful constants often don’t depend on our choice of units

For example, the speed of light is

$c = 299,792,458 m/s$

Here a meter, $m$, has dimension $L$. A second, $s$, has dimension $T$. The speed of light, $c$, has dimensions $L T^{-1}$. To make the dimensions match, it follows that the number $299,792,458$ must be dimensionless.

Now suppose someone comes and changes our units. Say they redefine the meter to be twice as long as it had been. Then $m$ doubles and the number $299,792,458$ gets halved, keeping $c$ the same. So we see:

• In a certain sense the dimensionless constant $299,792,458$ depends on our choice of units. Of course this number is what it is, regardless of our units. But if we say $c = C m/s$ then the dimensionless quantity $C$ depends on the definition of $m$ and $s$.
• The dimensionful constant $c$ does not depend on our choice of units. If we double $m$, we halve $C$, but $c$ stays the same.

All perfectly trivial - yet physicists like to run around saying the fine structure constant is more fundamental than the speed of light because it’s dimensionless and therefore doesn’t depend on our choice of units! They mean something sensible by this, but what they mean is not what they’re saying.

It’s good to compare two examples:

The fine structure constant $\alpha = e^2/4 \pi \epsilon_0 \hbar c \simeq 1/137.036$ is a dimensionless quantity built from quantities that seem very fundamental - the electron charge $-e$, the permittivity of the vacuum $\epsilon_0$, Planck’s constant $\hbar$ and the speed of light $c$. (Ultimately, Benjamin Franklin is responsible for the conventions that make the electron charge be called $-e$ instead of $e$. But that’s another story.)

The speed of light in meters per second: $C = c/(m/s) = 299,792,458$ is also dimensionless, but it’s built from quantities that seem less fundamental. $c$ seems fundamental, but $m$ and $s$ seem less so. After all, the definition of a second is “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of a caesium-133 atom at rest”. Like the speed of light, the value of this quantity is a fact about physics - but a more complicated fact. The length of the standard meter rod in Paris is an even more complicated fact, which has the disadvantage of being tied to a specific artifact! With this definition of $m$, the dimensionless quantity $C$ tells us something something funny about the universe: something about how the speed of light, the frequency of a specific kind of light emitted by caesium, and the length of the meter rod in Paris are related. It’s a bit like how $\alpha$ tells us some relationship between the electron charge, the permittivity of the vacuum, Planck’s constant and the speed of light - but it seems less “fundamental”, whatever that means.

But the definition of a meter no longer involves a rod in Paris - that’s obsolete; I mentioned it just to illustrate a point. The current definition says a meter is “1/299,792,458 times the distance light in a vacuum travels in one second”. And this makes a different point. Again the value of this quantity is a fact about physics - we could radio an alien civilization the definition of a meter, and if they knew enough physics, including the definition of a second they could build a rod the right length. But with this definition of $m$, the dimensionless quantity $C = c/(m/s)$ seems to tell us nothing about our universe!

(Actually it tells us some funny blend of information about the speed of light and the definition of $m$ and $s$.)

One might argue that $C$ is less fundamental than $\alpha$ because we could get any value of $C$ by changing our definitions of $m$ and $s$. But that can’t be the whole point, since we could also get any value of $\alpha$ by changing our definitions of $e, \epsilon_0, \hbar$ and $c$. So, there must be some other reason why $\alpha$ seems important and $C$ seems completely silly. What’s going on, exactly?

What are your most insightful thoughts on dimensional analysis? Your trickiest unsolved dilemmas? Your favorite references?

Note: I’ve summarized few insights from the following discussion in my second post on gnarly issues in physics: Dimensional Analysis and Coordinate Systems.

Posted at September 21, 2006 9:15 PM UTC

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### Re: Dimensional Analysis

I remember Eddington did a lot of fundamental thinking about dimensional analysis in his late years when he was working on his fundamental theory of physics. If I am right this theory wasn’t taken really seriously, but his thinking about dimensional analysis made sense. I will have to look it up but probably someone over here knows more about his work.

Posted by: lauret on September 21, 2006 11:06 PM | Permalink

### Re: Dimensional Analysis

The late Clive Kilmister was always interested in such matters and, for example, followed up Edddington’s work in a great deal of detail and not just historically either. Baez gives his current take on dimensional analysis in the blog. I find particularly attractive his concept of a ‘gnarly’ idea, which can help in both “top-down” and “bottom-up” approaches and probably to conceptualise ideas which are already there.
from: uv http://ttjohn.blogspot.com/

Posted by: John on September 23, 2006 11:26 AM | Permalink

### Re: Dimensional Analysis

The first thing which springs to my mind is the introductory chapter to Nigel Goldenfeld’s book on the renormalization group. (Pauses to unpack book from box.) The example Goldenfeld takes is the relation between the phase speed of shallow-water waves and the depth of the water. In the limit that the depth is small compared to the wavelength, he derives that the phase speed $c$ scales as the square root of the depth $h$. Given that the physically significant variables are $h$, the gravitational acceleration $g$ and the fluid density $\rho$, juggling the units gives

$c = \left(gh\right)^{1/2} f\left(\frac{h}{\lambda}\right),$

where $f$ is some function which dimensional analysis cannot determine. Taking the aforesaid limit gives $c \propto \sqrt{h}$. However:

Actually, this happy state of affairs is an illusion — we made a strong assumption in going from eqn. (1.4) to eqn. (1.5), namely that the limit process was regular. Usually, this can only be justified properly by considerations other than dimensional analysis. This point is by no means obscure mathematical pedantry: in fact, the cases were the regularity assumption breaks down constitute the central topic of this book.

The first homework problem in the book is to prove the Pythagorean theorem knowing that the area of a right triangle “can be expressed in terms of the hypotenuse and (e.g.) the smaller of the acute angles”. Hint: construct a single, wisely chosen line and then apply dimensional analysis. Part (b) then asks what happens when the triangle is drawn in Riemannian or Lobachevskian geometry. The second problem gives a table the blast wave radius as a function of time, $R(T)$, for the first atom bomb explosion; the task is to deduce a scaling law for $R(T)$ assuming that the shock-wave motion depends only upon the blast yield $E$ and the air density $\rho$.

• Goldenfeld, Nigel. Lectures on Phase Transitions and the Renormalization Group (Westview Press: 1992), ISBN 0-201-55409-7.
Posted by: Blake Stacey on September 22, 2006 2:10 AM | Permalink

### Re: Dimensional Analysis

Isn’t your ‘C’ not, well, fundamental? After all, 85.912384 is also a dimensionless number, but there’s nothing particularly interesting about it. How does that differ from your ‘C’? Neither show up in any theory that I know.

On the other hand, alpha shows up when one is doing QED. That seems to me to be the fundamental difference.

Posted by: Aaron Bergman on September 22, 2006 3:45 AM | Permalink

### Re: Dimensional Analysis

Just to muddy the waters —
by your reasoning, 3.1415+ is not a fundamental dimensionless constant either. It basically never shows up in theories.
6.283+, now, that’s plainly fundamental.

Posted by: Allen Knutson on September 22, 2006 4:35 AM | Permalink

### Re: Dimensional Analysis

Isn’t your ‘C’ not, well, fundamental? After all, 85.912384 is also a dimensionless number, but there’s nothing particularly interesting about it. How does that differ from your ‘C’? Neither show up in any theory that I know.

You’re right. And even if they did, my page on constants would refuse to call them “fundamental physical constants” because they are computable: like $\mathrm{ln}(2)$ or $\sqrt{163}$, we can write a computer program to crank out as many decimal places of these as we want. We need some criterion like this to avoid calling $4\pi$ a fundamental physical constant, or 3. Both these show up in a lot of physics equations - the latter being the number of dimensions of space, at least in certain theories. But for fundamental physical constants, the only way we know how to obtain an extra decimal place is by doing experiments.

Posted by: John Baez on September 22, 2006 7:57 AM | Permalink

### Re: Dimensional Analysis

Actually, it is relevant to look exactly where α appears for the first time: it is defined as the quotient between two angular momenta, namely the maximum angular momentum allowed by relativity and the minimum angular momentum allowed by quantum mechanics. It could be claimed that a dimensionless physical constant is of the kind that can be defined as the quotient of two physical quantities having the same units.

Now I think about, also π is a quotient between two lengths, and in fact the primitive notation was p/r, but people just started to say “but r is unity when measured in uniies of r”.

Posted by: Alejandro Rivero on September 22, 2006 10:59 AM | Permalink

### Re: Dimensional Analysis

Indeed, unless I am mistaken (sorry for the bad format, it’s late
and I have to wake up very early tomorrow; some other day I’ll try itex),

[alpha] = [C^2]/([F/m] [J s] [m/s])

= [C^2]/([C^2/(m J)] [J s] [m/s])

= [C^2]/([C^2 s^2/(kg m^3)] [kg m^2/s] [m/s])

([] indicates units, C=Coulomb, F=Farad, J=joule, etc)

The only fundamental unit above is the electric charge unit. The others, kg, s and m are our inventions and are not fundamental. One for instance could set a new unit of time as being the meter, so that one meter of time is the time it takes light to travel one meter. (Then c=1, dimensionless). It could be set as any unit of length one wishes in their wildest dreams.

If that is logical, what if one uses the Planck length as a new unit of time? Or more reasonably yet, choose the Planck units altogether, and you have that e = sqrt(alpha) in these units. The Planck length, etc, is a fundamental property of free space, and does not depend on any object or elementary particle arbitrarily chosen. Hence, if you use the Planck units, which are defined independently of the elementary charge, then a variation in the value of alpha would have to be considered due to a variation in the elementary charge.

Please let me know if I am wrong somewhere or whether I did not understand your question; perhaps it is more subtle than I realize.

Best regards,

Christine
PS: BTW thanks for the references, I’ll take a look at them.

Posted by: Christine Dantas on September 22, 2006 5:17 AM | Permalink

### Re: Dimensional Analysis

In a certain sense the dimensionless constant 299,792,458 depends on our choice of units. Of course this number is what it is, regardless of our units. But if we say c=Cm/s then the dimensionless quantity C depends on the definition of m and s.

The dimensionful constant c does not depend on our choice of units. If we double m, we halve C, but c stays the same.

This looks convincing at first, but I’m not quite sure if it’s right. If we define C as cs/m (as it seems that you do), then the value of C doesn’t depend on our choice of units. If we choose to measure things in feet instead of meters, then C retains its value, but c and m both “increase in value” by the same ratio, canceling out. When we change our units, we don’t change the dimensional quantity m, but just change the way we express it.

For exactly the reason you say that the dimensionful constant c doesn’t change its value when we change units, none of these constants (whether dimensionful or dimensionless) change their value when we change our units. Thus, this can’t really be the property that interests us in (traditional) dimensionless constants.

