the Planck set

The present writing is a collection of essays concerned with how Planck quantities are perceived. Certain features of the quantities will be emphasized. One feature is their involvement in everyday experience. Another is the opportunity they provide for a fresh look at physical laws and data. Another is their concreteness: One of the power-of-ten multiples of the Planck length happens to be about a mile and another is the thickness of a penny--if there is a penny in your pocket you have with you a decimal Planck cousin. Their concrete reality can be emphasized in the way we handle the quantities: decimal multiples (such as a penny's thickness) will be identified and used descriptively. Other aspects will be discussed later, after the quantities have been introduced. Much of what I have to say is apt to be familiar to anyone with a background in general physics, though some ramifications may have escaped notice. In any case, the focus here is not on the latest theories and discoveries, but on something which is considerably more modest and which runs in a contrary direction. The concern is with more thorough assimilation of what is long-standing and well known--and with broader awareness of the fundamental proportions in nature.

a range of quantities for comparison

The Planck quantities form a coherent set including one constant of each physical type: one length, one interval of time, one speed, and so on. Members are interrelated in simple, direct ways and play fundamental roles. The Planck speed, for instance, simply consists of traveling the Planck length during the Planck interval of time--and it is the speed of light.

There will be an as-yet-unspecified connection between our units and the Planck set, both of which will be described in greater detail later. What is provided in the next table is a range of approximately related quantities for comparison. Outline letters serve as a self-explanatory notation for members of the Planck set: is the force, the length, and the time. If needed, this provides ways of expressing other quantities symbolically: For example, although there already is a conventional symbol c for the speed of light, if necessary we could also represent it as / (that is, as Planck length per Planck time). Since the outline is already committed, a script is used for the Planck temperature. Some names are suggested which will help us to use these quantities descriptively later on, and rough equivalents are provided to aid in remembering them.

By coincidence the US statute mile (offically 1609.344 meters) is very nearly a power-of-ten multiple of the Planck length. In fact it comes within half a percent. The average of the US and British gallons is approximately 4.2 liters, close to a power-of-ten multiple of the Planck volume.

	quantity	conventional value	rough equivalent	possible name
times:
	10	0.539056(34) second	0.54 second	tick
	10	53.9056(34) second	0.9 minute
	10	5390.56(34) second	90 minutes
lengths:
	10	1.61605(10) millimeters	thickness of a penny	pennywidth
	10	16.1605(10) centimeters	thumb to forefinger	span
	10	1.61605(10) kilometers	statute mile	mile
volume:
	10	4220.50(81) cm	avg. US&Brit. gallons	gallon
forces:
	10	12.1056(15) newtons	2.72 pounds	stone
	10	12105.6(15) newtons	2720 pounds	ton
energy:
	10	1.95633(13) joules	2.0 joules	spanstone
temperature:
	10	1.41695(10) kelvin	1.4 kelvin	degree

The table derives from the listing of fundamental physical constants published in the Handbook of Physics and Chemistry, particularly from the values for Planck quantities given by the Handbook. Estimated standard deviations in the last two digits are shown parenthetically. For definiteness we will specify that a tick shall be exactly 0.54 seconds, and define a span as the distance light travels in a billionth of a tick. This is analogous to how the meter is defined in the metric system--as the distance traveled by light in a specified interval of time. Other lengths are defined accordingly--a mile as ten thousand span and a pennywidth as one hundredth of a span. This means that in one tick light travels a hundred thousand miles--it can be pictured as going four times around the earth in that time. It also means that our mile has an exact metric equivalent. There seems little need to spell this out in detail, or to belabor the issue of convertibility, but our mile is exactly 1618.8792732 meters--hardly something one would expect to need to know.

What is here called a tick corresponds to a recognizable tempo. Setting a metronome to beat at intervals of 0.54 second means setting it at or near 111 beats per minute, in a range designated tempo moderato on some metronomes. It is an easy rhythm to snap fingers in time with--one indication that the interval is humanly scaled. The speed of light can be pictured as a flash traveling four times around the earth with each snap of your fingers. The 0.54 second interval comes within two tenths of one percent of being a power-of-ten multiple of Planck time. Inasmuch as the tick is decimally related to the Planck time, so likewise are the longer intervals 54 seconds (0.9 minute) and 5400 seconds (90 minutes).

