Planck's Units

Given the universal constants G (Newton's gravitational constant), h (Planck's constant) and c (the speed of light), we can infer

known as Planck's mass, length and time. We can use these as our base units of measurement for lengths, masses, times and units derived entirely from these. In these units, the numerical values of c, h and G all come to 1, so these units are widely used. Personally, I prefer to leave at least G in plain view – it carries information about reality's self-interaction.

On the other hand, c does indeed simply describe the relationship between the units we used for measuring length and time, which the space-time manifold sees as being of the same kind, so choosing units in which c is 1 does seem reasonable. Ideally, we'd express things in some way that doesn't involve c even if it isn't 1: it's an artifact of our model, just like the change-of-units constant we'd have had to introduce if Newton had distinguished between inertial mass and gravitational mass; or the one we'd have needed to introduce if we measured vertical distance in fathoms and horizontal distance in furlongs, with one furlong equal to 110 fathoms. The naturality of c as unit binds units into families among which ratios are powers of c; thus time and length are members of one family, along with area/time and the inverse of accellerations; while mass, momentum and energy are members of another family.

Whether h should be construed as merely a conversion factor (between wave vector – of dimension inverse length – and momentum) or as carrying information (about the fuzziness of reality) is not so immediately clear; but a careful analysis (see below) of de Broglie's and Planck's results does encourage the former. Taken together with the speed of light, this declares length, time and the inverse of mass to all be quantities of the same kind: our habit of measuring them in different units is no more meaningful than archaic usage's habit of measuring vertical and horizontal distances in different units, or the weights of grain and gold in different units. In contrast, I construe G as encoding the physics of the system: even if we chose to use units which make its value a unit, it's still a real quantity.

Charge and Current

How about a unit of charge ? Quite a good unit of charge is the charge on the electron, or a third of it, give or take sign: this is clearly a genuine irreducible quantity of charge, making it ideal as a unit, hence widely used. However, the form of Planck's units thus far is based on the constants in the field equations themselves, rather than on the bodies controlled by these: though good (by virtue of its real irreducibility), it is defined in the same spirit as the atomic mass unit, rather than in terms of the field equations.

On the other hand, electrodynamics furnishes us with the impedence of free space, Z0 = √(μ00) = 376.730 Ohm, and √(h/Z0) is a charge: 1.33e−18 Coulombs or 8.278 positrons-worth of charge. Indeed, squaring 8.237 and doubling, we get the inverse of the fine structure constant: α = e.e/(4.π.ε0.c.ℏ); c is 1/√(ε00) so ε0 is 1/(Z0.c); and 2.π.ℏ is h; yielding α = Z0.e.e/(2.h) or 1/α = 2.(h/Z0)/e/e. (That μ0 = √(Z0/c) also seems worth mentioning.)

For currents, we thus obtain a Planck unit as charge/time: this is about 9.81e24 Amps; that's a pretty big current. Multiplying that by Z0, the unit of impedence (which is of the same kind as resistance), we get a unit of potential (i.e. Voltage) equal to 3.6978e27 Volts.

Relating the units to the real world

A droplet of water (e.g. in mist) between a third and a half of a millimetre across has volume of order a few dozen nanolitres, making its mass a few dozen nano-kg, i.e. roughly the Planck mass. Such a droplet of water is large enough to contain a lively diversity of life-forms: for example, water bears, a phylum of animals officially called tardigrades, have masses typically less than the Planck mass. The periot and blanc, two archaic units of mass supposedly used by jewelers, are smaller than the Planck mass; as are the masses of many life-forms. How many atoms of hydrogen are there in a Planck mass ? Sort of a Planck's version of Avogadro's number: 32.575 milliards of milliards (I think ten to the eighteen has some better name). As it happens, 9 Planck masses come pretty close to 268 times the standard atomic mass unit.