Posted by: Kenny Easwaran on September 22, 2006 6:36 AM | Permalink

### Re: Dimensional Analysis

I think part of the problem is that we are accustomed to viewing scalar quantities as numbers, when in fact they are often (multiplicative) torsors instead. A torsor can be multiplied by a number, but isn’t actually a number itself. One can multiply and divide torsors together but then one gets a torsor in a different space. The units only come in as a kind of non-canonical functor from the category of torsors (or torsor spaces) to the category of numbers (or of R).

So long as one is content to work with torsors and not numbers, there is no distinction between dimensionless or dimensionful physical quantities. There is still, however, a meaningful distinction between being dependent on units and being independent of units, but the latter can be seen simply by inspecting the definition of the quantity. For instance, the speed of light c, when viewed as a torsor, is independent of units, which is obvious from definition. However, the speed of light, divided by the unit length, times the unit speed (which in SI units would be the number 299792458) is blatantly dependent on the choice of units.

A quantity is dimensionless if its torsor can be canonically identified with a number. This has nothing to do with whether it depends on units or not. But: if the torsor quantity is independent of units, and the torsor is dimensionless, then clearly the numerical quantity associated to the torsor is also independent of units. That’s why the numerical value of (say) the fine structure constant is independent of units.

It turns out that identifying some torsors (such as the torsors for mass, length, and time) uniquely generate the identification for many other torsors too (leaving aside the role of charge, etc. for now). In particular, the multiplicative group R_+^3, which acts on the identifications between the mass torsors, length torsors, time torsors, and numbers (otherwise known as the unit mass, unit length, and unit time), then naturally acts on the identifications of all other torsors as well. The weight of that representation is the dimension of those torsors. One could of course choose other torsors to be the generators (provided they are a basis), and one would get a slightly different looking, but equivalent, notion of dimension.

Posted by: Terry on September 22, 2006 6:43 AM | Permalink

### Re: Dimensional Analysis

Ugh. As soon as I posted that, I realised that a torsor (much as I love the concept) is not quite the right term here: “one-dimensional vector space” would be more appropriate. (We also want to add quantities of the same type together, not just multiply and divide quantities of different types). And of course vector-valued physical quantities, such as velocity, also have dimension, though here another interesting thing comes in: to fully identify velocity with an element of R^3 (as we usually do) we don’t just need a unit time and a unit length scalar, but we actually need a unit time and _three_ unit length vectors, one for each coordinate direction. Units and coordinate systems are thus really part of the same thing. Actually, in order to identify spacetime (Galilean or Minkowski, it doesn’t really matter) with R^{3+1}, one actually needs a unit time four-vector and three unit length four-vectors, though we commonly call this “a time unit, a length unit, and an inertial reference frame” instead.

Posted by: Terry on September 22, 2006 6:52 AM | Permalink

### Re: Dimensional Analysis

Great posts, Terry! There are indeed many subtle distinctions that come in handy here, like “$\mathbb{R}$” versus “torsor for the additive group $\mathbb{R}$” versus “torsor for the multiplicative group $\mathbb{R}^*$” versus “1-dimensional real vector space” versus “1-dimensional affine space”. These are objects in different categories. However, the groupoid of torsors for the multiplicative group $\mathbb{R}^*$ is equivalent to the groupoid of 1-dimensional vector spaces, so your “mistake” of mixing them up in your first post was really quite negligible from a sufficiently lofty perspective.

(They are equivalent as groupoids but not as categories, since “multiplication by zero” is a morphism between vector spaces but not between torsors. We probably want to use the groupoids here - the categories would show up only if someone wanted to use a unit of length that’s zero, or infinite.)

Posted by: John Baez on September 22, 2006 7:24 AM | Permalink

### Re: Dimensional Analysis

I think part of the problem is that we are accustomed to viewing scalar quantities as numbers, when in fact they are often (multiplicative) torsors instead

I agree with this. I would go further and claim that the reason why certain quantities come to us in the form of (elements of) torsors is that these are really certain isomorphisms that happen to live in a linear $\mathrm{Hom}$-space.

Suppose we were to describe a 1-dimensional physical world, given by the 1-dimensional real vector space $V$.

We would tend to formulate our physical theories of that world in terms of working with a model of this world given by $\mathbb{R}$.

In order to compare these theories with the real world, we need to specify an isomorphism

(1)$\mathbb{R} \stackrel{\simeq}{\to} V \,.$

Fixing such an isomorphism is specifying a unit of length in $V$. And the space of all these isomorphisms happens to be an $\mathbb{R}$-torsor.

Posted by: urs on September 22, 2006 5:04 PM | Permalink
Read the post Dimensionless Constants
Weblog: Antimeta
Excerpt: Over at the n-category cafe, John Baez discusses dimensional analysis in physics, as the beginning of a "collection of wisdom on gnarly issues in physics". (It sounds like "gnarly issues" means "philosophical issues" - so philosophers of physics should...
Tracked: September 22, 2006 7:09 AM

### Re: Dimensional Analysis

As for “we could also get any value of á by changing our definitions of e,ϵ 0 ,ℏ and c”, I don’t understand exactly what is meant. If you refer to the meaning of the symbols, well, in the formula they are just shorthands. You could say “the square of the elementary charge over the product of Dirac’s constant, the surface area of a unit sphere, the dielectric constant of free space and the speed of light”. And then you can go further noticing that 1/(4pi*epsilon_0) is Coulomb’s constant, so e^2/(4pi*epsilon_0) is “the electrostatic force between two electrons over their distance squared”, and so on, until you have no longer arbitrary definitions. For example, “the ratio between the energy needed to bring two electrons from infinite distance to an arbitrarily choosen distance, and the energy of a photon whose wavelenght equals the radius of a circle whose circumference equals the distance to which the electrons were brought”.

Posted by: Army1987 on September 22, 2006 10:05 AM | Permalink

### … and numerology

related perhaps to this discussion, I was amazed about the different meanings of the word “numerology” between laymen, esoterists, and physicists. For laymen it just mean some nice trick to argue for personality archetypes in the same way that the Zodiac is used nowadays. For esoterists it is about relationships between dimensionful quantities, for instance the length of the corridor in the great pyramid when measured using a lenght unit of “greatpyramid foots”. For esoterism, of course, such units have been choosen on purpose by ancient knowledge or something so. Last, for physicists, numerology applies for relationships involving dimensionless quantities. So the attempts to get 1/137 or the assertion of the quotient between two masses (an adimensional quantity) is related to some rational value. Or all these, ahem, great findings about the Dirac large Number hypothesis, sometimes even rediscovering it.

Of course, there is also another question about numerology between pure numbers, such as “is the sequence 10, 18, 26… related to Bott periodicity”? But this is beyond dimensional analysis (pun intended here).

Posted by: Alejandro Rivero on September 22, 2006 10:52 AM | Permalink

### isomorphisms of models

I think units are data that specify isomorphisms of models (where I use model in a sense like here).

“Dimensionful constants” are quantities that do depend on a choice of representative in an isomorphism class of models.

“Dimensionless constants” are quantities that do not depend on a choice of representative in an isomorphism class of models.

Here is a simple example.

Assume for the moment that we are interested in the model of reality which says that the space we observe is a Euclidean 3-dimensional vector space

(1)$V$

containing exactly two Newtonian point masses.

Back in the past, somebody came along and figured out an isomorphic model. Namely, he drew two coplanar ellipses into $\mathbb{R}^3$ and claimed that this is the right model for the observed two point masses.

Apart from saying what kind of models we consider, we need to say what a morphism is between two models. In the present case, we would want to agree that a morphism of models is a morphism of vector spaces which respects angles.

Now, $\mathbb{R}^3$, equipped with its standard scalar product, is a 3-dimensional Euclidean vector space.

In the above sense, $\mathbb{R}$^3 is isomorphic to $V$ - but not canonically so.

A choice of isomorphism

(2)$\mathbb{R}^3 \stackrel{\simeq}{\to} V$

is precisely what we call a choice of unit of length.

Using the Euclidean scalar product on either $\mathbb{R}^3$ or on $V$, we may compute, say, the major diameter of our two ellipses. In general, the number we get for our ellipses in $\mathbb{R}^3$ and $V$ will differ. The difference (the quotient, actually) is measured by our isomorphism $\mathbb{R}^3 \stackrel{\simeq}{\to} V$, namely by our unit of length.

So the size of our ellipses is “dimensionful” (dimension of length in this case). It depends on a choice of isomorphism.

Other quantities of our model are independent of the choice of isomorphism. For instance the quotient of the major diameter of one ellipse with that of the other.

This quotient is hence a dimensionless quantity. In fact, it is the dimensionless quantity governing this model (encoding the relative mass of our two point masses).

So, in general, there is something called $R_o$, the observed reality. And there are models $M,\,M',\cdots$, of it in the sense that we have morphisms

(3)$R_o \to M \to M' \,.$

Some properties on $M$ may change if we pass to a different, yet isomorphic model $M'$. These are the dimensionful properties of $M$. Other properties are invariant in a given isomorphism class of models. These are the dimensionless properties.

Posted by: urs on September 22, 2006 12:28 PM | Permalink

### Re: Dimensional Analysis

Here’s a related question that may or may not shed some light on the issues:

When you do physics, including GR and field theory, via the mathematics of differential geometry – where and what are the units?

For a specific example: When considering the Schwarzschild solution, do the coordinates in the various manifold charts have units, including length and time? Or are the coordinates unit free, with the metric (or frame) degrees of freedom carrying the units?

It turns out, as far as I can tell, that the machinery of differential geometry works just fine with whatever units (or lack of units) one chooses to attach to manifold coordinates. It is a matter of taste where to bring them in. Personally, I like leaving the coordinates free of units, and having the frame (and thus the metric) carrying the units of time (and/or length), since these are what I think of as “physical” variables. Once you bring them in, the units get carried along and attached to all the usual suspects.

Any one else have thoughts on this?

Posted by: garrett on September 22, 2006 7:37 PM | Permalink

### Re: Dimensional Analysis

It turns out, as far as I can tell, that the machinery of differential geometry works just fine with whatever units (or lack of units) one chooses to attach to manifold coordinates. It is a matter of taste where to bring them in. […]

Any one else have thoughts on this?

I always get confused about this when I try to solve a GR problem with the help of dimensional analysis. I wind up spending so much time analysing the basic issues that I get bogged down before I reap the rewards of this work! So then I give up and act like a mathematician where everything is dimensionless… and then the next time this situation comes up, I’ve forgotten what I learned before.

So in fact, I need to start from scratch again here. Let me relive my blunderings in public - it could be educational.

When I see a coordinate like $x^i$ my instant gut feeling is to assign this units of length. But then I think “diffeomorphism invariance” and imagine a general coordinate transformation

$y^i = f^i(x^1, \dots, x^n)$

so I think “oh-oh, it’s forbidden to apply an arbitrary smooth function to a dimensionful quantity!” So then I start wanting the coordinates to be dimensionless.

