Natural Units for Scientists and Engineers

Definitions

The Planck quantities (restricted set) are those generated as algebraic combinations of G, c and hbar. The extended set consists of all the quantities arising from the first three generators with Boltzmann's k and the elemenatary charge e (the electron charge) adjoined.

The symbol W_P is used here for Planck angular frequency. Let me know if the Greek omega doesn't come through and I will replace it with a letter that does. To simplify things, angular frequency (radians rather than cycles) has been used consistently, so we need only one Planck's constant rather than two.

Planck force, F_P: c⁴/G

Planck power, P_P: c⁵/G

Planck frequency W_P: defined by hbarW_P² =c⁵/G

Planck time, t_P: 1/W_P

Planck length, l_P: c/W_P

Planck acceleration: cW_P

Planck energy, E_P: hbarW_P

Planck momentum: hbarW_P/c

Planck mass, m_P: hbarW_P/c² equivalently defined by F = ma, as Planck force over acceleration.

Adjoining k and e to the set of generators gives electrical quantities and quantities related to temperature:

Planck charge: e

Planck current, I_P: eW_P

Planck voltage, V_P: hbarW_P/e

Planck resistance, R_P: hbar/e²

Planck temperature, T_P: hbarW_P/k

These quantities are fundamental physical constants--among the most universal and basic features of nature--which constitute a natural system of units (a system useful for particle physics, string theory, and cosmology) but one which is unhandily scaled for application to everyday life.

Other fundamental physical constants are often simply expressible in Planck terms. For example the Couloumb constant, basic to electricity and magnetism, has the value alpha (approximately 1/137.036) in terms of Planck units and the Stefan-Boltzmann radiation constant has the value (pi²/60)

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Spectrum of values of fundamental constants

One of the best ways to get an idea of the sizes of Planck quantities (which range from very small to very large) is in terms of human-scale units (of force, distance, time, charge, and temperature) which make the Planck quantities take on values which are powers of 10⁵.

When they are expressed in these terms the Planck quantities (and some other closely allied fundamental constants) have the following values.

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Putting in the names of the units

It is not so important what the units are called. "Planck ton" is merely serving here as a nametag for 10^-40F_P and "pace", which happens to be around a thousandth of an ordinary mile, is serving as a nametag for 10³⁵l_P. The names work because the former is a roughly ton-sized force and the latter is about a five-foot length, the traditional length of a pace. It is a trivial exercise to insert the units in the above table, as follows:

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Decimal conversion factors

In physics the principal conversion factors (linking frequency to vacuum wavelength, mass to energy, energy to frequency, frequency to temperature, temperature to pressure and volume, inertial to gravitational mass, current to field, and so on in a great variety of circumstances) are, in practice, the fundamental constants. Those responsible for overseeing the metric system and redefining the units are prevented, for historical reasons, from giving the fundamental constants values which are powers of ten. They can give them exact values like 299 792 458 (as was done with the speed of light), implying on a logical level that they have the freedom to make arbitrary stipulations subject to the Conference's vote of approval. But because of the rough sizes of the units formally established in 1791, they cannot give them round exact values like 10⁹.

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Practical issues surrounding the use of Planck units in coursework

.It would be an interesting possibility for students in college physics courses to have access to Planck units for purposes of problem solving, especially since the main conversion factors used in calculating with nature (the speed of light, Planck's constant, Boltzmann's k, the elementary charge, and G) are unitary—have the value one—when expressed in Planck terms.

A consideration which deters the use of Planck units in calculation at the college physics level is the extreme sizes of some of the units, which are therefore difficult to visualize and remember. Elsewhere at this site (see for example Planckian Talent-Mile Units, or A Scale Model of the Planck Quantities) are attempts to bridge the gap in scale by setting out practical-sized power-of-ten scaled versions of the natural units.

Another obstacle to the direct use of Planck units in ordinary calculation is that many of them contain experimental uncertainty. An exact metric value of the speed of light (which is one of the Planck quantities) has been officially adopted. But for many of the other basic Planck quantities, metric values are known only to about four decimal places. Most of the uncertainty can be traced back to experimental uncertainty in the measurement of G.