Notice that the Planck momentum is c.√(c.h/G) which (if I've done my sums right) comes to 16.356 kg.m/s, an entirely sensible quantity on the scale of macroscopic creatures such as you and I – a fat cat running vigorously has roughly the Planck momentum, 5.76 stone mile per hour: is this the uncertainty of momentum of a cat shut in a box with a randomly dangerous device ?

Then, of course, we have the Planck energy which, at 4.9 giga Joules, is pretty big: converting to the standard nuclear unit of energy, the electron.Volt, we get 30 milliards of milliards of milliards of electron volts – the energy transfered to a mole of electrons as they pass through a potential difference of 50.82 kilo-volt; in the standard units of nuclear bomb yield, that's the energy released by the detonation of a little over one US ton of TNT.

Thus the …, mass, momentum, energy, … chain straddles real-world sized units in its familiar units. For contrast, note that the item on the …, time, length, … chain which is nearest to order 1 in SI units is .327 metre5 / second4: the Planck length and time are tiny even on the scale of nuclei (after all, a photon with wavelength the Planck length has the Planck mass – which is pretty huge by nuclear standards).

Just as c is a conversion between units of length and time, Boltzmann's thermodynamic constant, k, converts between those of energy and temperature; the consequent Planck temperature, energy / k, is about a third of 1033 Kelvin: (a.k.a. 1e33 K) which is very hot indeed !

Exploiting de Broglie

De Broglie's relationship between 3-momentum and wavelength (spacelike period) for matter combines with Plancks' relationship between energy and frequency (of light's quanta) to say, in Einstein's world, that the structure of (a particle of) matter is periodic along its own world-lines; and a mass m has period h/m/c.

Now, when m is Planck's mass, as discussed above, h/m/c is the Planck length. That means a Planck-mass mist drop's proper period is the Planck length (or time, as you wish); this is a factor of 1030 smaller than the physical dimensions of the mist drop, which has much to do with why one doesn't much notice quantum effects on raindrop-sized things. Smaller objects have longer periods; bigger objects have shorter periods. The Earth's period is of order 1e−66 metres, much shorter than the Planck length, and the Sun's is shorter yet: 1e−72 seconds. An electron's period, for comparison, is 2.4263 pico metre.

Back to our water-droplet: the smaller we make it, the larger its period will grow, and it's presently bigger than its period, so let's shrink it until they're equal. The volume of a sphere is cube of diameter times π/6; multiplying by density we get mass; solving for diameter and period both equal to d, we obtain 6.h/π/c/density equal to the fourth power of d, making the diameter 8 pico metres – which is rather smaller than a single water molecule (but bigger than an atomic nucleus). Doing the same sum for liquid hydrogen at 20 Kelvin, with a density of 70.99 gram per litre, I get 15.6 pico metre, which is a little under a sixth of a hydrogen atom's diameter. For gaseous hydrogen at zero Celsius, the matching calculation gives about 83 pm, which is smaller than the diameter of a lone hydrogen atom but larger than the separation of two hydrogen atoms in the H2 molecule; however, it's much smaller than the separation of hydrogen molecules in the gas at zero Celsius. A hydrogen atom's radius is about 40 thousand times its wavelength.

Hiding c

Scaling by c turns mass into momentum into energy; time into length; and each of these sequences is but the familiar portion of a chain stretching off at either end, as length.c = area/time, time/c = 1/acceleration, … and similarly for the momentum chain. On each chain it would be nice to chose a position to think of as the middle: the rest of the chain will then be the middle times successive powers of c, with the middle at zero power.

For the mass chain, on which we have 3 familiar quantities, the middle one of these looks a good place to chose as mid-point: and anyway I prefer to describe things in terms of momentum. Choosing the middle of the other chain is harder: I can argue for √(length.time) or possibly the fourth root of volume.time, and they could sound more reasonable than either of the two obvious candidates.