Then this desire gets heightened when I remember that the metric is what takes tangent vectors and spits out lengths (actually squares of lengths). So, I want to pack units of $length^2$ into my metric tensor somehow.

But then I think: the metric tensor is an element of $S^2 T^* M$, the symmetric square of the cotangent bundle. If this has units of $length^2$, I must want cotangent vectors to have units of length.

And then I think: no, it’s not the metric tensor $g$ that has units of $length^2$, it’s when we apply this tensor to a pair of tangent vectors, say $v$ and $w$, that we get something with dimensions of $length^2$, namely $g(v,w)$.

So where do I put the units of length? Do I put two of them in $g$, or one in $v$ and one in $w$? If I do the latter, I’m saying tangent vectors have units of length. But a minute ago I was wanting cotangent vectors to have units of length! How can I be so confused? I’m supposed to know something about physics, but apparently I don’t even know if tangent vectors or cotangent vectors have units of length!

It goes on like this for a while. At some point I start wishing I were doing particle physics on good ol’ Minkowski spacetime, where everyone has settled on standard conventions for the dimensions of everything…

And then, yes, at some point I start wanting to use the vierbein formalism, also called the tetrad or frame field formalism:

Personally, I like leaving the coordinates free of units, and having the frame (and thus the metric) carrying the units of time (and/or length), since these are what I think of as “physical” variables.

Yes, this formalism lets us define a coframe to be an isomorphism

$e : TM \to \mathrm{T}$

where the internal space $\mathrm{T}$ is a vector bundle equipped with a fixed metric, say $\eta$. Then we define the metric $g$ on the tangent bundle by

$g(v,w) = \eta(e(v), e(w)) .$

This is nice because it’s a great example of everything Terry and Urs have been saying, only working with a tangent space $T_x M$ instead of a 1-dimensional vector space.

In other words, we can say the tangent vectors $v$ and $w$ are dimensionless, while the internal vectors $e(v)$ and $e(w)$ have units of length. So, the units of length are sitting in the isomorphism $e$.

But, there are probably lots of other ways to set things up. Maybe we can put the units of $length^2$ in the internal metric $\eta$.

You seem to think it’s an acceptable choice to let the coordinates have units of length. How do you apply a general coordinate transformation and get new coordinates with units of length? People claim this is forbidden, but there could be some trick…

Posted by: John Baez on September 22, 2006 9:31 PM | Permalink

### Re: Dimensional Analysis

Yay! A couple of months ago I was feeling silly to even be thinking about this question, about units, which is supposed to be the first thing learned in high school physics. It makes me much happier not to be the only one to travel this same path. And, yep, for the most part I had the same line of thought, and feelings, as you’ve amusingly described.

I do think it’s acceptable to let coordinates have units – though I prefer not to, as it gives the false impression that coordinates are physical. The coordinates, when it comes down to it, are just labels; and it doesn’t really matter whether you use “10” for your label, or “2 meters,” or “2.2 bananas.” What matters is that when you use the frame, as you’ve described, and integrate along a timelike curve in spacetime you get a proper time in time units, i.e. seconds.

Say you have a coordinate patch in which the coordinates are labeled by numbers, and you have a path parameterized by numbers. Then things work if the components of your frame 1-form, or vierbein tensor, have units of time. Physically, Einstein says there is a physical reference frame, specified by the vierbein, that is locally like nice QFT Minkowski space. Now, if you apply a general coordinate transformation to a set of new coordinates that have units, like meters or bananas, then the vierbein tensor components over that same patch in those new coordinates are going to pick up these new units so that in the end, for your path, you calculate the same physical proper time in seconds. Similarly, if you decide to parameterize your paths with something that has units, you’ll also have to include this change in your vierbein.

It does work. And people do it: even giving Schwarzchild coordinates units like meters and seconds, because it’s familiar and comforting. But these coordinates are fundamentally meaningless parameters. One observer’s second is not another observer’s second. What’s physical is the proper time along paths – all will agree on that. So I think it’s better, philosophically, not to label coordinates with units. But, on the other hand, it’s sometimes fun to put units on the coordinates and match them up with the physical times that would be experienced by some hypothetical observers, such as free-fall times for observers falling into black holes. It’s comforting to know that these other possible label choices find their way into the units of the vierbein tensor and everything works out.

I wouldn’t want to put the units in the Minkowski tensor because, err, then it wouldn’t be the Minkowski tensor. Also, it might make sense to parameterize timelike paths with seconds, but I’d rather not, because paths make sense even when there’s no vierbein around to say what a second along the path is. So, after mulling it over and following these same lines of thinking, and wondering why I couldn’t find this issue addressed in GR books, I’ve come to think that coordinates and path parameters are best left with no units, but can handle arbitrary choices of unit labels, and that physical units – time (and possibly length) – are in the vierbein tensor, along with whatever units might be needed to cancel out the choices made for coordinate units.

(If anyone feels like it, they can see the math worked out explicitly over at my wiki:
http://deferentialgeometry.org
Look at the “proper time” note)

In any case, thanks for making me feel a little less silly. For the moment.

Umm, why would letting general coordinate transformations include changes to coordinates with units be bad? I thought people did this all the time.

Posted by: garrett on September 23, 2006 3:00 AM | Permalink

### Re: Dimensional Analysis

Garrett writes:

Umm, why would letting general coordinate transformations include changes to coordinates with units be bad? I thought people did this all the time.

No, there’s a widely-agreed-upon prohibition against applying functions other than homogeneous polynomials to quantities with dimensions.

Suppose our coordinates have units of meters - or, to make blogging more fun on a Friday night, let’s follow your suggestion and use bananas. We monkeys haven’t invented meter sticks, but we’ve got lots of bananas. So, we measure the position of a rock by seeing how many bananas away it is from the origin.

Suppose the rock’s position is 2 bananas. If we apply a linear transformation like doubling, its position becomes $2 \times 2$ bananas = 4 bananas. That makes sense: it’s twice as far away now. Or, we could translate it one banana away, so its position became 2 bananas + 1 banana = 3 bananas. That’s fine too. So, affine coordinate transformations are okay, even with bananas. That’s all we need for the Galilei group and the Poincaré group - so we can handle Newtonian physics and special relativity.

But now say we want to monkey around with general relativity. We want to do general coordinate transformations!

Say we apply a nonlinear transformation like squaring. We don’t get 4 bananas - we get 4 bananas2. This is a problem: a squared banana is a unit of area, not distance.

But things get even worse if we apply a more general coordinate transformation like $x' = x + \sin x .$ What the heck is a banana plus the sine of a banana? No monkey has ever figured that out.

I thought maybe you’d figured it out. I can imagine some trick like working in the ring of formal power series

$\mathbb{C}[[banana]]$

But the usual way is to only apply fancy functions to dimensionless quantities. For example, when some guy wants you to compute $\exp(iHt)$, where the quantity being exponentiated has units of action, you dream up a constant with units of action - call it $\hbar$ - and instead compute $\exp(iHt/\hbar)$.

Posted by: John Baez on September 23, 2006 6:51 AM | Permalink

### Re: Dimensional Analysis

The One monkey came out with the GR Matrix answer: There is no banana.

OK, so maybe I shouldn’t post after midnight, but I will anyway. What I’m saying is that the units attached to coordinates don’t really mean anything. In order to calculate a distance in GR, we have to integrate along a path between the two events we want to know the distance between. (We can’t trust objects not to be wobbly.) When we do this, the path velocity vector contracts with the frame (vierbein) 1-form to give our integrand. The units in this frame exactly cancel out whatever funky units we have in our coordinate system, and in our velocity, leaving us with a nice sensible answer. If we change coordinates, even to something as nonsensical as square bananas, the vierbein tensor transforms under the coordinate transformation, absorbing the banana units, such that it still gives the exact same sensible answer when contracted with velocities and integrated.

If you choose some even weirder coordinate transformation, like x -> x + x^2, as long as it’s a smooth, well defined and one-to-one coordinate transformation over the range you’re interested in it may give some weird units in the vierbein tensor – but these should be exactly what’s needed so that the consequently weird velocities you contract with it give the same sensible answer.

Of course, I agree completely that the sensible thing to do is just not assign units to coordinates. But I still don’t see how things break if you do. Though I’d like to, so I could rule out the bananas.

Posted by: garrett on September 23, 2006 9:20 AM | Permalink

### Re: Dimensional Analysis

I guess the argument against units in coordinates is that we want to allow arbitrary coordinate transformations, and not just those that handle the units nicely. If you say that’s true, I’ll buy it.

One implication of using dimensionless coordinates is we’re going to have to write in some arbitrary dimensionfull constant into things like the Schwarzschild metric, which traditionally was written with dimensions in the coordinates. This makes the solution uglier, and is why I’ve been inclined to allow units in coordinates and parameters.

Another implication of unitless coordinates is that the frame, and the metric, components are all going to have the same units – either temporal or spatial (but not a combination). I like choosing temporal units, to give nice proper times in seconds. But there seems to be a long standing preference out there for spatial geometric units.

Also, a change in these units is going to be like a conformal transformation.

Posted by: garrett on September 24, 2006 10:58 PM | Permalink

### Re: Dimensional Analysis

I’ve got confused about what it means to “choose” units! I’ll think I’ll go back to basics and then try and say something about $x+\sin (x)$.

Hmm, suppose we start with the speed of light. Imagine we’re in the Warehouse of Standard Speeds. All around us are standard speeds. For concreteness, let’s imagine a bunch of wheels (like grindstones) revolving so that a point on the circumference of a wheel moves at the wheel’s Standard Speed. So there’s a 1 m/s wheel, a 2 m/s wheel, a 1 mph wheel, a 1 kph wheel, a 2 foot per week wheel, a 12 parasang per day wheel, etc.

“Choosing a unit” means going and standing next to a wheel.

a) The speed of light (I mean the actual speed that light actually goes at in a vacuum) does not depend on which wheel you’re standing next to.

b) The ratio of the speed of light to the Standard Speed of the wheel you’re standing next to does depend on which wheel you’re standing next to.

c) The ratio of the speed of light to the Standard Speed of the 1 m/s wheel (i.e. 1 m/s) doesn’t depend on which wheel you’re standing next to.

Well, that made things clearer in my mind, anyway.

That’s units; what about dimensions? Comparing two speeds seems easy, but suppose we want to measure speed in kilograms? We need an operational way to define the right number of kilograms.

OK, let’s try this, which works for non-relativistic speeds. The number of kilograms corresponding to a speed is that number of kilograms of water which we need to pour into a certain rectangular tank we happen to own, such that water initially pours out of a little hole in the bottom at the given speed.

