Copyright © 2001,2002 by Leonard Cottrell. www.planck.com All rights reserved.
The Planck quantities are a system of natural units which arise purely from gravity and light. They don't depend on the mass of any one particular thing, like an electron or the universe.
Reference: Max Planck: 'Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899).
Last year a question came up here about basing natural units on the mass of the universe (the whole thing not just the "observable" part from which light has already reached us and whose extent changes daily.) According to prevailing models, the mass of the universe is infinite, as is its extent. Even if, contrary to the accumulating weight of evidence, the mass were finite, it's not clear how we would estimate it.
Of course one can always choose some favorite particle, like an electron or a hydrogen atom, put it on a pedestal, and make its mass "the" mass. This can provide a system with some natural basis which is appropriate to certain specialized contexts. Making one particle the kingpin turns out to be too narrow an approach, as Planck suggested in his original 1899 paper. By contrast, the Planck units are intrinsically present wherever there is light and gravity, and, to put it in a nutshell, they are only general-purpose set of natural units in town.
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The picture of light as a swarm of individual quanta each carrying a definite amount of energy was worked out in 1905 by Einstein to explain why different colors of light produce different voltages in a photoelectric tube. Light of the same wavelength always produces the same voltage whether it's bright or dim and shorter wavelength light produces a higher voltage. These effects had been observed but not explained. Einstein's Nobel prize was for explaining them and he managed it by modeling light as coming in small packages of energy.
The energy carried by an individual bundle is measurable in work terms—force×distance. A bit of light can, for instance, move an electron for a certain distance overcoming some definite force and the energy so delivered can be rated as work by multiplying the distance by the force. Indeed in the photoelectric tube this is exactly the sort of work a photon of light performs.
As far as we know, multiplying the wavelength of any quantum of light by the energy which it carries gives the same answer for every quantum of light. This is pretty remarkable. It means there is an energy×distance product that is pervasive all over the universe, just as there is one standard speed prevalent all over the universe. There are some technical conventions but they don't bear on the basic notion. Unless I specifically say otherwise I mean angular-format "vacuum" wavelength (the one the light has in empty space). What I'm saying here is true however one decides to standardize things.
Photons of light with shorter wavelengths carry proportionately more energy so there is a tradeoff and what you get by multiplying the two is the same for every photon in the universe, as far as we know. So this energy×distance quantity is an omnipresent universal thing, a celebrity among physical constants.
Not only does light determine a definite energy×distance quantity but any two masses determine a product of energy with distance as well! Picture two truck-sized round balls placed some distance apart and imagine the work of dragging them away from each other to infiinity or at least so far apart that their attraction is no longer of consequence. The distance and the work are inversely proportional—if the bodies are placed twice as far apart then the work of separating them.will be only half as much. The work of separating them out to infinity is called their binding energy. The product of separation with binding energy depends only on mass and is the same no matter how near or far apart the two are.
Gravity associates to every amount of mass the energy×distance product you would measure using a pair of twin bodies each having that mass. It's something that is not hard to calculate. The energy×distance associated with a mass M is GM2, the square of the mass multiplied by the gravity constant G.
Conversely, gravity associates with any energy×distance product the mass twin copies of which would have that product for their binding energy×separation. To find the mass associated to some energy×length quantity, you just take the quantity, divide it by G, and do square root.
Planck mass is the mass associated by gravity to the energy×length constant present everywhere in all light.
It turns out to be flea-sized: roughly a millionth of an ounce.
A billion times the Planck mass is the mass of a large dog.
It's about 48 pounds. We could invent a unit of mass called the dog, and then the Planck mass---this quantity built into nature all over the universe, would be a billionth of a dog. But it happens to be roughly equal to a traditional mass unit called the talent. On the whole it seems safer to call it by the traditional name.
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There is more than one approach to the Planck quantities. We've just encountered the Planck mass and we could get all the other units from that one: Mc2 defines the natural unit of energy and h-bar is a universal ratio of frequency to energy so that ratio then gives us an intrinsic frequency thoughout all nature. That frequency can be taken as our clock and define a time unit for us, the Planck time. The Planck length is the distance light travels in unit time, and so on. All the units arise in this way from Planck mass.
