In physics, Planck units are a system of units of measurement going back to Max Planck that is an early definition of natural units. The system is defined only using the following fundamental physical constants and is "natural" in the sense that the numerical values of these five universal constants become 1 when expressed in units of this system.
|speed of light in vacuum||L1T-1|
|"reduced Planck's constant" or Dirac's constant||where is Planck's constant||ML2T-1|
|Coulomb force constant||where is the permittivity in vacuum||Q-2 M 1 L3 T-2|
The Planck units are often semi-humorously referred to by physicists as "God's units". They eliminate anthropocentric arbitrariness from the system of units: some physicists believe that an extra-terrestrial intelligence might be expected to use the same system.
Natural units can help physicists reframe questions. Perhaps Frank Wilczek said it best (June 2001 Physics Today) :
- ...We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in Natural (Planck) Units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)]...
The strength of gravity is simply what it is and the strength of the electromagnetic force simply is what it is. The electromagnetic force operates on a different physical quantity (electric charge) than gravity (mass) so it cannot be compared directly to gravity. To note that gravity is an extremely weak force is, from the point-of-view of natural units, like comparing apples to oranges. It is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, and that is because the charge on the protons are approximately a natural unit of charge but the mass of the protons are far, far less than the natural unit of mass.
Newton's Law of universal gravitation
- using Planck units.
The energy of a particle or photon with radian frequency in its wave function
- (i.e. a body with a mass of 5000 Planck Mass units will have an intrinsic energy of 5000 Planck Energy units) and the full form
The unit of temperature is defined so that the mean amount of thermal kinetic energy carried per particle per degree of freedom of motion
- when using Planck Units. (The factors would have been eliminated if would have been normalized instead of the Coulomb Force Constant .)
Base Planck units
By constraining the numerical values of the above 5 fundamental constants to be 1, then 5 base units for time, length, mass, charge, and temperature are defined.
|Name||Dimension||Expression||Approx. SI equivalent measure|
|Planck time||Time (T)||5.39121 × 10-44 s|
|Planck length||Length (L)||1.61624 × 10-35 m|
|Planck mass||Mass (M)||2.17645 × 10-8 kg|
|Planck charge||Electric charge (Q)||1.8755459 × 10-18 C|
|Planck temperature||Temperature (ML2T-2/k)||1.41679 × 1032 K|
Derived Planck units
As in other systems of units, the following units of physical quantity are defined in terms of the base Planck units.
|Name||Dimension||Expression||Approx. SI equivalent measure|
|Planck energy||Energy (ML2T-2)||1.9561 × 109 J|
|Planck force||Force (MLT-2)||1.21027 × 1044 N|
|Planck power||Power (ML2T-3)||3.62831 × 1052 W|
|Planck density||Density (ML-3)||5.15500 × 1096 kg/m3|
|Planck angular frequency||Frequency (T-1)||1.85487 × 1043 rad/s|
|Planck pressure||Pressure (ML-1T-2)||4.63309 × 10113 Pa|
|Planck current||Electric current (QT-1)||3.4789 × 1025 A|
|Planck voltage||Voltage (ML2T-2Q-1)||1.04295 × 1027 V|
|Planck impedance||Resistance (ML2T-1Q-2)||2.99792458 × 101 Ω|
At the "Planck scales" in length, time, density, or temperature, one must consider both the effects of quantum mechanics and general relativity. Unfortunately this requires a theory of quantum gravity which does not yet exist.
Most of the Planck units are either too small or too large for practical use, unless prefixed with large powers of ten. They also suffer from uncertainties in the measurement of some of the constants on which they are based, especially of the gravitational constant (which has an uncertainty of 1 to 7000).
It might be interesting to note that the elementary charge measured in terms of the Planck charge comes out to be
where is the fine-structure constant
The dimensionless Fine-structure constant can be thought of as taking on the value that it does because of the amount of charge, measured in natural units (Planck charge), that electrons, protons, and other charged particles happen to have been assigned by nature herself. Because the electromagnetic force between two particles is proportional to the product of the charges of each particle (each which would, in Planck units, be proportional to ), the strength of the electromagnetic force relative to other forces is proportional to .
The Planck impedance comes out to be the characteristic impedance of free space scaled down by meaning that, in terms of Planck Units, that . This factor comes from the fact that it is the Coulomb Force Constant in Coulomb's law that is normalized to 1, as is done in the cgs system of units, rather than the permittivity of free space . This, and that fact that the gravitational constant is normalized rather than , could be considered to be an arbitrary definition and perhaps a non-optimal one from the perspective of defining the most natural physical units as the choice for Planck Units.
Planck units and the invariant scaling of nature
Referring to Duff, Okun, and Veneziano Trialogue on the number of fundamental constants (The operationally indistinguishable world of Mr. Tompkins), if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. (When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, all physical quantities are measured relative to some other like dimensioned values.) We can notice a difference if some dimensionless physical quantity such as or the proton/electron mass ratio changes (atomic structures would change) but if all dimensionless physical quantities remained constant, we could not tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our existence if a dimensionful quantity such as c has changed, even drastically.
If the speed of light c were somehow suddenly cut in half and changed to c/2 (but with all dimensionless physical quantities continuing to remain constant), then the Planck Length would increase by a factor of from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant:
Then atoms would be bigger (in one dimension) by , each of us would be taller by , and so would our meter sticks be taller (and wider and thicker) by a factor of and we would not know the difference.
Our clocks would tick slower by a factor of (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by , but we would not know the difference. This hypothetical god-like observer on the outside might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299792458 of our new meters in the time elapsed by one of our new seconds. We would not notice any difference.
This conceptually contradicts George Gamow in Mr. Tompkins who suggests that if a dimensionful universal constant such as c changed, we would easily notice the difference. We must then ask him, how would we measure the difference if our measuring standards also changed in the same way?
Max Planck's discovery of the natural units
Max Planck first listed his set of units (and gave values for them remarkably close to those used today) in May of 1899 in a paper presented to the Prussian Academy of Sciences. Max Planck: 'Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899)
At the time he presented the units, quantum mechanics had not been invented. He himself had not yet discovered the theory of black-body radiation (first published December 1900) in which the Planck's Constant made its first appearance and for which Planck was later awarded the Nobel prize. The relevant parts of Planck's 1899 paper leave some confusion as to how he managed to come up with the units of time, length, mass, temperature etc. which today we define using Dirac's Constant and motivate by references to quantum physics before things like and quantum physics were known. Here's a quote from the 1899 paper that gives an idea of how Planck thought about the set of units.
- ...ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und aussermenschliche Kulturen notwendig behalten und welche daher als "natürliche Masseinheiten" bezeichnet werden können...
- ...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"...
- The NIST website (National Institute of Standards and Technology) is a convenient source of data on the commonly recognized constants, including Planck units.
- Planck's original paper