Dimensional Analysis

September 21, 2006

Posted by John Baez

$MathML-enabled post (click for more details).$

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.

$MathML-enabled post (click for more details).$

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.

So, let’s start with dimensional analysis!

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 $M^{0} L^{0} T^{0}$ ? 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 $α = e^{2} / 4 π ϵ_{0} ℏ c ≃ 1 / 137.036$ is a dimensionless quantity built from quantities that seem very fundamental - the electron charge $- e$ , the permittivity of the vacuum $ϵ_{0}$ , Planck’s constant $ℏ$ 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 $α$ 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 $α$ 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 $α$ by changing our definitions of $e, ϵ_{0}, ℏ$ and $c$ . So, there must be some other reason why $α$ 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

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/944

69 Comments & 3 Trackbacks

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

$MathML-enabled post (click for more details).$

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 $ρ$ , juggling the units gives

$c = {(gh)}^{1 / 2} f (\frac{h}{λ}),$

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 $ρ$ .

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

$MathML-enabled post (click for more details).$

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 $\ln (2)$ 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 $4 π$ 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

$MathML-enabled post (click for more details).$

Great posts, Terry! There are indeed many subtle distinctions that come in handy here, like “ $ℝ$ ” versus “torsor for the additive group $ℝ$ ” versus “torsor for the multiplicative group $ℝ^{*}$ ” 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 $ℝ^{*}$ 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

$MathML-enabled post (click for more details).$

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 $Hom$ -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 $ℝ$ .

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

(1)

ℝ \overset{≃}{\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 $ℝ$ -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

$MathML-enabled post (click for more details).$

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

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

A choice of isomorphism

(2)

ℝ^{3} \overset{≃}{\to} V

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

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

$MathML-enabled post (click for more details).$

Garrett the FQXi grant winner writes:

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

$e : TM \to T$

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

$g (v, w) = η (e (v), e (w)) .$

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 $η$ .

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

$MathML-enabled post (click for more details).$

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

$ℂ [[banana]]$

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 $ℏ$ - and instead compute $\exp (iHt / ℏ)$ .

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

$MathML-enabled post (click for more details).$

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)

$A \frac{ρ v^{2} + P_{atm}}{g}$

where $A$ is the cross-sectional area of the tank, $g$ is the local acceleration of gravity, $ρ$ 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

$MathML-enabled post (click for more details).$

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 $ℝ^{n}$ ).

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

$MathML-enabled post (click for more details).$

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 $\tilde{r} = r \frac{c^{2}}{G m}, \tilde{t} = t \frac{c^{3}}{G m}$ . 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

$MathML-enabled post (click for more details).$

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 $η$ .

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

$MathML-enabled post (click for more details).$

Arun wrote:

…since we could also get any value of $α$ by changing our definitions of e, $ϵ_{0}$ , $ℏ$ and c

A demonstration of this claim would add clarity.

I didn’t mean anything profound or nonobvious by this. Right now $ℏ$ is defined to be $h / 2 π$ . Say we change the definition to $h / π$ . Then $α = e^{2} / 4 π ϵ_{0} ℏ c$ gets halved.

This may seem very silly. Ones first reaction is “But now you’re talking about a completely different concept of $α$ - 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 $α$ .

Both $α = e^{2} / 4 π ϵ_{0} ℏ c ≃ 1 / 137$ 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 $α$ seem more “fundamental” than $C$ ? Of course it’s because $e, ϵ_{0}, ℏ$ 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

$MathML-enabled post (click for more details).$

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

$MathML-enabled post (click for more details).$

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

$MathML-enabled post (click for more details).$

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

$MathML-enabled post (click for more details).$

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 $ℝ^{3}$ .

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

$MathML-enabled post (click for more details).$

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 $ℝ^{3}$ (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 $ℏ = 1$ .”

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

$MathML-enabled post (click for more details).$

They are not necessary if I can ensure that once I have fixed that isomorphism of the world around me with $ℝ^{3}$ (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

$MathML-enabled post (click for more details).$

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
google
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

$MathML-enabled post (click for more details).$

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 $ℏ$ , $c$ 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 $ℏ$ ? Presumably this means changing the unit of action that we use to quote the size of the action $\frac{e^{2}}{4 π ϵ_{0} c}$ . It obviously doesn’t change the actual physical quantity of action currently denoted by $ℏ$ .

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 $α$ to be 2 times the old $α$ , or 2.9 times the old $α$ , or 299,792,458 times the old $α$ , 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 $α$ , 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

$MathML-enabled post (click for more details).$

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 $k \in ℝ$ . But being careful reveals that all $k > 0$ and all $k < 0$ 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 a

The n-Category Café

Skip to the Main Content

September 21, 2006