The 2.7 pound weight mentioned is roughly that of a rounded fist-sized stone, the sort you sometimes see in dry riverbeds. It is a humanly scaled Planck relative that will help acquaint us with the vastly greater universal force.

why look at Planck quantities?

The Planck set amalgamates the most basic proportions of the universe into a single coherent entity. It is normal for members of this set to occur as proportions in nature, and it is also common for them to be extremely large or small in human terms. The speed belonging to the set--the speed of light--is typical both in its extremity and its widespread occurrence as a basic proportion.

At this point it may be worthwhile to emphasize the distinction between physical quantities and numbers. The Planck set consists of actual quantities which, when scaled to match human senses, can be seen, felt, or directly experienced in some other type-specific way. They often appear in the company of basic numbers such as the electrodynamic constant . The fundamental quantities, together with a few basic numbers, endow nature with internal structure. We will be able to trace examples of that structure, and also to measure aspects of experience such as temperature, energy, and force on scales derived from Planck quantities.

These quantities and associated numbers are ancient universals of unknown origin. They seem favorable to the emergence of life, at least to some extent. Perhaps we can attribute the apparent tranquility around us to the fact that they are not more favorable. We do not know how different the constants could have been: I indicate some alternative fantasies merely to suggest a range of possibility.

necessity:

What we see is the only possible fabric of physical reality. A comprehensive system of formal rules, unknown to us but knowable in principle, forces all parameters, including the number of spatial dimensions, to be as they are.

accident:

Radically different faces of nature emerge through uncontrolled random variation in the parameters. Universes proliferate by a kind of budding at junctions where constancy fails and the regular fabric breaks down. Amidst a welter of diversity, this particular universe happens, by good fortune, to have satisfactory characteristics.

design:

Conscious choice of constants has occurred.

Reality has an enigmatic basis of which physical constants form a conspicuous part. At the moment, Planck quantities and allied numbers such as provide a direct encounter with the unexplained which may be sufficient reason to become acquainted with them. Many aspects of the world have been understood by members of our species, but these have not.

profoundly ordinary

To consider yet a sixth fundamental constant, whether or not you find the room around you comfortable depends largely on the flow of radiant warmth from the walls--and the Stefan-Boltzmann coefficient determining this thermal radiance is multiplied by still another Planck quantity.

One way to look at the Planck set is to see it as arising from five primitive quantities: a force, a length, a time, a charge, and a temperature. These five basic quantities--appropriately scaled versions of which can be directly experienced--algebraically generate the Planck set. What this means is that the set consists of these five together with whatever you can make by combining them using multiplication and division as applied to quantities. These five primitive quantities can be thought of as seeds of the set, because it grows out of them, or as its "generators". Describing a set by designating generators is a frequently used mathematical technique. The set, in this case, is a complete listing of all the combinations that can be made from the generators by multiplication and division.

We will be using outline letters , , and as self-explanatory notation for Planck force, length, and time. Planck temperature will be written because the outline letter is already committed to time. The elementary charge--a natural unit found on constituents of matter such as the electron--is conventionally denoted by the symbol e. So the five generators: the force, charge, length, time, and temperature, can be written {, e, , , }. The order was chosen as mnemonic.

Although for definiteness we focus on these, several other fivesomes would serve equally well. We would, for instance, get the same set if we started with {c, , e, , }. This is easy to see in view of the facts that = /c and c = /, so that either of the two sets of generators can be made using the other.

By combining generators we immediately come up with whatsoever other Planck quantities come to mind--for instance the Planck energy is equal to : the result of applying Planck force for the extent of the Planck length. As a further illustration, Planck pressure is /: unit force per unit area. Among much else the commonplace constants mentioned earlier are also generated, in some cases with the help of auxilliary numbers such as .