Hmm, well, some scribblings on a scrap of paper reveal that this mass M is (in a useful approximation)

$A\frac{\rho v^{2}+P_{atm}}{g}$

where $A$ is the cross-sectional area of the tank, $g$ is the local acceleration of gravity, $\rho$ is the density of water, $P_{atm}$ is atmospheric pressure, and $v$ is the speed in question

So that won’t work because of

a) the offset due to atmospheric pressure

b) the fact that v is squared

But if we move the zero to get rid of the atmospheric pressure term and then take the square root, we get to correctly measure speed in $\sqrt{kg}$. Then ratios, sums and scalar multiples of speeds will all work out correctly.

That was more enlightening than I expected!

OK, so now I think I have a point to make about coordinates.

We choose a dimension for a coordinate, e.g. length, time, angle, or mass. This means we have thought of an operational way to (in principle) compare coordinate values to some physical quantity which “naturally” has the appropriate dimension (whatever that means…). Then we can pick a unit.

For example, let’s have a coordinate $x$ with dimension length and unit metres.

Suppose we now wish to go from

$x$ to $x+\sin (x)$

Taking the sin means we need a dimensionless argument, so we need to divide $x$ by a length $L$. After we’ve calculated the sin of this, we get another dimensionless number. To add this to $x$, we need to multiply by another length $R$. Implicitly, I think you used your unit (e.g. 1 metre) for both these lengths $L$ and $R$.

Well, that’s quite trite, but here is the crucial bit: in real life, trying to solve a real-life GR problem, there’s surely a good reason why you chose to give $x$ the dimension of length, and there’s surely a good reason why $x+\sin (x)$ is a natural coordinate transformation. The good reasons, whatever they may be, will presumably relate to the nature of the problem you’re trying to solve. So almost certainly, you have some natural length scales floating around in your problem!

In fact, it’s quite likely there’s just one really fundamental length scale in this particular problem. Your coordinate $x$ will have been most likely defined in terms of it in some way, and the lengths $L$ and $R$ will have natural definitions in terms of it.

If you’re trying to do completely general coordinate transformations, why would you choose units? What actual benefit would they give? There aren’t any natural scales of any physical quantity to compare them to. I think that’s the problem.

I’m not sure about vectors and covectors. It’s often convenient to give vectors the dimension of length, and covectors the dimension of reciprocal length. Then the inner product comes out dimensionless. The significance of this escapes me. :-(

Posted by: Tim Silverman on September 23, 2006 11:56 PM | Permalink

### Re: Dimensional Analysis

This is a brilliantly funny post - even in the middle of the afternoon! “No monkey has ever figured that out” - this line has made my day!

Posted by: Postgrad on September 28, 2006 1:35 PM | Permalink

### Re: Dimensional Analysis

By the way, there is maybe an instructive “deep” reason why it is such a bad idea to try to assign units of length to coordinates:

unlike isomorphisms of vector spaces, which themselves live in a linear space, coordinates are isomorphisms of smooth spaces (namely diffeomorphisms between pieces of manifolds and $\mathbb{R}^n$).

These do not live in a linear Hom-space, nor can they be in any good way be related to isomorphisms of vector spaces, in general.

Coordinates are an example of “nonlinear” isomorphisms. They are themselves “units”, in a sense - but “nonlinear units”.

Posted by: urs on September 28, 2006 6:24 PM | Permalink

### Re: Dimensional Analysis

Yes, Urs, I agree. Units in coordinates mess up diffeomorphisms.

The other reason they’re bad is because there’s no notion of length on a bare manifold – the notion of length (or duration) has to do with the metric (or frame).

So I’m very happy to exclude units from coordinates and parameters. And this should help with doing dimensional analysis correctly.

Nevertheless, there is a lot of historical precedent for having units attached to coordinates, so it makes sense to convert these descriptions by moving these units into the metric – which always works. Or, we can do these conversions in our heads, and keep these units in the coordinates with the understanding that, really, they’re not there but are in the metric.

I’m trying to have my cake and eat it too: Now that we know the coordinates don’t have units, I still want to be able to handle the cases when they do!

If we don’t accept this approach, there’s going to be a big fight on wikipedia when we try to uglify the Schwarzschild and other GR solutions by moving the units out of the coordinates.

Posted by: garrett on September 28, 2006 9:32 PM | Permalink

### Re: Dimensional Analysis

If we don’t accept this approach, there’s going to be a big fight on wikipedia when we try to uglify the Schwarzschild and other GR solutions by moving the units out of the coordinates.

Oh, c’mon. No there won’t.

Take the standard Schwarzschild metric and, instead of the dimensionful coordinate, $r$, write it in terms of the dimensionless coordinates $\tilde{r}= r \tfrac{c^2}{G m},\, \tilde{t} = t \tfrac{c^3}{G m}$. Then everything looks pretty much the same, but with a factor of $(G m/c^2)^2$ sitting out in front of the whole metric.

I don’t think switching between $r$ and the rescaled $\tilde{r}$ is particularly profound.

Posted by: Jacques Distler on September 28, 2006 11:00 PM | Permalink | PGP Sig

### Re: Dimensional Analysis

John wrote that fixing a unit of length (and of time in general relativity) in our world $M$ is choosing

an isomorphism

(1)$e : T M \to T$

where the internal space $T$ is a vector bundle equipped with a fixed metric, say $\eta$.

That sounds good. My simple example should be the Newtonian absolute space non-relativistic version of that.

And this presciption is pretty close to what we actually do in practice.

I have a vector here, some stick say. I want to know how long it is in meters. What can I do?

Well, I should parallel transport that stick to Paris until it sits in the same vector space as the Urmeter. Then I compare. This means: then I compute the image of my vector under a fixed isomorphism $e$.

Of course what really happens is that big companies parallel transport a stick to Paris, do the above comparison, and then produce many copies of this stick, trying to convince us that they didn’t cheat in the process.

And of course what really, really happens is that nobody parallel transports anything to Paris. But everybody does something equivalent.

Posted by: urs on September 23, 2006 1:27 PM | Permalink

### Re: Dimensional Analysis

I don’t agree that c can be considered dimensionless. c is defined as ‘the speed of light in a vacuum’ - therefore it has the dimensions of L/T.

In my view dimensional analysis is a wonderful tool, but the risk is to take it too far. One of the best uses for dimensional analysis is a quick check on your math. If you derive an expression for velocity which has dimensions other than L/T you’ve made a mistake. You can also make educated guesses that can save a lot of time, especially when you only need to know how a quantity scales.

For example how does speed of sound in a gas scale with pressure? If it also depends on gas pressure a quick look at the units shows that you can combine these two quantities to get a velocity by making the speed of sound proportional to sqrt(pressure/density).

So what does it mean to say the fine structure constant is ‘more fundamental’? If you can’t define that, then it doesn’t mean much. Yes, beings from another planet would have a different number for c in their theories but the same numbers for fine structure constant, pi and e. So?

Posted by: davros on September 23, 2006 7:12 AM | Permalink

### Re: Dimensional Analysis

…since we could also get any value of α by changing our definitions of e, ε0, hbar and c

A demonstration of this claim would add clarity.

Posted by: Arun on September 23, 2006 1:44 PM | Permalink

### Re: Dimensional Analysis

Arun wrote:

…since we could also get any value of $\alpha$ by changing our definitions of e, $\epsilon_0$, $\hbar$ and c

A demonstration of this claim would add clarity.

I didn’t mean anything profound or nonobvious by this. Right now $\hbar$ is defined to be $h/2\pi$. Say we change the definition to $h/\pi$. Then $\alpha = e^2/4\pi \epsilon_0 \hbar c$ gets halved.

This may seem very silly. Ones first reaction is “But now you’re talking about a completely different concept of $\alpha$ - of course it has a different value!” And this is true. But the same thing would happen if we redefined the meter to be twice as long: the dimensionless quantity $C = c/(m/s)$ would be halved, but now we’re talking about a completely different concept of $C$. We still haven’t gotten to the root of what makes $C$ different from $\alpha$.

Both $\alpha = e^2/4\pi \epsilon_0 \hbar c\simeq 1/137$ and $C = c/(m/s) = 299,792,458$ are examples of dimensionless quantities that (unsurprisingly) would change if we changed the definitions of the quantities they’re built from. Why does $\alpha$ seem more “fundamental” than $C$? Of course it’s because $e,\epsilon_0, \hbar$ and $c$ seem more “fundamental” than $m$ and $s$. But we still need to explain what we mean by “fundamental” here - and we need a concept of “fundamental” that applies to dimensionful physical constants to do this. We can’t just resort to a simplistic philosophy that says only dimensionless constants can be fundamental!

Posted by: John Baez on September 23, 2006 3:37 PM | Permalink

### Re: Dimensional Analysis

We can’t just resort to a simplistic philosophy that says only dimensionless constants can be fundamental!

Why not?

Posted by: Aaron Bergman on September 23, 2006 4:47 PM | Permalink

### Re: Dimensional Analysis

John wrote:

We can’t just resort to a simplistic philosophy that says only dimensionless constants can be fundamental!

Aaron wrote:

Why not?

That’s what I just explained in the buildup to that final shocker.

Briefly: not all dimensionless quantities are fundamental, and determining which ones are requires that we decide which dimensionful quantities are fundamental.

I should emphasize that this is a complete turnaround from the position I’ve had for years. Until Kehrli emailed me, I’d never noticed that the dimensionful quantity $c$ doesn’t depend of our choice of units, and that the dimensionless quantity $C$ does.

Posted by: John Baez on September 23, 2006 5:07 PM | Permalink

### Re: Dimensional Analysis

I still don’t see it. One never measures a dimensionful quantity. Units are nice and helpful, but at the risk of going all instrumentalist, when in the philosophical weeds I think we should try to speak of things that we can measure. With that, I don’t see where these fundamental dimensionful quantities you’re talking about are.

Posted by: Aaron Bergman on September 23, 2006 5:19 PM | Permalink

### Re: Dimensional Analysis

One never measures a dimensionful quantity.

I don’t think that’s true.

You always measure a pure number - but with a specified measuring device.

So you have either the pure number 3 - measured with a yardstick (which, say, has meters on it),

or you have the pure number 12 - measured with bananas (assuming the bananas satisfy EU-norm).

Gauging your measurement device corresponds to establishing the isomorphism that allows you to relate the pure number you read off its display to reality (or to any other measurement device for the same quantity).

Units are nice and helpful, but at the risk of going all instrumentalist, when in the philosophical weeds I think we should try to speak of things that we can measure.

Every measurement comes with a concept that relates it to something. Just measuring the number “3” is not sufficient. You need to know what you did to obtain that number. That’s where the units come in.

Posted by: urs on September 25, 2006 1:16 PM | Permalink

### Re: Dimensional Analysis

I didn’t say unitless (which is trickier); I said dimensionless.

