Built on Facts

In the United States generally and to a much lesser extent in the UK and a few other countries you’ll see some very old-school units of measurements. Miles, yards, pounds, fortnights, pints, gallons, and numerous others. Most of the rest of the world uses metric units, the primary variant of which is called SI units used almost universally in the physical sciences. You know, meters, kilograms, that sort of thing. My American patriotism aside, SI is of course a lot easier to work with. Kilometers to meters is a lot easier than fathoms to nautical miles (or whatever).

But it’s not perfect. In a sense it would be more natural to define units of measurement in terms of physical constants rather than physical constants in terms of whatever human-sized conveniences happen to be in use. For instance in electrostatics we have (as an example) the force between two charges:

Where the q are the charges in the usual SI unit of coulombs, and that epsilon is the electric constant (itself often called the vacuum permittivity). The permittivity is some emperically measure decimal value with no clean form.

Or you can pick out a different system of units. In Gaussian units the basic unit of charge is finagled in such a way so as to cancel out that ugly proportionality constant, like this:

Cleaner, no?

This is a choice much beloved of theorists, for good reason. Nothing of theoretical interest is gained by carrying around awkward proportionality constants, and sometimes they even manage to obscure the fundamentals of the theory. My E&M instructor this semester is using Gaussian units, and in the classical field theory course he teaches he goes farther and sets c = 1. Newer editions of Jackson apparently use standard SI, much to his dismay.

In other branches of physics this sometimes gets taken to extremes. You’ll see things like G = 1, h = 1, e = 1 (the electron charge, not Euler’s number), α = 1 (incompatible with c = 1, however), and a host of other possibilities.

I should note that all this is still technically metric, but it’s not SI. In classes like this one we so rarely use actual numbers that it makes little difference. For the purposes of actually getting numbers in order to compare to experiment you simply do the conversions as the very last step. No point in dragging in weird constants before that. The classical field theory class I mentioned was actually something I took last semester – I used a calculator exactly once.

Comments

#1 MartinB
January 23, 2009

Although used to setting every constant to 1, it has one drawback: There’s no way to check for consistency using units on both sides of some equations.
#2 ObsessiveMathsFreak
January 23, 2009

Custom units and unit systems are useful, and indeed prudent, when one is working in a single field. However, they lead to problems if one is working in more than one field, or if one is attempting to cross-field work.

The best example of this is the physics lecture itself. Undergraduates are exposed to all branches of physics during their modules and having a consistent framework throughout is I think an advantage. Yes, you do end up with constants everywhere, but you also have surety that mass will always be in kilograms, distance in meters, etc. It’s a great thing to have.
#3 Uncle Al
January 23, 2009

English units discriminate against women. Their thumb joints are not inches, their shoes are not feet, arms-extended fingers to noses are not yards, and there is never a Ridgid calendar on the wall. If you don’t see that calendar don’t expect anybody knows a chamfer from a herpolhode.

(To be fair, when a skosh is too big there is only one source for a fiduciary RCH.)
#4 Matt Springer
January 23, 2009

That’s very true. In that field theory class last semester I did feel disoriented more than once by the inability to easily check to see that the units worked outl
#5 dWj
January 23, 2009

I once saw a string theorist announce, at the beginning of a talk, that he would be setting G, h-bar, c, and “occasionally 2 pi” equal to 1. I didn’t follow the details of the talk carefully enough to figure out whether he was kidding.
#6 Brad
January 23, 2009

@#1 – that’s not quite true. Given that the equation as shown is an equality, you’d have to measure force in units of charge^2/d^2; similarly gravitational attraction would be in units of mass^2/d^2.

I suspect that this is one of those areas where thinking about it a little or a lot doesn’t make your head hurt, it’s the middle ground of thinking about it a fair bit that hurts.
#7 Winter Toad
January 23, 2009

When I was in graduate school I sometimes had to check my measurements against theoretical models written 40 years earlier. I spent quite a lot of time making sure I could safely and consistently travel back and forth between lab measurements and theoretical papers that computed similar but not identical things in either SI, CGS-ESU, or CGS-EMU. I did not enjoy that.
#8 Eric Lund
January 23, 2009

The choice of units is always a matter of convenience. Most of the time, it is convenient to use the same units the rest of the world uses, hence the standard SI. But sometimes it is more convenient to choose your units so that awkward constants drop out. Just make sure you know how to convert between theoretical units and SI units if you ever need to put real numbers in your equations.

Going between CGS and SI units in E&M is even more tricky. It’s not just keeping track of the odd factors of c, ε₀, and μ₀. (A useful identity here: c²ε₀μ₀ = 1.) Sometimes there is a factor of 4π to keep track of.
#9 llewelly
January 23, 2009

Newer editions of Jackson apparently use standard SI, much to his dismay.

Sweet. Perhaps that will encourage people to do the smart thing and keep using the old editions.
#10 Chris' Wills
January 24, 2009

Well I’ll stick with the three Fs + S

Furlongs, Fortnights, Farenheit and Scruple.
#11 Alex
January 24, 2009

Working in theoretical optics, I prefer Heaviside-Lorentz. Sure, there’s still a factor of 1/4pi in Coulomb’s Law, but I never use Coulomb’s Law. I do, however, use D = E + P and H = B – M at times. And putting 1/c in front of time derivatives means that when I go to harmonic waves everything is just k_0, making life easier.
#12 Pseudonym
January 28, 2009

Speaking with my engineer hat on, there’s one extremely good reason why we don’t work in Planck units as a matter of course: We don’t know what they are to a high enough precision.

The Planck length is known to a relative uncertainty of about 10^-5. By contrast, the metre is known to a relative uncertainty of about 10^-11. This means that if we worked in Planck units, the measured length of the Golden Gate Bridge would be plus or minus about half an inch.

That’s more than enough to cause serious problems in the sorts of experiments we have to do today.
#13 Pseudonym
January 28, 2009

Oh, one more thing: If you really want to trick someone, ask them this question: Which weighs more, a pound of gold or a pound of feathers?

(Extreme kudos to anyone who gets the right answer.)