The universe has a set of special quantities, technically known as Planck quantities, which serve as built-in natural scales of measurement: a top speed, a main force, a core frequency, a full power, and so on. We can adjust our own quantity senses to the natural units by using familiar-sized, power-of-ten versions of them — a set of human-scale Planck units.
What it's like to use human-scale Planck units is explored in some collections of brief fables: Nature Fables 1, Nature Fables 2, and Nature Fables 3. Earlier versions of some of them were collected as the Planckian Fables.
The practical convenience of units attuned to nature can be suggested by example. If you are forced to use metric units, whose sizes were established in the late 18th century, then your figure for the speed of light is 299,792,458 meters per second. This obscures natural relationships, like those between mass and energy, or frequency and wavelength, and involves unnecessary arithmetic. On the other hand, in the alternatives to metric introduced here, the speed of light is exactly a power of ten. Other basic proportions, such as hbar and the gravity constant, have equally simple values.
Because these units are the same as nature's (though possibly scaled up or down by powers of ten), you can count on the most commonly used conversion factors being powers of ten. If you need to use them you don't get as bogged in arithmetic as you do with metric— indeed for much of the time instead of calculating you just change the exponent or move the decimal point.
Some alternative Postmetric systems of units are outlined here with links to pages giving more detail. The first set consists of the raw, unscaled Planck units. The other sets have been adjusted by powers of ten to have practical human-scale sizes.
A. RAW PLANCK UNITS JUST AS WE GET THEM FROM NATURE
Just as there is a standard reference speed c intrinsic to light and the geometry of the universe as a whole, there is a standard force FPlanck. This force is the main coefficient in the equation which models how gravity works (the so-called "Einstein equation" central to General Relativity), where it relates the curvature of a region to the concentration of mass-energy there. Planck recognized FPlanck = c4/G as the force unit belonging to his system of natural units—this was in 1899, 16 years before Einstein discovered it was intrinsic to how gravity works—but could not at that time convince others of his units' physical importance.
Natural units of speed and force imply those of power and frequency as well. Unit power is that delivered by unit force pushing at unit speed—a familiar connection present in all coherent systems of units—and power is related to (the square of) frequency by a universal ratio, Planck's constant.
The speed and force are very great—essentially at the limit their respective scales—and consequently the units of power and frequency are extreme as well. In fact the sizes of most of the raw natural units are extreme by human standards—either very large as in the case of the force, the power, the speed, and the frequency or very small as in the case of the natural time and length.
The natural units, as we find them in nature, also lack precise convertibility with metric and other conventional units, chiefly because of imprecision in measuring the gravitational constant G.
One can however define human scale units which are convertible with other systems and which approximate power-of-ten scaled versions of the natural units as closely as desired or as the natural units are measured.
B. PLANCKIAN TALENT-MILE UNITS
The following familiar-sized versions of the Planck units can be defined if no restriction is placed on the powers of ten used to scale the natural units. Time is told in hours and conventional minutes, with a special tenpercent reduced minute available for physics. Here are the exact definition of the most basic units:
minute—54 atomic clock seconds, 1/1600 of a day
mile—exactly one tenmillionth of a light minute
talent—exactly 1040 hbar minute per square mile.
In this system an effort was made across the board to get versions of the Planck units as close as possible to practical units used in everyday life, and to give them traditional names. What proved the hardest was finding a traditional unit corresponding to the 12 newton (2.7 pound) force. The only traditional unit of this size is the oc, my simplified spelling of the ocque or oke used in parts of eastern Europe and the Mediterranean. Several of the units are listed here with rough indications of their size.
minute—1/1600 day, tenpercent reduced minute
mile—one tenmillionth of a lightminute, slightly longer than convenional
mile, about 1619 meters
talent—48 pounds, about 22 kilograms
oc—2.7 pounds of force, gives unit acceleration to the unit mass
ocmile (energy)—about 5 food Calories, delivered by a mile-long oc
push
degree—1.4145 kelvin
mole—1023 items
dram (charge)—1023e, one mole of the elementary charge,
16 conventional kilocoulombs.
