Planck units
30Es9
Essays
A look at a complete system of units based on the Planck units.

Planck units.
What would a system of units based not on totally arbitrary criteria be like, but rather instead based on fundamental quantities of the Universe?

For an idea, we pick the Planck units (with other constants thrown in). For those who don't know, the Planck units [footnote 1] are:

name symbol expression value
Planck mass mP (hbar c/G)(1/2) 2.177 x 10-8 kg
Planck length lP (hbar G/c3)(1/2) 1.616 x 10-35 m
Planck time tP (hbar G/c5)(1/2) 5.391 x 10-44 s
Note that these values, like the fundamental constants h, G, and c, which compose them, are independent of the system of units in which you express them. Thus they make a good base for a system of units.

A system of units needs a set of fundamental units, from which all the other units can be derived. In SI, these fundamental units are:

quantity unit symbol
length metre m
mass kilogram kg
time second s
electric current ampere A
thermodynamic temperature kelvin K
luminous intensity candela cd
amount of substance mole mol
plane angle [footnote 2] radian (rad)
solid angle [footnote 3] steradian (sr)

Now for our fundamental length, mass, and time units we can just use the Planck units directly. We will call these units L, M, and T respectively.

For electric current, we can simply use charge divided by the Planck time, but we need a charge. An obvious (unit independent) choice is the magnitude of the charge on an electron, e.

For thermodynamic temperature, we need to relate a fundamental constant concerning temperature to the rest of the units. An obvious choice is the Boltzmann constant, which has units of energy divided by temperature. So we can choose our energy unit (which we will discover later) divided by the Boltzmann constant (k) as our temperature unit.

Luminous intensity is a tricky one. There are only three luminosity units in SI (cd, lm, lx), only one of which is fundamental (cd). Since these are specialized units that are only intended to be used under extremely narrow situations (this is clear when you look at the definition of the candela), we can make the sweeping generalization that all applications using these units can in fact use the derived units for energy and power (combined with other units). For instance, the Planck equivalent of the candela would just be the power unit; the lumen equivalent would be the power unit times the steradian, and the lux equivalent would be the power unit times the steradian divided by the square of the Planck length.

Then we have amount of substance. This one's easy; we eliminate the unit. Wherever mol would have appeared, it is just replaced with the appropriate number (6.02 x 1023).

So now we have our fundamental units (less the luminosity units), with SI conversions:

quanity symbol value
length L 1.616 x 10-35 m
mass M 2.177 x 10-8 kg
time T 5.391 x 10-44 s
current C == e/T 2.972 x 1024 A
temperature E == M L2/T2/k 1.415 x 1032 K
plane angle rad 1 rad
solid angle sr 1 sr

Now we can start getting derived units. There are an awful lot of derived units, but only some of them are named. (For instance, area, volume, density, momentum, entropy, viscosity, intensity, etc., are all perfectly good -- and even frequently-encountered -- units, but they happen to have no unique names in SI.) For simplicity's sake, we'll only go over the derived units which have names in SI (with conversion factors). First, the mechanical units:

quanity symbol value
force M L T-2 1.210 x 1044 N
energy M L2 T-2 1.956 x 109 J
power M L2 T-3 3.629 x 1052 W
frequency T-1 1.855 x 1043 Hz
pressure M L-1 T-2 4.635 x 10113 Pa (!)
And the electromagnetic units:
quanity symbol value
capacitance M-1 L-2 T4 C2 1.312 x 10-47 F
charge T C 1.602 x 10-19 C [footnote 4]
electric conductance M-1 L-2 T3 C2 2.434 x 10-4 S
inductance M L2 T-2 C-2 2.215 x 10-40 H
magnetic flux M L2 T-2 C-1 6.582 x 10-16 Wb
magnetic flux density M T-2 C-1 2.520 x 1054 T
resistance M L2 T-3 C-2 4.108 x 103 O [footnote 5]
voltage M L2 T-3 C-1 1.221 x 1028 V
And, finally, if you want to get really silly, how about the radiation units?
quanity symbol value
activity T-1 1.855 x 1043 Bq
absorbed dose M L2 T-2 E-1 1.382 x 10-23 Gy
dose equivalent M L2 T-2 E-1 [footnote 6] 1.382 x 10-23 Sv
Note that we've left out the luminosity units (cd, lm, lx) for the reasons stated above.

Note what we've really done here: We've made a system of units out of the fundamental constants hbar, G, c, k, and e, with some choices of units, but nothing so arbitrary as picking random values for units. (The SI kilogram is still dictated by the mass of an archetype bar in France.)

Footnotes.
1.
Note that the Planck units are traditionally expressed in terms of hbar, where hbar == h/(2 pi).

footnote 1
2.
A unit system can either choose charge or current (charge divided by time) as the fundamental unit. SI arbitrarily chooses current, and so for simplicitly and analogy we will use current as well.

footnote 2
3.
The radian and the steradian are technically supplementary units, since they have meaning outside of the unit system.

footnote 3
4.
By design, we get back the electronic charge e as our base charge unit.

footnote 4
5.
The SI symbol for ohm is a capital Omega, but with ASCII I'm forced to write O.

footnote 5
6.
Absorbed dose and dose equivalent (Gy and Sv in SI, respectively) have the same units but different by a specified dimensionless weight, which represents the damage done to biological tissue.

footnote 6
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