Two concise derivations of the Planck units are given here, aimed at making the defining formulas easy to remember. Familiarity with the constants G, h-bar, and c is assumed. In one case, use is made of the fact that c4/G is a force. This is the "Planck force", FPlanck, the force belonging to the set of Planck units. It is roughly 1040 tons and more precisely 12×1043newtons.
Einstein's 1916 equations of General Relativity provide our primary model of how gravity works and the central constant there is the force c4/G. In this exerpt from a sample online scientific paper (included here to give a bit of the flavor) you will see the reciprocal G/c4 of the force.
[[...The field equations of general relativity form a system of ten second-order partial differential equations obeyed by the space-time metric,
G = 8piG/c4T
where the Einstein curvature tensor G is generated, through the gravitational coupling k = 8piG/c4, by the matter stress-energy tensor T...]]
The sample quote is from a
paper
on Gravitational Radiation by Luc Blanchet of the Institut d'Astrophysique
de Paris,
his email < blanchet@iap.fr>
about Luc <http://www.iap.fr/users/blanchet>
If FPlanck is our unit force, then unit power must be PPlanck = cFPlanck = c5/G. Think of it as the power delivered by the Force pushing at the speed of light. If you have a calculator handy and know the metric values of c and G can easily find out what the natural power unit is in watts.
In its turn a natural unit of power implies one of frequency: power is related to (the square of) frequency by a universal ratio, Planck's constant.
PLANCK FREQUENCY ZPlanck is the square root of PPlanck/hbar— in other words ZPlanck = (c5/Ghbar)½
PLANCK TIME is the reciprocal of the frequency—namely tPlanck = (Gh-bar/c5)½
PLANCK ENERGY is the energy of a photon of light with Planck frequency—namely hbarZPlanck = (c5h-bar/G)½
PLANCK LENGTH is the wavelength of such a photon, and so on.
The natural unit of energy×length, hbar×c, is present in every quantum of light as the product of its energy with its vacuum angular wavelength (see footnote).
But corresponding to any mass M there is also an energy×length product GM2. This is the energy binding two separate compact bodies of that mass multiplied by the distance separating them. Because the binding energy is reciprocally less for greater separations, this product is the same at all distances. And conversely any energy×length product determines a corresponding mass. The natural unit or Planck mass is simply the one corresponding to the natural unit of energy×length. It is the solution to the equation.
GM2 = h-bar×c
That determines what the other units have to be as well. Once the natural unit of mass M = MPlanck is fixed at (h-bar×c/G)½, the solution to the above equation, there is a natural unit of energy Mc2. Then, since we already have a natural unit of energy×length, there is only one possible choice for the unit length. The natural or Planck length unit must equal the energy×length unit divided by the energy unit:
h-bar×c/(h-bar×c/G)½c2.
After some cancelation, the length works out to be (h-barG/c3)½
So one can say that (1) there is a natural unit of energy×length present in all light as the product of quantum energy and wavelength—every photon in the universe exhibits this constant product as far as we know—and (2) the Planck mass is simply the mass which gravity associates with that natural product of energy and length. The other units follow routinely once the units of mass (and consequently energy) have been established.
CODATA recommended values for some basic Planck units are as follows. Relative standard uncertainty in each case is given as 7.5 × 10-4 which is about 1/13 of a percent.
Planck mass 2.1767 × 10-8 kilogram
Planck length 1.6160 × 10-35 meter
Planck time 5.3906 × 10-44 second
From a practical standpoint, much of the convenience of using Planck units comes from the fact that the values of the main natural constants, G, h-bar, c, are one when expressed in terms of natural units.
To some extent this convenience carries over to several human-scale Planck systems in which the natural units have been scaled by powers of ten to make them handier to use. Here are links to a couple of examples:
Index page for Planckian Talent-Mile
units
Index page for practical Planck
units
In human-scale Planck systems the main natural constants take on values which (instead of being one) are powers of ten. In one such system, for instance, the values of G, c, and h-bar are one millionth, one billion and 10-33, the last two figures being exact.
Footnote: Why is it that in the case of each quantum of light the
product of its energy E with lambda, its vacuum angular
wavelength, is h-bar×c? This depends on two standard textbook facts
relating these things to the quantum's angular frequency omega.
E=h-bar omega
lambda=c/omega
Multiplying energy and wavelength together, the omegas cancel, and one is just
left with h-bar×c.