Posted by: Aaron Bergman on September 25, 2006 3:17 PM | Permalink

### Re: Dimensional Analysis

I didn’t say unitless (which is trickier); I said dimensionless.

What is a dimensionful quantity, if not a quantity that needs to be expressed as a number times a unit?

(Or, as I would say: as a number times a choice of isomorphism.)

Posted by: urs on September 25, 2006 3:23 PM | Permalink

### Re: Dimensional Analysis

Dimensionful quantities are related to dimensions.

I think it was Moshe who has this whole story, right. There is some abstract space of theories on which we can choose coordinates. The rest is just a distraction.

Posted by: Aaron Bergman on September 25, 2006 3:31 PM | Permalink

### Re: Dimensional Analysis

Dimensionful quantities are related to dimensions.

Like, for instance - let’s see - the mean diameter of the hydrogen atom: a length, right?.

It’s a fundamental constant (in as far the charges and masses involved are fundamental - they are in QM), and it can be measured. It is expressible as a number times a unit.

Posted by: urs on September 25, 2006 3:39 PM | Permalink

### Re: Dimensional Analysis

It’s a fundamental constant (in as far the charges and masses involved are fundamental - they are in QM), and it can be measured. It is expressible as a number times a unit.

No, it can’t be measured. You never measure dimensionful quantities.

Describe any experiment you choose, and I’ll tell you the dimensionless quantity you’re measuring.

Posted by: Aaron Bergman on September 25, 2006 3:47 PM | Permalink

### Re: Dimensional Analysis

Describe any experiment you choose, and I’ll tell you the dimensionless quantity you’re measuring.

I measure the length of my office table.

I use a yardstick to do so and follow what I said above.

The dimensionless quantity I am reading off is the ratio of the table’s length to the distance of the marks on my yardstick.

But taken together it’s a length: a number and a unit, namely a specified procedure which maps the world around me to $\mathbb{R}^3$.

Posted by: urs on September 25, 2006 4:09 PM | Permalink

### Re: Dimensional Analysis

You can interpret it as a length, but you didn’t measure a length. The units you’ve added simply reflect this reinterpretation. They are not necessary, however; they just make life easier in a lot of places.

Posted by: Aaron Bergman on September 25, 2006 4:15 PM | Permalink

### Re: Dimensional Analysis

They are not necessary, however;

They are not necessary if I can ensure that once I have fixed that isomorphism of the world around me with $\mathbb{R}^3$ (using my yardstick) I will never have to face anyone using another isomorphism.

If so, I can say: “The length of my table is 3/2”.

That’s what we do when we say things like:

“In all of the following we set $\hbar = 1$.”

This means really: we choose one isomorphism once and for all, and whoever dares to change it will get into trouble.

I know I am splitting hairs. But the hair I am splitting is the one whose splitting this thread is about.

Posted by: urs on September 25, 2006 4:25 PM | Permalink

### Re: Dimensional Analysis

They are not necessary if I can ensure that once I have fixed that isomorphism of the world around me with $\mathbb{R}^3$ (using my yardstick) I will never have to face anyone using another isomorphism.

If so, I can say: “The length of my table is 3/2”.

You never have to say such a thing. You can do all of physics in terms of dimensionless quantities.

Posted by: Aaron Bergman on September 25, 2006 4:30 PM | Permalink

### Re: Dimensional Analysis

If so, I can say: “The length of my table is 3/2”.

You never have to say such a thing. You can do all of physics in terms of dimensionless quantities.

Now let me ask a question:

Assume you have developed a theory which predicts some new particle or some new bound state of sorts - whatever. Assume you want to tell some experimentalist at what energy the new object is predicted.

How do you do that?

Posted by: urs on September 25, 2006 4:47 PM | Permalink

### Re: Dimensional Analysis

I thought I had successfully posted something here. Apparently not. Let me try again sorry if it is somehow repeated.

To recap what you were saying about the speed of light, but with a simpler example. The length of my table is 5 feet. The 5 is a dimensionless number expressing the ratio of the table length to the length of my ruler, and the dimensional quantity is the “feet”. If I were to use a different unit, say, 152.4 centimeters, my dimensionless number has changed along with the units, the table hasn’t changed, etc.

So far, so good. The height of my table is 3 feet, and a similar spiel goes with that measurement.

The ratio of the length of my table to the height of my table is dimensionless, and more over is the same no matter what units I use. I could arbitrarily put in a 4 Pi or a 1/2 into this ratio, but either I’m making a useless convention in using different units to measure length and height, or else, I’m trying to deny the isotropy of space.

The dimensionless ratio of length to height of any object is not something fundamental, it is a result of its history or design, and is not very interesting. Nor is the length of my table very interesting. It doesn’t even capture something universal about tables.

The speed of light (as opposed to the speed of say, sound), the electron charge, the quantum of action - these are interesting quantities because they appear universally. The fact that they appear universally is an extremely interesting physical discovery, and are hard won experimental and theoretical facts. All electrons (at all times, we think) have the charge -e, the speed of light in a vacuum anywhere and anywhen is c, etc., etc., not like my table.

A diversion on the number of dimensional quantities. That we are able to reduce them to M,L,T is a result of having physical laws. E.g. I could define a standard voltaic battery - stick copper and zinc in an acidic solution, etc., - and measure current by the amount of silver deposited on a cathode per unit time, and thus define a measure of conductance. Without any further theoretical development, conductance would be another fundamental dimension. It is because of the various laws of electromagnetism that I can relate conductance to the M,L,T dimensions. It is not some arbitrary invention.

Thus, e.g, Coloumb’s law relates e^2/omega0 to the M,L,T units. You can put in various constants into Coloumb’s law, but this is essentially a redefinition of units, a convention.

You can likewise decide to call hbar hbar/2 but it would be a redefinition of units, that the energy of a photon of angular frequency omega is hbar omega however fixes the meaning of hbar, and all you are doing is changing units or introducing useless additional conventions.

Alpha is quite independent of the units we use, exactly like the ratio of length to height of my table, and moreover the quantities that appear in it, c, e, hbar, omega0 are fundamental unlike my table’s dimensions. You could introduce arbitrary factors into c, e, hbar, etc., but that would be like introducing arbitrary factors into the ratio of length to height of my table. It may actually be worse, because, c, e, hbar etc., have much more definite and profound meanings via their appearance in physical law than anything about my table.

Alpha thus is independent of our units of measurement and is built from quantities that are fundamental themselves. It cannot be arbitrarily redefined, e.g., by setting h to h/2. You are either creating an unnecessary convention or the universe itself changes by that redefinition.

BTW, I expected you to say why not alpha^2 or (alpha)^(1/2) instead of alpha, when you said alpha can be redefined arbitrarily. I think the same argument holds as given before. If physical quantities depended on a power series in alpha^(1/2) and not alpha, then alpha^(1/2) would be the “right” fundamental constant rather than alpha.

Posted by: Arun on September 24, 2006 5:25 AM | Permalink

### Re: Dimensional Analysis

What about topological measurements? For instance the euler characteristic of a surface? They are not related to any unit, in principle.

But… some of these topological invariants can be got from dimensionful operations, for instance integrating some local quantity along all the surface. The trick of units can be played again here: measure the local quantity in inverse of square foots but use square meters for the area of the surface… you will get Euler characteristic expressed in a consistent way as “xxx (meter/foot)^2”.

You can feel happy doing this kind of trick with non-topological quantities, as the fine structure constant. But here?

Posted by: Alejandro Rivero on September 23, 2006 6:19 PM | Permalink

### Re: Dimensional Analysis

C will change when we go to a cgs system as opposed to SI units. alpha won’t. It’s not that complicated.

Posted by: folosopy on September 24, 2006 4:39 AM | Permalink

### Re: Dimensional Analysis

What I want to know is why
doesn’t seem to know about planck length.

Posted by: Jesse on September 27, 2006 4:16 PM | Permalink

### Re: Dimensional Analysis

Perhaps because you typed in “speed of light in furlongs per fortnight”. To learn about the Planck length you have to type in “Planck length”.

Pretty soon I will delete your post and this reply, because they’re sort of digressive and silly.

Posted by: John Baez on September 28, 2006 8:14 AM | Permalink

### Re: Dimensional Analysis

I have an elementary grumble about a bit of terminological confusion here (which I see also bedevils the Wikipedia discussion). Let me get this off my chest:

Something that seems a bit peculiar in the original post is the business about “changing the definition” of the metre (or, in a later post, “changing the definition” of $\hbar$, $c$ etc). This isn’t normally how we change units. Normally, changing units means going from e.g. metres to inches. Different unit: different name. Obviously the ratio of the speed of light to the metre-per-second is different from the ratio of the speed of light to the inch-per-second. On the other hand, “changing the definition of the metre” means using the same name for a different unit. Obviously this is a recipe for confusion.

What about “changing the definition” of $\hbar$? Presumably this means changing the unit of action that we use to quote the size of the action $\frac{e^2}{4\pi\epsilon_0c}$. It obviously doesn’t change the actual physical quantity of action currently denoted by $\hbar$.

To give a linguistic parallel, consider that classic piece of love poetry:

Roses are red,
Violets are blue,
Sugar is sweet,
And so are you.

We are used to the fact that the word “you”, e.g. as it appears in the last line, has a context-dependent referent: who it refers to depends on who the speaker is talking to. There’s a technical linguistics term for this which I’m blanking on. Anyway, it means that if we “measure” the truth value of the last clause, it depends on our “unit”, i.e. which standard we are measuring sweetness against, i.e. who we are talking to.

This clearly makes the word “you” different from “roses”, “violets” and “sugar”. It is true that somebody could say, “Aha, but suppose you change the meaning of ‘roses’ to refer to violets, ‘violets’ to refer to roses, and ‘sugar’ to refer to manure? Then the truth-value of the other clauses would change, too.” Well, yeah, but that’s just being confusing for the sake of it.

Exactly the same confusion could be introduced into the definition of the fine structure constant. We could define a new $\alpha$ to be 2 times the old $\alpha$, or 2.9 times the old $\alpha$, or 299,792,458 times the old $\alpha$, which would apparently demolish one of the disputants on Wikipedia, but obviously wouldn’t say anything substantive about the issue of units. It we have a new definition, we should have a new symbol instead of $\alpha$, and use that new symbol in our equations.

Although if you did that, people would probably look at you funny and grumble about huge, meaningless numbers cluttering up their QED calculations.

Posted by: Tim Silverman on September 24, 2006 2:55 PM | Permalink

### Re: Dimensional Analysis

Hey, this looks like fun, let me have a try…instead of talking about what is “fundamental”, somehow an emotionally charged issue always, we can concentrate on what it means to change dimensionful vs dimensionless quantity.