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A Planckian CGS-style System

I want to try developing a Planckian system rather similar to the old CGS that was standard in laboratories for roughly 1900-1970 and is still used in lots of places. It had a small mass unit (gram) and delicate force and energy units which were a hundredthousandth and a tenmillionth of today's metric newton and joule.

The numbers for such a "modernized CGS-style" system look extremely nice. We can have G, c, e, k, and hbar equal to E-9, E9, E-15, E-18, E-30. Rough sizes of the units:

length unit---centipace---1.6 cm
mass unit---ounce---22 gram (a thousandth of the other system's talent)
force unit---dyne---1/8 of metric newton
energy unit---erg---2 metric millijoules
charge unit---quad---quadrillion (E15) electrons, a sixthousandth of a metric coulomb.
voltage unit---wolt---erg/quad = 12 conventional volts
current unit---vamp---quad/trice = 3 milliamps.
power unit---fott---erg/trice = 36 milliwatts.

Method of definition

You may remember the technical minute of 54 seconds used to define the Talent-Mile system. Here we start with a time unit that is a thousandth of that---a milliminute. Since milliminute is a four-syllable mouthful I call it a trice for short. It is 0.054 second, or between 18 and 19 to a second. The definitions proceed from the time unit by specifying exact power-of-ten values of the basic physics constants:

c = E9 centipace/trice
hbar = E-30 erg trice
e = E-15 quad
k = E-18 erg/grade
mole = E24 items

Unit force dyne = ounce centipace/sq.trice (the force giving unit acceleration to unit mass)
Unit energy erg = dyne centipace (the work performed by unit push for unit distance).

Corollaries

Atomic mass unit = 1/13 ×E-24 ounce, when more precision is needed replace 13 by 13.0855474 (from the CODATA figure for Hz in amu.)
One mole of the atomic mass unit = 1/13 ounce.
G = 1.00E-9 cubic centipace/sq.trice per ounce.
StefanBoltzmann sigma = (pi²/60) erg/cp²t per grade⁴
Norm for sealevel gravity—1.76 cp/t²
Norm for sealevel pressure—219 dyne/sq.centipace
Norm for seawater density (avg. surface temp)—0.20 oz/cp³.

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The spectrum of values in Planckian "CGS-style" terms

To see how it looks, I'll redo the above table as follows. Remember that a trice is short expression for milliminute, a thousandth of a Planckian minute.

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Two other Planck-based systems for use in coursework

Slightly different versions of Planckian T-M units can be defined, depending on what is specified as the time unit. If this is specified to be an exact number of seconds like 53.908 or some exact fraction of a minute like 1/1.113, then the units can be made to match the Planck quantities to within experimental accuracy, to within the precision with which Planck quantities are currently determined.

On the other hand if the technical "reduced minute" used to define the system is specified to be exactly 54 seconds or 9/10 of a conventional minute then the system is easier to describe, although its fit with the Planck quantities is less precise.

In either case, many of the natural constants (such as hbar, e, k, Avogadro, gas constant, and others) take on exact power-of-ten values. However in the latter case (54 second minute) the value of G is only accurate to about 3 significant figures—1.00E-15 mile³/minute² per talent.

Because of the ease of description, I will discuss this version first and quote briefly from the T-M index page at this site.

[The Talent-Mile system uses variants of traditional units—such as the mile, the gallon, and the classical (approximately 50-pound) talent mass unit—which happen to be power-of-ten scale-ups of the Planck units intrinsic to the physical universe.

Talent-Mile is a fully decimal system in which the main natural constants take on values which are exact—in one case approximate—powers of ten. It is intended as a non-metric alternative for educational and scientific use. When expressed in T-M units the speed of light is E7, Planck's hbar is E-40, Boltzmann's k is E-25, the Avogadro number is E23, the electron charge is E-23. These power of ten values are exact and that of G is approximate—1.00E-15.

As a convenience, for use in some specialized contexts, the basic system definitions employ a technical minute of 54 ordinary seconds—exactly 90 percent of the ordinary minute. No reform of time-telling is intended, one continues to use conventional units of time (hours-minutes-seconds) in general, but has one additional unit available for optional use in problem-solving. The definitions lead to a mile of 1618.88 meters, a gallon of 4.24 liters, and a talent mass unit of 21.73 kilo—these being approximately equal to power-of-ten scale-ups of the corresponding Planck units.