When these decimal versions of the Planck units are used, the fundamental constants take on simple power-of-ten values. In most cases these power-of-ten values are exact:
c = 107 miles per minute(exact)
hbar = 10-40 ocmile minute (exact)
G = 1.00×10-15 oc mile2 per
talent2
elementary charge e = 10-23 dram (exact)
eevee = 10-23 ocmile (exact)
Coulomb's const. = alpha × 1013 oc
mile2 per dram2 (exact)
Boltzmann's k = 10-4 eevee per degree (exact)
alpha is the fine structure constant of physics, approximately 1/137.036.
The conventional norm for sealevel gravity is 17.6 mile per minute2 in this system. In normal gravity the weight of a talent is 17.6 ocque.
The link is to a page describing a system called the Practical Planck Units, which results if one adjusts familiar-sized units so that the fundamental physical constants acquire values which are not any old powers of ten but powers of a thousand. That means 103 is the base for all the ratios, which become easier to remember and use. In this system, time is told in the same conventional hours and minutes as always, but for physics and engineering course-work a small interval roughly a thousandth of a minute is defined. Overall the units are rather close to the familiar-sized units. The same mass unit, the 48 pound talent, is used. Since it is a billion times Planck mass it is already in a power-of-thousand relation to the natural unit.
D. TON-PACE-MINUTE (TPM) PLANCK UNITS
Ton-pace-minute units, which are power-of-105 scaled versions of the Planck quantities adapted for use as units, are described on several pages of this site, including A Human-scale Model of the Planck Quantities. TPM units result if one adjusts familiar-sized units so that the fundamental physical constants acquire values which are not merely powers of ten but more particularly are powers of 105. In this system, time is told in the same conventional seconds and hours as always, but a slightly different minute is used for college-level physics and engineering course-work. Unit force in this system is ton-sized and the unit of length is the traditional two-step pace, about 5 feet.
ADAPTING TRADITIONAL NAMES
The metric system was an early (1790s) attempt to arrange practical-sized units so that some basic constants were powers of ten. The job needs to be redone using more fundamental and universal things than the density of water (like the speed of light, Planck's constant, and the elementary charge). Naming human-scale Planck units presumably requires some arbitrary choices. Your guess is as good as mine as to which names will stick. Fortunately several are close to traditional units so that there is a possibility of adapting the traditional names.
HOW THE PLANCK MASS IS IMPLICATED IN LIGHT AND GRAVITY
Several of the natural or Planck units have obvious natural meanings. The unit of speed in the system is, well, the speed light travels in empty space. There is a lot of empty space and a lot of light, so that is a fairly obvious feature of the world. And the natural unit of force is intrinsic to all the gravity around us, so it has an everpresent meaning.
There is a less familiar member of the set of natural units—unit energy × unit length—which is all around us, in every photon of light. Every photon carries an amount of energy and has a (vacuum angular) wavelength and because of a built-in trade-off between these two things, the two multiplied together is the same for all photons. We can even say what the energy×length product that is present in all light happens to be—it is hbar×c.
Every mass M also defines an energy×length product, GM², which is the distance separating two compact bodies of that mass multiplied by their bond energy—the work required to draw them indefinitely far apart. Because of a tradeoff intrinsic to gravity—the greater the separation the less the binding energy—this product of energy and length is independent of the separation and depends only on the mass.
The Planck mass is simply the unique mass which gravity links to the energy×length product intrinsic to all light, in other words the M defined by setting GM² equal to h-bar×c
The 1998 codata recommended value for Mplanck is 21.767 micrograms with an uncertainty of 1/13 percent. Essentially this means that a billion times the Planck mass is 21.767 kilograms or, if you prefer, 48 pounds. And since it is a solution of GM²=h-bar×c, the formula for it is (hbar×c/G)½.
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