I think that mathematical theories with different values of the dimensionful constants are isomorphic: you can establish a mapping between all their observables simply by rescaling (a.k.a “dimensional analysis”). On the other hand theories with different values of “fundamental” dimensionless constants are really different from each other- if alpha of QED is any different the theory is different, possibly qualitatively different. It is then reasonable for me to call such constants, ones that parametrize the set of inequivalent theories, “fundamental”, but it really doesn’t matter.

Note that this criterion also distinguishes physical from mathematical constants, assuming one does not want to solve all cosmological problems by positing time variation of \pi or something…

Posted by: Moshe on September 24, 2006 4:55 PM | Permalink

### Re: Dimensional Analysis

Some further thought leads me to basically agree with John, a good example would be to consider \alpha of the strong interaction (measured at some scale, say at the Z0 peak). It is dimensionless and all, but we know that through the magic of dimensional transmutation it is really nothing but an arbitrary scale- theories with different values of that dimensionless constant are isomorphic.

So, I guess my criterion would be to choose a parametrization of the space of inequivalent physical theories, and then to carefully distinguish between what are real changes (moving in that space), and what are merely coordinate changes (which are just conventions). Choosing coordniates that are independent of human conventions would aid in avoiding such confusion.

Posted by: Moshe on September 24, 2006 5:17 PM | Permalink

### Re: Dimensional Analysis

Moshe wrote #:

I think that mathematical theories with different values of the dimensionful constants are isomorphic

[…]

On the other hand theories with different values of “fundamental” dimensionless constants are really different from each other

I think that’s precisely the point. As I wrote above:

“Dimensionful constants” are quantities that do depend on a choice of representative in an isomorphism class of models.

“Dimensionless constants” are quantities that do not depend on a choice of representative in an isomorphism class of models.

The remaining subtlety is hence to agree on what precisely it means for two theories to be isomorphic. (We need to agree not just on the objects, but also the morphisms of the category that we are talking about).

For instance, naively it may seem that the FRW model of cosmology depends on a dimensionless constant $k \in \mathbb{R}$. But being careful reveals that all $k \gt 0$ and all $k \lt 0$ give rise to models in the same isomorphism class, respectively. Hence in the end the FRW solution depends on a dimensionless parameter

(1)$k \in \{-1,0,1\} \,.$

Similar comments might apply to the fine structure constant. I am not sure about that, but this seems to be what Moshe is implying here #:

It is dimensionless and all, but we know that through the magic of dimensional transmutation it is really nothing but an arbitrary scale- theories with different values of that dimensionless constant are isomorphic.

Posted by: urs on September 25, 2006 12:23 PM | Permalink

### Re: Dimensional Analysis

I’m fast becoming the proverbial fat lady of the blogosphere, so just to clarify, I think the non-trivial object here is the set of non-equivalent theories. Change in dimensional parameters is “morphism” because it does not say anything new and interesting, in fact I doubt one can find non-circular way to define such changes, only making reference to measurements. On the other hand, change in dimensionless parameters may or may not lead to new physics. We seem to find all kinds of examples when it doesn’t…

Posted by: Moshe on September 25, 2006 3:42 PM | Permalink

### Re: Dimensional Analysis

Oh. This has grown enormously. Since my previous comment received no follow-ups, I am unsure whether I simply missed the point or I wrote something completely trivial, naive or (not?) even wrong! Please let me know!

My point is:

The Planck units reflect a fundamental property of free space, and do not depend on any object or elementary particle arbitrarily chosen (necessarily used to define, eg, the meter, the second, etc).

If you measure all quantities in Planck units, then they are all dimensionless, so what is the deal of being dimensionless or not?

We only say that alpha is “fundamental” because alpha = e^2 (in Planck units), and therefore the strength of the electromagnetic force relative to all other *fundamental* forces is proportional to alpha, a number. This is a number given by Nature that have to be explained. It is a fundamental number.

So the point is: alpha is not fundamental because it is dimensionless. It is so because it is a number that happens to set the relative strength of the electromagnectic force to the other fundamental forces.

Ok, I’ll no longer insist on my possibly naive understanding of all this. Perhaps there is much more to it. When you gentlemen find out all about this, please do post a summary on your conclusions!

Christine

PS: But what happens if you change c, etc? Would we notice? (Read the section on Planck units and the invariant scaling of nature here: http://en.wikipedia.org/wiki/Planck_units).

Posted by: Christine Dantas on September 26, 2006 5:31 PM | Permalink

### Re: Dimensional Analysis

My point is:

The Planck units reflect a fundamental property of free space, and do not depend on any object or elementary particle arbitrarily chosen (necessarily used to define, eg, the meter, the second, etc).

If we all agree once and for all to stick with one fixed choice of units (choice of isomorphism of the world with our model of it) then - indeed - we may stop mentioning that choice of units, trusting that, following your suggestion, whenever we say

the table has a length of $10^{35}$

everybody understands that we implicitly use Planck units (or some other fixed unit that we all agreed on), to say that the table has a length of $\sim 3/2$ meters.

That’s perfectly fine. What is not correct is to conclude from the fact that by agreement we may never mention our units, that there are no units involved.

Posted by: urs on September 26, 2006 5:52 PM | Permalink

### Re: Dimensional Analysis

What makes this whole business interesting, and confusing, is our ability to switch at will between theories with a greater or smaller supply of special (or canonical) isomorphisms. I think you’ve already said something like this further up. But I always like to add lots of simple (or silly) examples…

So, for instance, imagine we have a set of scales (I mean an actual balance for weighing things). We can weigh different things with it. For instance, we can weigh apples, and express this in a unit “pounds of apples”. We might also weigh cheese but, noticing that cheese is not the same as apples, create a different unit “pounds of cheese”.

Obviously, the set of scales provides an isomorphism between pounds of apples and pounds of cheese, so we then get a unit, “1 pound of apples per pound of cheese”. If we didn’t understand very much about weighing, we might think this was a purely “accidental” isomorphism, tied to the particular set of scales. But as long as we agreed to use those scales as a standard, we could express other ratios of apples to cheese in terms of the same unit. Alternative, we could create some other unit using this unit as a starting point (e.g. 2 pounds of apples per pound of cheese) also based on the same set of scales. Or we could implicitly omit the unit, and say that the ratio is dimensionless.

But then, as our theories of physics advance, maybe we decide that actually, there was a single underlying property, “weight”, which the scales measures. Suddenly the accidental isomorphism provided by the scales starts looking like a canonical isomorphism of more widespread validity – or even like an identity. Then the unit “pounds of apples per pound of cheese” starts to look a bit silly – kind of like a fancy name for the number 1.

This sort of thing seems a bit less trivial when we actually start out with two very different measurement procedures, using different kinds of apparatus. E.g. the electrostatic measurement equipment that gave us electrostatic units versus the magnetostatic measurements that gave us electromagnetic units; or the rulers and clocks that gave us metres and seconds. It’s easy to establish an accidental isomorphism, e.g. by picking an arbitrary velocity to compare all other velocities to. But suppose a new theory comes along and tells us there’s a canonical isomorphism, because the (apparently) different things we are measuring are really the “same kind” of thing. That feels different. One of the velocities is special – and at the same time particularly dull! It provides the canonical isomorphism, and at the same time sets itself to 1! Of course, we are still allowed to pretend the original quantities are different if we want, and so to go on using different units for them. Sometimes we really care about the difference between apples and cheese, and we want this to be reflected even in the units we use for their weights.

This doesn’t answer the puzzle about what makes the fine structure constant different, though. Perhaps the lesson is that electromagnetism doesn’t count as “the same sort of thing” as quantum mechanics. I’m not convinced that’s it, though. Perhaps there’s something to be said here about the difference between theories that are isomorphic and those that are merely equivalent.

Posted by: Tim Silverman on September 26, 2006 11:16 PM | Permalink

### Re: Dimensional Analysis

I really like Tim Silverman’s comment. Units seem similar to types. After years (decades) of looking at programming languages and trying to work out what a “type” was, I eventually decided that the type of a value (i.e. of any expression in the language) is just “everything the compiler knows about the value” and type checking is looking for proofs of inconsistency [it is not in general feasible to prove consistency].

My other little point concerns clifford algebras. Consider the 2d case. An element of the real clifford algebra has a real part. Ostensibly you could add a real to an element of the clifford algebra to get a new element. But that would almost invariably be a type error / units error. The real part of our real clifford algebra comes [typically] from the dot product of 2 vectors and its natural unit is length squared. Books like Lasenby and Doran “Geometric Algebra for Physicists”, when looking at a problem, get rid of all the unit/type information, do complex algebraic manipulation, then put the units back at the end to give the result a physical significance. I can’t help feeling that it would all be better if we kept track of the units as we went along.

Posted by: Robert on September 27, 2006 3:37 AM | Permalink

### Re: Dimensional Analysis

John wrote:

But with this definition of m, the dimensionless quantity C=c/(m/s) seems to tell us nothing about our universe!

As long as you think about one constant at a time it looks like they tell us nothing because you can redefine their value by changing units. But as soon as you your physical theory has more constants than dimensions you no longer have complete freedom and you’re making meaningful statements.

Posted by: Dan Piponi on September 26, 2006 7:20 PM | Permalink

### Re: Dimensional Analysis

Dan Piponi wrote:

John wrote:

But with this definition of $m$, the dimensionless quantity $C = c/(m/s)$ seems to tell us nothing about our universe!

As long as you think about one constant at a time it looks like they tell us nothing because you can redefine their value by changing units. But as soon as you your physical theory has more constants than dimensions you no longer have complete freedom and you’re making meaningful statements.

Right! That’s why I used the phrase “seems to” in the passage you quote, and then said:

Actually it tells us some funny blend of information about the speed of light and the definition of $m$ and $s$.

A bunch of statements, each of which individually seems to say nothing substantial about the universe, can add up to something that does. As you note, it’s all about linear dependence.

For those who don’t see what Dan is getting at:

Suppose you want to know the value of some variable $x$. Instead of satisfying your curiosity directly, I introduce some extra variables $y$ and $z$, and tell you that they satisfy $y = x + 1$ $z = x + 2$ and $z = 2y$ Each one of these statements, or even any pair of them, seem to say nothing about what $x$ equals, since they don’t constrain its value at all. However, all three taken together imply $x = 0$.

This is completely obvious. But, people tend to get tripped up when it happens in dimensional analysis. A bunch of statements, each of which by itself can be taken as “true by convention”, can imply a statement whose truth can only be determined by experiment!

In fancy jargon, we’re seeing a “failure of subadditivity of invariant information” - a bunch of statements taken together can convey more information invariant under some symmetry group than the sum of the amounts of invariant information contained in each separate statement.

