# Maru the Cat does dimensional analysis

Here is a picture of (I think) Maru the cat playing in a bag. He loves bags.

Here is the same picture of Maru, at half the size:

Now imagine that Maru is a physicist and the pictures are not pictures but instead windows into the universe he occupies, separate from ours with (possibly) its own unique set of physical laws. The only difference between the two universes is that one has the lengths of everything reduced by a factor of 2.

Can the parallel versions of Maru tell which universe they're in - the smaller or the larger? Or if you want to imagine what you might do, suppose that in some Kafkaesque way you find that you wake up one morning and everything if twice the size it was the day before, including you. Could you tell? After all, all the rulers are twice as big too.

Physicists don't generally directly worry about questions like this, but they're actually very similar to questions about whether fundamental constants like the speed of light are truly constant. If the speed of light doubled overnight, what would change? Maybe lots of things - the famous Einstein relation E = mc^2 would seem to imply that the ratio between a fistfull of matter and the corresponding quantity of energy would change by a factor of four, which would certainly have an observable effect in things like the nuclear reactions that power the sun.

Or, maybe simultaneously everything suddenly got less massive by a compensating factor and the change in m balances the change in c. Or, maybe the rate of flow of time (whatever that might be) changed or maybe length scales changed, or who knows?

This isn't idle dorm-room philosophy. Physics needs to be able to deal with quantities that change in time, be they truly fundamental or not. There's actually some real subtlety involved, but the essentials revolve around just what we mean by units and dimension.

Units are specifically defined increments of a physical quantity. Meters, feet, angstroms, miles, astronomical units, and parsecs are all measuring the same thing in different increments. Likewise liters, gallons, and barn-megaparsecs. And teslas and gauss, and so forth. You can use whatever units are convenient, and keeping track of units is one of the first things you learn in freshman physics. In theoretical physics it's very common to pick a system of units where various fundamental constants are equal to 1. For instance, with c = 1, Einstein's mass-energy relation becomes the rather zen-like E = m.

Dimensions are the thing being measured, independent of units. Length in space, duration in time, volume, magnetic field strength, momentum, and so forth.

The crucial test of measurement in physics is the variation in ratios of numbers with identical dimensions. If everything in the universe doubles in size, the ratio of your height to the height of a ruler will stay constant. If only you double in size, the ratio of your height to the ruler will double also, leading to a possible NBA career. (As a naturalized Texan, I believe I should now proffer support for Dallas. Go Dallas!) Ratios of numbers with identical dimensions are said to be dimensionless, and they're the interesting quantities.

So we don't look for variation in the speed of light as such. We look for variation in dimensionless numbers involving the speed of light. The most famous of these numbers is probably the fine structure constant, which is the square of the electron charge divided by the product of Planck's constant and the speed of light. (Maybe with another constant in there too, depending on the system of units you're using for electric charge. The value of the constant, like all dimensionless constants, is that its numerical value is identical regardless of the system of units.)

The fine structure constant is approximately α = 0.0729735..., which happens to be a hair under 1/137. Why this particular value? Nobody knows, but as a dimensionless number appearing all over physics it's a great test subject for investigating the possible change in the laws of physics (including the speed of light) over time. So far there's no real indication that it has ever changed, which is a nice thing to know.

But if it ever did, we could learn about it along these lines without having to worry much about changes in our lab equipment.

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Of course, the fine structure constant is not quite dimensionless, since it has an anomalous dimension and changes under the renormalization group flow. The 1/137 value is what it takes in the far infrared - ie at macroscopic scales.

There's a nice and quite elementary discussion of "dimensional analysis and field theory" by Stevenson, which is discussed at ParticlePhD.

Also interesting is the discussion at the nCafe (continued here) which shows that being dimensionless is not the property that makes the fundamental dimensionless quantities special.

I was always given the example of sausages hanging from strings in a butcher shop.

The strings would now have 4x the area holding 8x the weight.

Unless you can come up with a systematic way to compensate for that?

It's most definitely that classic ham, Maru-cat. :)

Cool post, thanks!