If you are curious about specifics, you might wish to see how some of those freshman physics constants are generated. We can begin with the Coulomb constant, basic to electricity and magnetism. The units appropriate to this constant are force length per charge--in Planck terms the combination /e. This is obviously made from the generators by multiplying and dividing, so it belongs to the set. The Coulomb constant, here denoted , is equal to that quantity multiplied by --what we have, in effect, is = /e. You might say that what is conventionally called the fine structure constant is the value of the Coulomb constant expressed in Planck terms. Here are a few other members of the set. In two cases an alternative expression using the Planck energy E has been included.

c = /
= =
G = c/
k = / = /

G. J. Stoney discovered and described the Planck quantities, referring to them by a different name, in 1874. According to our standards his definition of the Planck length was off by a factor of the square root of , and so were some others, but that seems no reason to deny him credit. Some writers omit the elementary charge and so arrive at a more restricted collection. But Stoney included the elementary charge (the existence of which he was the first person to postulate and to which he gave the name electron). Stoney's paper "On the Physical Units of Nature" was presented to an 1874 meeting of the British Association and published in the May 1881 issue of The Philosophical Magazine. Max Planck described a collection of Planck quantities to the Prussian Academy in 1899. in a lecture on the thermodynamics of radiation. Like Stoney, he referred to them as "natural units". (footnote to Barrow and Silk, re G.J.Stoney)

involvement with everyday experience

A few examples involving the force F can serve to illustrate. One can become acquainted with the Planck force through its humanly scaled relative, the 2.7 pound weight mentioned earlier, and through other definite small fractions which play a part in everyday experience. The Planck force is basic to gravity: Standing upright, you feel a small easily calculated fraction of F in the soles of your feet. Indeed it is in gravitation that the force is most nakedly apparent. But the scaled-down version is also basic to electricity and magnetism, which will be our focus for the moment.

The electrostatic force between charged particles is the force that contracts your muscles and that drives the chemistry in your body. A fairly common separation at molecular level is 10 , which is one billionth of a span. The force between unit charges at that separation is on the order of one billionth of a stone. The relevant law is remarkably concise--the force is simply equal to divided by the square of the size of the separation (expressed as a multiple of ). If the size of the separation is 10, in Planck terms, then its square is 10. To find the force in that case one simply divides by 10. You feel countless copies of this all the time, involving pairs of unit charges which are varying distances apart.

The magnetic forces which drive and guide much of technology arise between charges in motion and particularly between variously configured electric currents. To mention only one example, in the lurch of acceleration you feel riding the metro many copies of this force are simultaneously at work. In parallel linear wires, on a segment whose length is half the separation between the wires, this force is equal to multiplied by both the sizes of the two currents--sizes being rated in comparison with the Planck current e/. A fairly common current in appliance motors is 10 e/, so its size would be 10-24. If parallel currents are both of that size, calculating the force evidently means multiplying by 10.

For the sake of brevity, I will conclude by describing a way in which another Planck quantity, the speed of light, participates in everyday affairs. Of course much of what is constantly happening around us, whether it is natural or artificial, consists of transmissions of energy and information at the speed of light. This is the speed at which much of what is important to us happens. Its general involvement in everyday life does not require discussion. But in addition to its more obvious roles, there is a special one. The Planck speed has become the standard by which other speeds are measured, and has quietly crept into the foundations of our language. If you analyze the operational meaning of such phrases as "miles per hour" and "meters per second" you will see that they involve the natural unit as a reference. Speed is not the only category in which this kind of evolution is taking place.

Many of the ranging instruments by which distances are established have themselves come to measure in light-time, so that references to distance traveled per unit time involve an implicit comparison with light. Moreover, the Doppler instruments so often used to make practical measurements of speed actually detect it in terms of the natural unit. For practical purposes, in the case of an approaching source, a fraction of a percent rise in frequency indicates that the speed of approach is that same fraction of a percent of the speed of light. (A slightly more complicated rule applies at very high speeds. It ought also to be mentioned that light travels slower in denser media such as water and glass, while "the speed of light" customarily refers to its speed in vacuum taken as the standard.)