The corresponding force unit (oc = talent mile per sq.minute) is about 12 newtons or 2.7 pounds, and the ocmile is 4.7 food Calories, i.e. 4.7 kilocalories. The definitions are logically equivalent to requiring c, hbar, k, e, and the Avogadro number take on certain exact power-of-ten values.

c = E7 miles per minute
hbar = E-40 ocmile minute
k = E-25 ocmile per grade
mole = E23 items
e = E-23 charge unit

With T-M so-defined, G = 1.00E-15 cubmile/sqminute per talent.]

It may be worth noting some immediate exact consequences of the definitions:

eevee = E-23 ocmile
hbar = E-17 eevee minute
k = E-2 eevee per grade

The atomic mass unit (carbon-12 twelfthmass) plays an important role and is approximately 1/13 × E-27 talent. The talent energy equivalent is E14 ocmile or E37 eevee, which makes the amu energy equivalent approximately 1/13×E10 eevee. The approximate 13 is calculated more precisely as 13.0855474...in a section deriving T-M metric conversion factors.

The move in metrology towards exact fundamental constants adopted by convention

Units in the metric system are redefined from time to time by the General Conference (CGPM) as recommended by the International Committee on Weights and Measures (CIPM). In 1983 the meter was formally redefined so as to make the speed of light an exact unmeasurable quantity, determined by convention. The same thing is in the process of happening to Planck's constant and the elementary charge. In 1988 the CIPM recommended clock-based electrical standards, taking effect as of January 1990, which (among other things) make the Coulomb unit charge exactly equal to 6 241 509 629 152 650 000 times the elementary charge. The 1990 electrical standards have proven satisfactory and are part of a trend towards adopting exact values of the fundamental constants.

In 1999 the CGPM's resolution 7 called for a redefinition of the kilogram along these lines — developing a clock-based standard making the old block of platinum-iridium metal obsolete. The most promising approach uses a electrical force-measuring device called the watt balance and is logically equivalent to specifying an exact conventional value for Planck's constant. In line with the General Conference's resolution, the NIST's Mohr and Taylor, the current and previous chair of CODATA, have suggested that the greater Planck's constant (2pi h-bar) be made exactly equal to:

299 792 458²/135 639 274 × 10^-42 joule second.

This is long overdue. For more on-line about their proposals, see their August, 2000 article in the Electronic Journal of Differential Equations: Quantum electrodynamics and the fundamental constants, by Peter J. Mohr and Barry N. Taylor.

http://ejde.math.swt.edu/conf-proc/04/m1/mohr.pdf

It should be noted that the metric system's 1990 electrical standards also determine an exact value for Planck's constant. But the exact value proposed by Mohr and Taylor is not identical to the 1990 value and would presumably supersede it. To become official every such proposal must finally be ratified by the General Conference.

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Precise Planck Counterparts: Getting Rid of Experimental Uncertainty

In routine calculation it is awkward to be required to carry along inherited experimental uncertainties (which may have nothing to do with the problem) and related confidence intervals. To prepare versions of the natural quantities which are suitable for routine use as units, we can simply truncate the uncertainty and fix on a set of precise counterparts which are virtually the same as the experimentally determined ones, but which have exact metric equivalents.

To substitute for W_P, the Planck (angular) frequency, we have its precise counterpart W_P* defined as 1.855×10⁴³ events (radians) per atom clock second. In basing certain other units on the atomic clock, the 1990 exact values adopted by the CIPM (International Committee on Weights and Measures) for the (angular) Josephson (K_J-bar = 2e/h-bar) and the von Klitzing (R_K = 2pi h-bar/e²) constants are assumed. The precise counterpart to Planck voltage is V_P*, defined as 2 W_P*/K_J-bar. The precise counterpart to Planck current is I_P* defined from V_P*using the resistance quantity R_K.

W_P* is essentially equal to the Planck frequency, agreeing with it to the accuracy with which the Planck frequency is measured. Likewise the unit time t_P*, defined as 1/ W_P*, is equal to the Planck time to the accuracy with which that quantity is measured. The number 1.855 (also written 371/200 in some calculations) is exact and chosen to produce a fit with experimental measurements of G as closely as possible given their current degree of precision.

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