Fix, then, on momentum: and examine Newton's equation re-arranged as G = r.r.F/(m.M). We need to re-express the product of masses, m.M, as the inner product of two 4-momenta, p·P/c/c. Now, F is a rate of change of momentum so r.F has the units of speed×momentum. This gives us G/pow(c,3) as a length/momentum quantity: call this D. Combining with h, which is a length.momentum, we obtain √(h/D) as a momentum, √(h.D) as a length. (We can equally use D/c = G/pow(c,4), a time/momentum, and h/c, time.momentum, if we want to use time rather than length.)

Now, the definition of h is as the constant in the law of proportionality, E = h.f, between the energy of a photon and its frequency, f. But I want to work in terms of momenta, so consider p = h.f/c and notice that f/c is a 1/length, which means a gradient (equally p = h/wavelength). This fits well with the view, in quantum mechanics, of momentum as the differential operator in space-time (which has the dimensions of a gradient).

That gives me a hint that it'll be worth working in terms of length and momentum: but, of course, if I use h/c in place of h, I'd equally be working in terms of time and momentum. This is just a choice of whether h or h/c gets to be regarded as the fundamental constant, the other being derived, and is exactly equivalent to the choice of whether to treat length or time as middle on their chain.

Perhaps we can clarify the issue by considering what another factor of h (and some factors of c) will get us on the other side of length and time. We have momentum divided by h as a 1/length; dividing by h again we get the inverse of the product of area and momentum. One factor of c turns the product of area and momentum into a product of volume and force; a second turns it into a product of volume and power.

Refinements and variations

Note that taking Z0 and c as units makes ε0 and μ0 units also. Now, ε0 is the constant in the field equation of electrostatics; it's the constant of proportionality between the gradient of the field and the charge density. Contrast this with G, which is the constant of proportionality in Newton's gravitational law, which describes a special case – the two-body problem – whose analogue in electrostatics is the Coulomb law, which uses 1/(4.π.ε0) in place of G. When Newtonian gravitation is described by a field equation, the constant of proportionality between gradient of the field and mass density (analogous to ε0 in electrostatics) is 4.π.G (or its inverse). We might thus argue for replacing G in our system of units with 4.π.G. In Einstein's field equation for gravitation, the constant which shows up is 8.π.G, give or take some factors of c, so we might sensibly use this in place of G. Either way, we perturb the units that result by some factors of two, π and their square roots.

Likewise, we could try to justify using Dirac's constant (a.k.a. ℏ = h / 2 / π) in place of Planck's constant; or half of Dirac's constant (the spin of those Fermions with the least non-zero spin of all particles). These choises, likewise, throw in some stray factors of two, π and their square roots. However, these choices are more in the spirit of using a unit based on the actual properties of observed particles (the least-spin Fermions), whereas Planck's constant arises naturally in a field equation. We do get Dirac's constant in field equations – it appears naturally in Schrödinger's equation, and it's the constant of proportionality between the gradient and momentum operators in the field equations of Quantum Mechanics – so this isn't necessarily a fatal objection.

Here, then, are a few of the alternative systems (C = Coulomb, e is the charge on the positron) of Planck-like units we can come up with, using choices I could justify as above:

c, Z0, G, h
c, Z0/4/π, G, h
c, Z0, 8.π.G, h
c, Z0, 8.π.G, ℏ/2
c, Z0, 4.π.G, ℏ

See also

Sabine Hossenfelder gives a good account of the sense in which the Planck length is minimal; it's not a granularity of space-time, just a length-scale below which we're unlikely to be able to make any kind of measurement – since the energy a thing must have, for its wave-nature to have a resolving power as fine as the Planck length, is of course the Planck energy, whose Swarzchild radius is (of course) the Planck length, so you would destroy any information that did exist on such a length-scale by hitting it with an evaporating black hole. She also makes the good point that it's a length scale way below our present experimentally accessible range so, by the time we get anywhere near to being able to probe it, we may have found other complications to the story.

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