So one intuitive approach is to assimilate the notion of a 48 pound dog-sized talent and think of the natural unit of mass as a billionth of a talent, and imagine deriving the other natural units from the mass. Or, since the natural unit is the mass of a flea, begin by simply imagining a flea.
But there is another way that begins by studying the force unit belonging to the system. This huge force — Planck force — is also implicit throughout nature and comparatively accessible so I am going to discuss it next. What we now call the Planck force was identified and described as early as 1874 by the Irish physicist George Johnstone Stoney (who also first identified the natural unit of electric charge and named it the electron). Planck did not work out his system of units until 1899.
This universal standard force is fundamental to all gravity. In general relativity (where Planck force is usually written c4/G) it is the constant appearing in the main equation which relates the curvature of space-time to the energy density causing it. But, happily for us, we can picture this force as it appears in immediately accessible things without getting involved with the equations of general relativity!
Once you make the speed of light your scale and start expressing speeds as fractions of it, the force appears at the surface of every planet and in every binary star-pair. This doesn't take a lot of algebra to show—in fact something like a nursery rhyme sums it up:
A planet sets an upper bound
On speeds of orbits circling round.
Raise her limit to the fourth,
Divide her weight, and get the Force.
Assuming enough roundness, any planet has a well-defined weight (in its own surface gravity) and defines a top speed for circular orbits: namely the speed of a surface-skimming orbit. No circular orbit around the planet can be faster than one skimming the surface. This speed limit is also a practical limit on surface travel. You can't exceed it without leaving the surface and swinging out into space. Setting the speed of light equal to one gets us the planet's characteristic surface speed as a pure number. Dividing the planet's weight by the fourth power of the surface speed limit gives the Planck force.
Also in readily understandable fashion, the Planck force is present in all binary star-pairs and is particularly visible in circular binaries. Again, a nursery rhyme can sum it up if we prefer that to an algebraic formula:
Between a circling pair, the pull,
Divided fourthly by their gait,
And by both fractions of the whole,
Is universal standard weight.
Any two bodies in circular orbit have a constant combined speed which is the speed it looks to each one that the other is traveling and which, to an outsider, looks like the sum of their two individual speeds. Each body has a definite share of this combined speed—if the two have equal mass they split it half and half, otherwise the body with smaller mass has the larger share of speed. The pull between the two, divided by that speed's fourth power and by each one's share, is the Planck force.
In our units this force always comes out to be a fixed definite (1040) number of tons. It is all-pervasive, indispensible to how the universe is put together, and (like the speed of light and Planck's h-bar) stands out plainly in nature. Defined in terms of G and c the Planck force is c4/G. It plays the usual role of unit force in the Planck units: Planck force applied to Planck unit mass produces unit acceleration.
Once we know the force and the speed of light there are obvious units of power (force × speed) and frequency (from power courtesy h-bar), and the rest of the natural units flow from these. So this is a second approach. The first approach was by way of Planck mass—once we find the natural mass unit in gravity and light we can get all the others from it—and the second was by way of the force: it arises in ordinary orbits of things around us and the rest of the natural units can be derived from it too.
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I'm reluctant to appeal to expert practice (even the practice of the very smart people known as string theorists) because we ought to be able to see these things for ourselves. But there is probably something to the fact that anyone who has anything really basic to say about the universe nowadays is pretty much forced to use Planck quantities as units. Speaking in Planck terms just seems to make things more intuitive, easier to say, and less trouble to compute. It is as if nature "wanted" us to use those units in discussing time, space, matter and their beginnings.
As it is in modern cosmology, so it is in string theory, currently the most promising attempt to understand matter and its forces.
This has prompted some of us, myself included, to try out Planck quantities, particularly certain power-of-ten variants, to see how they work both at the highschool and college science level, and in everyday life. They work unexpectedly well, partly because of accidents like the fact that a Planck mile — ten-to-thirtyeight Planck lengths — happens to be within half a percent of the ordinary US statute mile. There simply is no subjective difference between Planck miles and what we use. When you see a roadsign listing miles to places, you may as well assume it is in Planck miles —1038 natural lengths each. Several such fortunate accidents (the Planck ton's similarity to our ton force, the Planck minute's similarity to our traditional minute) help to make human-scale power-of-ten variants of the Planck quantities easy to visualize.
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