By speedwell (not verified) on 12 Jun 2011 #permalink

Marion: if everything doubled in size, including the constituent atoms, you still have the same number of atoms comprising the sausage. Assuming the mass per atom doesn't change, the weight won't change either.

If you double all of your lengths, you would have to double all of your times, since in special relativity the position 4-vector is (ct, x1, x2, x3). If you didn't modify the 0 component the same way that you modified the other three, then you could Lorentz boost into a frame where you could tell what had happened. That means you must increase either the value of c or the scale of time (or some combination thereof). As pointed out in the original post, changing the value of c will have observable effects, so t is what must change.

But I still think you will run into trouble when you go into the spectroscopy lab. The Bohr radius is given by a0 = α/4πR where R is the Rydberg constant. If you want your Bohr radius to remain numerically the same in the new universe (which must be the case if you want the number of atoms to remain the same; otherwise Marion's objection @1 would apply), then you must change either the fine structure constant (which will have observable effects elsewhere) or the Rydberg constant, which will alter the hydrogen spectrum.

By Eric Lund (not verified) on 12 Jun 2011 #permalink

I accept the basic point about the relational nature of dimension-holding constants in general. However: it is pretty clear that G can be decoupled from other factors and could reasonably vary with respect to other constants. Think: if "G were half the current value" we can indeed imagine other things, all of atomic physics etc. just being the same, but for example gravity at Earth's surface (well, supposing it kept it's same diameter given slight relaxing effect) would be half of current value, and so on. So the purist approach to the idea is misleading, isn't it?

And after all that we have discovered that the speed of light has never changed, Planck's constant has never changed, the electron charge has never changed, and so on. They really are all physical constants.

So, what did change? Why the value of pi. There's a reason the Old Testament referred to pi has having a value of three. Back then, it did. By the time of Archimedes, it had grown much closer to its present value, but was still closer to his lower bound than the value accepted today. Scientists estimate that the value of pi was barely one when the dinosaurs stalked the earth, which is perhaps why they never developed plane geometry.

According to cosmologists, at the dawn of the universe the value of pi was actually much smaller than Planck's constant which is why theorists speculate that circles and spheres did not exist until well after the inflationary period and circular motion may have been impossible until the first neutrinos were formed.

In fact, we can view the evolution of the universe as a consequence of the increasing value of pi. So, will we be celebrating Pi Day on March 15th any time soon? Probably not, but maybe. It turns out that we know a lot less about this familiar "constant" than we thought.

So, what did change? Why the value of pi. There's a reason the Old Testament referred to pi has having a value of three. Back then, it did.

Now that is funny.

Wouldn't you be able to tell by dropping something from eye level or by falling down? Unless you're changing gravity in some way too it'd take longer for the object to hit the ground if you're now twice the height.

By Lynxreign (not verified) on 13 Jun 2011 #permalink

@Kaleburg - :D :D

Lynxreign, that gets right to the heart of what I said about "G." We really could likely tell that G was weaker by the lower gravity on Earth. But if time and space itself *simply as coordinate descriptions* changed, then it wouldn't matter. It's like showing the same movie on a bigger screen and/or sped up or slowed down, the relative events are all the same - indeed most thinkers say that in Nature there isn't really a "screen standard" to define the "bigger" or "slower" at all! (Compare "absolute space", "absolute motion" etc.) But of course it might not be that simple.

BTW Matt, could you add "remember personal info" so we don't have to keep typing? tx, or am I missing something because of using Linux SeaMonkey?

Ya da, belki aynÄ± anda her Åey aniden bir telafi faktÃ¶rÃ¼ ve m deÄiÅikliÄinden daha az bÃ¼yÃ¼k var c. deÄiÅim dengeleriYa da, belki zamanÄ±n akÄ±Å hÄ±zÄ± (yani ne olursa olsun) deÄiÅtirilmiÅ ya da belki uzunluk Ã¶lÃ§eklerinde deÄiÅmiÅ, ya da kim bilir?

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