In 1983, by redefining the meter as the distance light travels in a certain exact fraction of a second, an official international body determined that one meter per second, the unit for measuring speed, shall be that same exact fraction of c. By definition of the meter, one meter per second is now identical to and the natural unit has now become our basic speed of reference. It has therefore become logically impossible to measure the speed of light in vacuo (which at one time was a regular thing for physicists to do).

In terms of the 1990 electrical standards of the Intenational Bureau of Weights and Measures (BIPM) the unit of charge, which can be called the BIPM coulomb to distinguish it from the older version, is an exact multiple of the Planck charge namely 6 241 509 629 152 650 000 e. How this comes about is described later, in the chapter on conversion factors. Using the BIPM electrical standards, which are those of choice for precise work, it is logically impossible to measure the charge on the electron (which also used to be a regular thing for physicists to do). Speed and electrical charge are not the only categories in which a move towards the use of natural, or Planck, units is taking place. There is more about this in a later chapter on exact constants.

In the next chapter the concept of gravitational length is discussed--gravitational length and the Planck quantities illuminate and confirm each other. To take one illustration, the sun's gravitational length of 0.9 mile is directly visible in the motion of its planets--in their distances and speeds. Yet that 0.9 mile is also equal to the distance which the Planck force would need to push in order to create" the sun--to deliver the energy embodied in the sun's material being. Consult the next chapter if there is immediate need for more information about gravitational length.

reaching the billion mark

Typically around the time a youngster turns seven years of age he or she reaches a "billion mark" in Planck terms. As an exercise in imagination, suppose that you are a seven-year-old and are reaching that mark. Depending on the circumstances, you might have occasion to reflect on some of the following:

• Your gravitational length is a billion times the Planck length . The fact that you are not round, like a small planet, does not really matter. We could imagine you out in space with things orbiting you. Their orbits, if wide enough not to be influenced by your lack of roundness, would tell your gravitational length.
• Your rest energy is a billion times Planck energy . This is the energy you embody as substance. It provides an idea of the effort which would be required to create an equal amount of material from energy alone.
• Escape energy for anyone on the earth's surface is 0.7 billionths of that person's rest energy. Your rest energy is a billion and the result of multiplying that by 0.7 billionths is 0.7 . This energy is a kind of bond between you and earth--it would take just seven tenths of the Planck energy to release you into independent orbit around the sun.
• Any quantity of energy has associated with it a frequency and a wavelength. The greater the energy, the higher the associated frequency and the shorter the wavelength. Your associated wavelength is one billionth of .
• The Planck length is the geometric mean of your gravitational length and your associated wavelength--one is a billion times and the other is a billionth of . Throughout your life the Planck length will always be the geometric mean of those two lengths. Whatever the two lengths become they will always be related to each other by in exactly this way. This is a special case of something more general which is true even for a proton or for an individual quantum of light.
• As a swimmer you displace 5 gallons.
• You comprise 13 billion billion billion baryons (the heavy particles such as protons and neutrons that constitute most of the weight of any ordinary material.) If you prefer exponential notation, your baryon number is 1310.
• Your inertia is a billion times the Planck quantity of inertia.

Some of this will be discussed at more leisure later on. Of course the general points apply mutatis mutandis regardless of your age. If your gravitational length is 3 billion (thrice that of the youngster), then under ordinary swimming conditions you will displace 15 gallons. Like the youngster, you have two different lengths associated with the energy you embody. If one length is three billion the other must of necessity be . Whatever multiple of the one length is, the other must be the reciprocal multiple. The Planck length links your two lengths, just as it does those of the youngster, just as it relates the analogous two lengths associated with a proton, and just as it links the analogous two lengths associated with a quantum of light.

other cultures from an 1899 perspective

At this point it might be appropriate to recall a remark made by Max Planck in a paper on the thermodynamics of radiation, which he delivered to the Prussian Academy in May 1899. He was talking about the set of quantities now known by his name. I believe it is to Planck's credit that he made the point about universality in a courageous and forceful way, leaving little room for secondhand reformulation.

These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as natural units"...

...ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und ausser menschliche Culturen nothwendig behalten und welche daher als "natürliche Maasseinheiten" bezeichnet werden können.