Posted by: john baez on September 29, 2006 1:41 AM | Permalink

### Re: Dimensional Analysis

John, you just said:

I should emphasize that this is a complete turnaround from the position I’ve had for years. Until Kehrli emailed me, I’d never noticed that the dimensionful quantity c doesn’t depend of our choice of units, and that the dimensionless quantity C does.

first, i agree that it is apparent that this is a complete turnaround and am somewhat astounded by this since i was trying to represent the position taken by you in sci.physics.research as well as Barrow, Duff (both of whom i had email chats with, and weirdly, it seems that Barrow has flipped his position from the quote of his “Constants of Nature” book - Duff is still very consistent in his position). nothing Kehrli has written is the least bit persuasive. as an engineer, i feel pretty confident that i understand the meaning of dimension of physical quantity and of units. i think i even understand the difference between the terms “unitless” and “dimensionless”. measureing an angle in degrees or sound intensity in dB are not unitless, but they are dimensionless. measuring the same quantities in radians and nepers is both unitless and dimensionless. anyway, all Kehrli is doing is repeating a fact we all know (that a dimensionful physical quantity is represented as a “product” of a dimensionless value with a unit) and then some less coherent ramblings where she says some clearly false things (which you now seem to give credence to) and makes appeals to some “IUPAC green book” which i think is non sequitur, it doesn’t speak to this issue in the least.

the reason that human beings measure $c$ to be 299792458 m/s is because (before they redefined the meter to fix $c$ to 299792458 m/s) the distance between the two little scratch marks of the platinum-iridium International Prototype Metre was, to the same precision, very nearly 6.18718916 x 1034 Planck lengths and the number of Planck times in one cycle of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom is, to the same precision, very nearly 2.01778195 x 1033 and that humans have anthropometrically decided that 9192631770 of those cycles make up one second (and that $c$ is always 1 Planck length per Planck time). that is a human construct. and that is solely why $c$ = 299792458 m/s.

we could get more historical about it when we remember that the meter was chosen so that there would be exactly 10 million of them on an arc of the earth from the north pole to the equator going through Paris (making the meridional circumference 40 million meters - they got that wrong by 0.02%, but it’s not bad for the 18th century). and, given that the mean solar day was originally 86400 seconds (before the atomic clock and leap seconds), the human POV of the measure of $c$ is that light is fast enough to make 647551.7 revolutions around our planet at the mean surface radius and over the poles in the period of time between consecutive instances that the sun is directly above the same given meridan. that is what $c$ = 299792458 m/s means to humans, and that is purely anthropometric. the aliens on the planet Zog could give a rat’s ass about it.

now we could communicate to the Zogs a dimensionful quantity by referencing to another like-dimensioned quantity (call it a standard, if you choose). but how do we communicate the magnitude of the standard to them? we could use something like Planck units as a standard to communicate the size of anything to them. so we say that our meter is 6.1872 x 1034 Planck lengths and that our second is 1.85487 x 1043 Planck times, but as soon as we say “we measured the speed of light to be 299792458 m/s” to them, that means, from our definitions of the meter and second, precisely the same as “we measured the speed of light to be 1 Planck length per Planck time.” in other words, it says nothing (new) at all.

however we can ask them to measure and report to us what the value of $\alpha$ is and if the number the report is substantively different, we know they live in a different world. i believe this is the basis of the dispute at Wikipedia about Physical constants. there is nothing profound i can see in Kehrli’s insistence that there is no qualitative difference between the dimensionful physical constants like $c$, $G$, $\hbar$ and the dimensionless constants like $\alpha$ or $m_p/m_e$ (essentially that dimensionless is just another dimension). it’s just wrong.

Posted by: robert bristow-johnson on September 29, 2006 1:28 AM | Permalink

### Re: Dimensional Analysis

robert bristow-johnson write:

there is nothing profound i can see in Kehrli’s insistence that there is no qualitative difference between the dimensionful physical constants like $c$, $G$, $\hbar$ and the dimensionless constants like $\alpha$ or $m_p/m_e$ (essentially that dimensionless is just another dimension). it’s just wrong.

I don’t recall where Kehrli said there is “no qualitative difference” between dimensionless constants and dimensionful ones. I think you tend to misinterpret her remarks. If can point to a specific passage written by her, that might help.

As I mentioned before, it would be a strange thing to say there’s “no qualitative difference” between dimensionless and dimensionful quantities, because there obviously is: the former are dimensionless, while the latter are not. That’s pretty obvious, right? Of course which quantities are dimensionless depends on ones system of dimensions: any given quantity can be made dimensional or dimensionless, depending on that choice. But, given that choice, the distinction is meaningful - and I think we agree that it plays an important role in physics!

To my mind, the interesting points Kehrli made were these:

• Dimensionless constants can depend on our choice of units - for example, the constant $C = c/(m/s) = 299,792,458$.
• Dimensionful constants often don’t depend on our choice of units - for example, the constant $c = C m/s = 299,792,458 m/s$.

They seem obvious in retrospect, but they came as a bit of shock to me.

You’ve written a bunch of stuff, most of which seems fine to me. Nowhere do I see you disagreeing with the two statements above. In fact at one point you write that “all Kehrli is doing is repeating a fact we all know”. Does that mean you agree with the two statements above? If so, you may have no quarrel with me - except perhaps that I should have known these things sooner, for which I apologize.

True, you say I “seem to give credence” to some “clearly false things”. But, you don’t point to anything false I actually said. So, that doesn’t constitute an actual disagreement.

You obviously have a beef with Kehrli. However, this blog is not the place for that. So, I will delete some of your grumpier comments about her. I’m trying to discuss dimensional analysis, not people, nor even the best way to write that Wikipedia article.

Posted by: john baez on September 29, 2006 2:38 AM | Permalink

### Re: Dimensional Analysis

FYI, the actions and justifications that started the whole mess with Physical constants is:

I removed the following paragraph because I think it is wrong:

“Unless the system of natural units is used, the numerical values of dimensionful physical constants are artifacts of the unit system used, such as SI or cgs; that is, they are essentially conversion factors of human construct.”

The point is that constant quantities do not depend on the units used. To give an example: the height of the Eiffel tower is a physical constant. The height of the tower, however, is always the same independent on the units in use. The sentence confuses the physical quantity Q with its numerical factor {Q} which depends on the units.

I also removed/changed the following paragraph:

“The fine-structure constant α is probably the most well-known dimensionless fundamental physical constant. The dimensionless ratios of masses (or other like-dimensioned properties) of fundamental particles are also fundamental physical constants, as are the measure of these properties in terms of natural units.”

I do not think that dimensionless constants are more fundamental than other constants. This is a misunderstanding coming from the believe that quantities depend on systems of units. However, all quantities do not depend on the system of unit, independent wether they have dimensions or not. The ratio of the Eiffel tower to the empire state building is always the same and does not depend on units, as is the height of the Eiffel tower.

–Kehrli 01:44, 12 August 2006 (UTC)

both of the statements deleted were correct and both of her reasons for deleting them were anemic. then in a section title “is α an artifact too?” she, in some weird convolution of language she says “that the ‘lack of units’ of α is also somewhat arbitrary and can be removed by choosing an ‘inconvenient’ system of units” and that “For α it is exactly the same story as for c. … Therefore there is no big difference between dimensionless units and units with dimensions… and that the difference is not large enough to justify calling all dimensionful quantites ‘not funtamental’.” later (using the semantic ambiguity as justification): “And NIST seems to agree with me since they list all those quantities like c, G … as fundamental quantities. The notion that only dimensionless quantities are fundamental is only backed by a non-reviewed web articel of Baez. Therefore I think it is POV.”

it just goes on and on. sorry if it’s difficult to differentiate what i’m saying from what i’m saying that Kehrli is saying from what Kehrli is saying what i or someone else is saying. i can’t figure out the markup rules for this blog.

r b-j

Posted by: robert bristow-johnson on September 29, 2006 3:59 AM | Permalink

### Re: Dimensional Analysis

robert bristow-johnson wrote:

i can’t figure out the markup rules for this blog.

Yeah, I’m sorry. They depend on the “Text Filter” you choose when you post a comment. If you don’t choose anything, you’ll get the default text filter called “Convert Line Breaks”. With this or the filter “itex to MathML with parbreaks”, here’s how quoting works:

You need to skip a line before and after the “blockquote” command, which must be in lower case, like this:

John Baez said:

<blockquote>

In the 60’s, Grothendieck led the reggae group shown in this rare photo:

</blockquote>

This would not:

John Baez said:

<blockquote>
In the 60’s, Grothendieck led the reggae group
shown in this rare photo:
</blockquote>

Other text filters may make your life easier - see what Urs and Jacques wrote in the forum devoted to these annoying TeXnical issues. If anyone has further questions, that’s the best place to ask them.

I’ve tried to make the quotes in your post a bit easier to follow. If I screwed up, let me know.

Posted by: John Baez on September 29, 2006 10:35 PM | Permalink

### Dimensional Relativity

Let me see if I understand the main conflict in this thorny thread, and how I think it’s resolved.

r b-j is saying (from the “Physical constant” wikipedia talk page) [square brackets mine]:

“You [Kehrli] are fully mistaken in thinking that dimensionless physical constants (like [the fine structure constant] α) are not more fundamental than the dimensionful physical constants [like the speed of light, c]. The former are numbers that are properties of the universe (and the aliens on the planet Zog will come up with the same number) and the latter are human constructs.”

Kehrli is saying (I paraphrase):

“Thhhhpppppttt.”

I have to side with Kehrli on this one. The physical constants with dimensions are just as fundamental as the dimensionless ones. Take the speed of light, c. We conventionally express c in terms of meters per second. And r b-j says these units are arbitrary and c is thus physically meaningless. But we could go measure (via scattering or something) the expected radius of a hydrogen atom, and measure the expected decay time for a 2p -> 1s transition, and then convert c to these speed units of hydrogen radii per hydrogen decay time. This way hydrogen atoms (the most common atoms in the universe) could act as standard rulers and clocks. And we could indeed compare our value for c, in these hydrogen units, with those found by the Zogians. The value will be the same if the speed of light is the same and hydrogen atoms are the same. In this way we can understand c to indeed be a physical constant.

But here is where things get tricky. What if the speed of light at planet Zog and hydrogen atoms at planet Zog were both different in just such a way as to make the Zogian’s measurement of c in Zog hydrogen units the same as our measurement of c in our hydrogen units? We’d get a false impression that our c’s were the same! Does this mean r b-j was right, and α is more fundamental? No, because it’s just as possible to imagine the physical measurements behind the definition of α to change and balance in a similar way to give the same numerical value. This brings us to the key revelation that John Baez made a couple of posts ago:

Physical constants, even dimensionful ones, are fundamentally about relations between physical measurements.

I’ll go ahead and call this “dimensional relativity.” It means that what really matters, as I think Urs has been saying, is whether one whole shebang of physical measurements is the same (isomorphic) as another. The physical constants, even dimensional ones, provide information in the form of relationships between measurements. The speed of light means nothing by itself – to be meaningful it has to be related with the rest of the universe, such as to hydrogen atoms, or to meters which are compared to hydrogen atoms. That’s pretty neat.

