Planck units

Natural units or Planck units are a natural system of units of measurement based on the fundamental constants:

Constant	Symbol	Dimensions
gravitational constant	${\boldsymbol {G}}$	M−1L3T−2
Dirac's constant	$\hbar =h/2\pi$, where $h$ is Planck's constant	ML2T−1
speed of light in vacuum	${\boldsymbol {c}}$	L1T−1
Boltzmann constant	${\boldsymbol {k}}$	ML2T−2K−1
permittivity of vacuum	${\boldsymbol {\epsilon }}_{0}$	Q2M−1L−3T2
electric charge	${\boldsymbol {e}}$	Q

The Planck units are often semi-humorously referred to by physicists as "God's units". They eliminate all arbitrariness from the system of units: some physicists believe that an extra-terrestrial intelligence might be expected to use the same system.

These units have the advantage of simplifying many equations in physics by removing conversion factors; for example, Einstein's famous equation $E=mc^{2}\,\!$ becomes simply $E=m\cdot 1^{2}\,\!$ (or effectively $E=m\,\!$), i.e. a body with mass = 5000 Planck Mass units will have an intrinsic energy of 5000 Planck Energy units. For this reason, the units are popular in quantum gravity research.

However, they are 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 $G\,\!$.

The Planck units are:

Name	Dimensions	Expression	Approx. SI equivalent measure
Planck length	Length (L)	$l_{p}=(\hbar G/c^{3})^{\frac {1}{2}}$	1.616 × 10-35 m
Planck mass	Mass (M)	$m_{p}=(c\hbar /G)^{\frac {1}{2}}$	2.177 × 10-8 kg
Planck time	Time (T)	$t_{p}=l_{p}/c=(\hbar G/c^{5})^{\frac {1}{2}}$	5.391 × 10-44 s
Planck current	Electric current (Q/T)	$I_{p}=e/(\hbar G/c^{5})^{\frac {1}{2}}$	2.972 × 1024 A
Planck temperature	Temperature (ML2T−2/k)	$T_{p}=m_{p}c^{2}/k=(c^{5}\hbar /G)^{\frac {1}{2}}/k$	1.415 × 1032 K
	Angle		1 rad
	Solid angle		1 sr
Name	Dimensions	Expression	Approx. SI equivalent measure
Planck force	Force (MLT−2)	$F_{p}={c^{4}}/G\,\!$	1.210 × 1044 N
Planck energy	Energy (ML2T−2)	$E_{p}=c^{2}(c\hbar /G)^{\frac {1}{2}}$	1019 GeV = 1.956 × 109 J
Planck power	Power (ML2T−3)	$P_{p}={c^{5}}/G\,\!$	3.629 × 1052 W
Planck density	Density (ML-3)	$\rho _{p}=c^{5}/(\hbar G^{2})$	5.1 × 1096 kg/m3
Planck frequency	Frequency (T−1)	$f_{p}=(c^{5}/(\hbar G)^{\frac {1}{2}}$	1.855 × 1043 Hz
Planck pressure	Pressure (ML−1T−2)	$Pr_{p}=c^{10}/(\hbar G^{2})$	4.635 × 10113 Pa

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 exist.

The Planck mass is credible, indeed many living things (such as some fleas) are smaller than it; the issue is that general relativity predicts that smaller black holes can exist within the event horizon (with a radius less than the Planck length), while quantum mechanics predicts that the mass would probably be outside the event horizon.

Max Planck's creation 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 $h\,\!$ 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 $\hbar$ and motivate by references to quantum physics before things like $\hbar$ 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 ausser menschliche Culturen nothwendig behalten und welche daher als 'natürliche Maasseinheiten' 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'...

Other Natural units

Although not formally part of the system of Planck units, the following SI derived units are defined naturally.

• Radian (rad) - a dimensionless measure of angle
• Steradian (sr) - a dimensionless measure of solid angle

See also

• dimensional analysis
• physical constants

External link

• 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