It seems pointless to speculate about proper degrees of qualification and detachment, or argue about whether Planck overstated his case. Although we do not know of any cultures elsewhere than on earth, if some did exist and if there were any crosscultural or shared features, then awareness of these quantities would likely be among them. By comparison, the number ten and ideas like "billion" seem specifically human. We must confess to being decimal numerates--the phrase unaccountably reminds me of "ungulates"--and make the best of it.

As Planck observed in 1899, the set of quantities he called "natürliche Maasseinheiten" might arise outside our immediate context:

[The same quantities] must appear over and over, measured by the most different intelligences in accordance with the most different methods.

...sie müssen also, von den verschiedensten Intelligenzen nach den verschiedensten Methoden gemessen, sich immer wieder als die nämlichen ergeben.

At some point during your seventh year you embody an energy which could be recognized by another decimal numerate (if there are any) even at a considerable cultural remove.

the universal nudge

There is a Planck quantity of impulse or "boost" which is human-scale. It can be directly experienced by anyone who is free to move--you could impart a push that size to a child in a swing. The Planck boost can be formally represented as an inconceivably abrupt nudge lasting for time and exerting force . But it is more understandable pictured as a push lasting 10 and exerting 10. The longer duration compensates for the weaker force making the push equivalent.

To a close approximation, the Planck quantity of impulse is delivered in a push lasting for one tick and exerting the 2.7 pound force earlier referred to as a stone. Such a "tickstone" push is quite modest by our standards. You might easily give someone in a swing several times that much momentum, and something on the order of ten times the Planck impulse would be needed to get you up to the speed of a run. As an imaginary young person reaching the billion mark, you might consider celebrating with a high-speed ride. Merely to reach highway speed calls for around a hundred "tickstone", or a hundred Planck units. These could be given you by the seatback of a drag-racer in a push exerting ten stone (27 pounds) and lasting ten tick--the time it takes to snap your fingers ten times in tempo moderato--or less abruptly in a more conventional vehicle.

There is a related idea of par boost. In this connection it helps to express speed as a fraction of the speed of light. To start a person moving at some ordinary speed requires a push which is times that person's par but for practical purposes is indistinguishable from times that persons's par. As someone celebrating the one billion mark, your par is one billion Planck units.

To illustrate, suppose you want to be boosted up to cold air sonic: the speed of sound in the cold typical of cruising altitude. Expressed as a fraction of the speed of light, cold air sonic is a millionth. So is one millionth and the required boost is one millionth of your par, which we already said was one billion Planck nudges. As a matter of simple arithmetic a millionth of one billion is one thousand--you require a boost of a thousand Planck units to reach cold air sonic.

It may bear remarking that, on the day you reach the billion mark, your weight in stone matches the earth's gravity expressed in humanly scaled Planck terms. At sealevel the acceleration due to gravity is ordinarily about 17 span per tick and your sealevel weight is some 17 stone.

the role of humanly scaled versions

In an earlier table a few names were suggested for some approximate power-of-ten multiples of Planck quantities. These humanly scaled versions, as was indicated at the outset, are meant to give the universal quantities a kind of immediacy or at least to let them be perceived more concretely than they otherwise would be.

One of these humanly scaled quantities is the approximately 2.7 pound weight of a fist-sized stone, and in fact the name suggested for that amount of force was "stone". It is a heftable fraction of the Planck force, 10 or thereabouts. A second humanly scaled quantity is the length depicted rather approximately here:

(thumb to forefinger span--width of both hands--width of one hand with thumb extended)

Still another is the gallon-sized volume of a cube with the depicted length as its edge.

Along these lines, typical weight at birth is in the neighborhood of 3 stone. The length of infants at birth is around 3 span. (Our local authority says the normal range is 43 to 48 centimeters.) Three span is commonly the distance from fingertips to elbow--a biblical "cubit"--or thereabouts. A newborn's volume on the order of a gallon. A deep breath of air is on the order of a gallon. Each of us was once three stone, three span, and a gallon. Why bother to note this? It confirms that the indicated multiples of Planck quantities really are "human scale" and it affords some practice thinking in these terms.