From this point of view, we can draw an analogy between physics and economics. Without money, you could always barter (trade) some items for other items. There are value relationships, expressible as ratios, between items. Money makes things easier – by introducing an item of arbitrary but universal value, you get an intermediary by which you can value and compare many items, including new ones. Now it should be clear: physical units are the money used to describe physical quantities.

To drive this home: If you have a hydrogen atom, you can describe its radius and decay time in terms of (arbitrary) meters and seconds. Now, to describe the speed of light, you can describe it in hydrogen units or you can describe it in meters per second – it’s the same information, and the same fundamental physical constant, either way. It makes no difference, economically, whether you say “a banana is worth a dollar and an apple is worth a dollar” or you say “a banana is worth an apple” – either way you’re defining the banana value.

You can also always use some physical measurements to stand in for units – this (nondimensionalization) is like going to the barter system and getting rid of money. What’s physically important is the web of relations between measurements.

I hope I’ve helped make this more clear instead of confusing things.

Posted by: garrett on September 29, 2006 8:11 AM | Permalink

### Re: Dimensional Analysis

I would like to make a bunch of comments and then close down comments on this entry. Then sometime I’ll write a new entry summarizing what I learned, and the discussion can continue.

Let me start with the argument between Robert Bristow-Johnson and Vera Kehrli which began this discussion.

Since the issues involved are a bit subtle, it’s important to quote people precisely when disagreeing with them. So, I’m glad r b-j has shown us the statements that Kehrli deleted from an old version of the Wikipedia article on physical constants, and her reasons for deleting them. According to r b-j, “both of the statements deleted were correct”.

Let’s take a look at them and see if they’re correct. First:

Unless the system of natural units is used, the numerical values of dimensionful physical constants are artifacts of the unit system used, such as SI or cgs […]

I think this statement is almost correct, as long as it’s referring not to:

1) a given experimentally measurable dimensionful physical quantity

2) the “numerical value” of that quantity in a given system of units.

The example I’ve given over and over is: 1) the speed of light, $c$, versus 2) the numerical value of the speed of light in meters per second, $C = c/(m/s)$.

Quantities of type 1) are independent of our units, while quantities of type 2) depend on our units.

So, if the Wikipedia passage above was talking about quantities of type 2), it’s correct. But Kehrli felt the offending passage was referring to quantities of type 1). In her words:

The point is that constant quantities do not depend on the units used. […] The sentence confuses the physical quantity Q with its numerical factor {Q} which depends on the units.

The right solution would have been not to delete the offending statement but to add a sentence clarifying it, or discuss it on the talk page.

Why did I say the Wikipedia statement was almost correct? Because of the stuff about natural units:

Unless the system of natural units is used, the numerical values of dimensionful physical constants are artifacts of the unit system used, such as SI or cgs […]

This suggests there’s something special about one system of units, presumably the one with $\hbar = c = G = 1$. That’s not right. Even if we pick this system, the values of dimensionful constants are “artifacts of the system chosen”. Someone else could pick $\hbar = c = 8\pi G = 1$, and then they’d get different values.

That’s why I suggested deleting the portion “Unless the system of natural units is used…”.

Next, the statement goes on to say:

Unless the system of natural units is used, the numerical values of dimensionful physical constants are artifacts of the unit system used, such as SI or cgs; that is, they are essentially conversion factors of human construct.

There’s certainly something sensible about the final bit, but if we examine it with a microscope it gets a bit worrying. If we define the meter and second in terms of independent physical quantities, like a meter rod and the vibration of cesium atoms, the quantity $C = c/(m/s)$ is a relation between these two physical quantities and the speed of light. Is this a “conversion factor of human construct” or not? That’s actually a bit tricky! The meter rod could have been made longer or shorter, true - it’s a “human construct”. But if we defined the meter in terms of (say) the radius of a hydrogen atom, that would no longer be so - except insofar as any definition is a “human construct”.

Conversely, if we define the meter to be “1/299,792,458 times the distance light in a vacuum travels in one second”, as we now indeed do, the number 299,792,458 ceases to be something we measure, and becomes given by definition.

So, the precise significance of $C = c/(m/s)$ depends on the precise definitions not just of $c$ but also $m$ and $s$. Depending on these, it could be either an experimentally measurable “conversion factor between human constructs”, a physical constant, or a number given by definition.

Next passage:

The fine-structure constant $\alpha$ is probably the most well-known dimensionless fundamental physical constant. The dimensionless ratios of masses (or other like-dimensioned properties) of fundamental particles are also fundamental physical constants, as are the measure of these properties in terms of natural units.

I think this is correct, and something like it deserves to be in the Wikipedia. But, I’m happier with how the Wikipedia is now - perhaps unsurprisingly, since I helped it reach this state.

Nonetheless, I found Kehrli’s insistence on the distinction between quantities of types 1) and 2) to be illuminating - not for the Wikipedia article, but for my own thinking.

In particular, I think it’s important to realize that the speed of light, $c$, is independent of units. It’s the dimensionless constant $C$ that depends on units. For this I thank Kehrli.

Posted by: John Baez on September 30, 2006 12:11 AM | Permalink

### Re: Dimensional Analysis

Tim Silverman wrote:

Something that seems a bit peculiar in the original post is the business about “changing the definition” of the metre (or, in a later post, “changing the definition” of $\hbar, c$ etc). This isn’t normally how we change units. Normally, changing units means going from e.g. metres to inches. Different unit: different name. Obviously the ratio of the speed of light to the metre-per-second is different from the ratio of the speed of light to the inch-per-second. On the other hand, “changing the definition of the metre” means using the same name for a different unit. Obviously this is a recipe for confusion.

That’s a good point, though I don’t think it should paralyze us.

In fact, when mathematicians first invented letters for variables, they ran into just this confusion. They had real arguments about how a single letter $x$ could stand for different values. They had serious fights about how a “variable quantity” differed in kind from an ordinary number, yet could equal it! Precisely what act are you performing when you “set $x$ equal to $1$”? What are the rules governing such acts?

When I teach calculus, I run into this issue when students compute

${d \over d x} x^2$

at

$x = 5.$

The good ones compute

${d\over d x} x^2 = 2x$

and then set $x = 5$ to get $10$. The bad ones set $x = 5$ to get

$x^2 = 25$

and then compute

${d\over d x} 25 = 0.$

The work of early 20th-century logicians provided very precise answers to all these puzzles. Later work by Lawvere and others on topos theory provided different answers, which take the concept of “variable quantity” more seriously. But, most people live merrily on without ever reading any of this stuff. They either get it instinctively and do well in math and physics, or don’t get it and flunk out.

It’s true that normally when we change units we give them a different name. But sometimes we give them the same name! For example, the meter has been given 8 different definitions so far. Most of us don’t remember when the meter was the length of a pendulum whose half-period was one second, or when it was $10^{-7}$ times the Earth’s meridian along a quadrant from the equator to the North Pole through Paris, or the length of a brass rod, or 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the $2p^{10}$ and $5d^{5}$ quantum levels of the krypton-86 atom. If we were in a certain fussy mood we would give each of these meters a separate name, like $m_1, m_2, \dots, m_8$. But usually we call them all $m$.

If we allow ourselves the notion of a variable unit of length $m$, and a variable unit of time $s$, we get a dimensionless variable

$C = c/(m/s)$

which changes as we change $m$ and $s$. If instead we demand a separate name for each unit of length and time, say $m_i$ and $s_i$ for the $i$th one, we get a bunch of dimensionless constants

$C_i = c/(m_i/s_i)$

Either approach is okay. Either we say $C$ depends on $m$ and $s$, or we say $C_i$ depends on $m_i$ and $s_i$.

Where, of course, $i$ is a variable.

Posted by: John Baez on September 30, 2006 12:44 AM | Permalink

### Re: Dimensional Analysis

Christine Dantas wrote:

Since my previous comment received no follow-ups, I am unsure whether I simply missed the point or I wrote something completely trivial, naive or (not?) even wrong! Please let me know!

Sorry to take so long to reply. Last week I flew back from Shanghai, and yesterday I taught my first course in this fall’s Quantum Gravity Seminar. So, blogging had to take a back seat for a while.

Nothing you said seemed wrong to me. So, I’ll make up some very small objections, just to annoy you , or maybe give you something to think about.

You write:

The Planck units reflect a fundamental property of free space, and do not depend on any object or elementary particle arbitrarily chosen (necessarily used to define, eg, the meter, the second, etc).

That’s true, but it’s worth noting that the Planck units treat gravity as special, and many modern physicists like to question that special role. For example, string theorists like to treat gravity as part of a package including other fields. Sakharov and Jacobson have considered the possibility that gravity is just a small side-effect of other fields.

Even without any theory like these, one might argue that the electron field is just as “fundamental” as the metric tensor. Sure, the way gravity couples to energy-momentum is very special! But, it’s not as if god made the metric tensor very carefully on Monday, and made the electron field by accident on Wednesday. They’re both fields defined on all of spacetime. So, it’s a bit funny to declare that gravity is a “fundamental property of free space”, while the electron is not.

If we used units where the electron charge, the electron mass, the permittivity of free space, and the speed of light were all equal to 1, then we’d think $\alpha$ was telling us something about Planck’s constant. This seems stupid, but it’s interesting to ponder why it’s stupid. Somehow it’s a less enlightening way to slice the pie.

You conclude:

So the point is: alpha is not fundamental because it is dimensionless. It is so because it is a number that happens to set the relative strength of the electromagnetic force to the other fundamental forces.

I agree that being dimensionless is not a sufficient condition for $\alpha$ to be “fundamental”. It’s fundamental because it says something about the strength of the electromagnetic force - at least if we pick units where $c = -e = \hbar = 1$. If we pick units where $c = 4\pi\epsilon_0 = \hbar = 1$, it says something about the charge of the electron. Either way, it’s saying something about electromagnetism. There are other ways to slice the pie (see above), but these are stupider.

(I wouldn’t say $\alpha$ sets the strength of electromagnetism “relative to the other forces”, since we aren’t bringing them in here. For example, we don’t need to set $G = 1$ in this discussion.)

Posted by: John Baez on September 30, 2006 1:36 AM | Permalink
Read the post Dimensional Analysis and Coordinate Systems
Weblog: The n-Category Café
Excerpt: More gnarly issues: dimensional analysis and coordinate systems.
Tracked: September 30, 2006 7:46 AM
Read the post Light Mills
Weblog: The n-Category Café
Excerpt: The Crookes radiometer is also known as a 'light mill' --- a little glass bulb with a windmill in it, with vanes black on one side and white on the other. It puzzled Reynolds, Maxwell and even Einstein. Do we really understand it yet?
Tracked: July 29, 2008 8:18 AM