Planck units

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 all those constants become 1 if expressed in this system. (In keeping with the cgs system of units, it is the Coulomb Force Constant in Coulomb's law that is normallized to 1, rather than the permittivity of free space. This is analogous to normalizing Gravitational constant in Newton's law as is the original purpose of Planck in defining Planck Units.)

Constant	Symbol	Dimension
Gravitational constant	G	M^-1L³T^-2
Dirac's constant	$\hbar$ (=h/2 $\pi$ , where h is Planck's constant)	ML²T^-1
speed of light in vacuum	c	L¹T^-1
Boltzmann constant	k	ML²T^-2K^-1
permittivity in vacuum	$\varepsilon _{0}$	Q² M ^-1 L^-3 T²

The Planck units are often semi-humorously referred to by physicists as "God's units". They eliminate anthrocentric 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 this reason, they are popular in quantum gravity research. For example, Einstein's famous equation E=m·c² becomes simply E=m, i.e. a body with a mass of 5000 Planck Mass units will have an intrinsic energy of 5000 Planck Energy units.

However, the units 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.

Base Planck units

Name	Dimension	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 temperature	Temperature (ML²T^-2/k)	$T_{p}=m_{p}c^{2}/k=(c^{5}\hbar /G)^{\frac {1}{2}}/k$	1.415 × 10³² K
Planck charge	Electric charge (Q)	$q_{p}=(c\hbar 4\pi \varepsilon _{0})^{\frac {1}{2}}$	1.875 × 10^-18 C

Derived Planck units

Name	Dimension	Expression	Approx. SI equivalent measure
Planck force	Force (MLT^-2)	$F_{p}=m_{p}l_{p}/t_{p}^{2}={c^{4}}/G\,\!$	1.210 × 10⁴⁴ N
Planck energy	Energy (ML²T^-2)	$E_{p}=F_{p}l_{p}=c^{2}(c\hbar /G)^{\frac {1}{2}}$	10¹⁹ GeV = 1.956 × 10⁹ J
Planck power	Power (ML²T^-3)	$P_{p}=E_{p}/t_{p}={c^{5}}/G\,\!$	3.629 × 10⁵² W
Planck density	Density (ML^-3)	$\rho _{p}=m_{p}/l_{p}^{3}=c^{5}/(\hbar G^{2})$	5.1 × 10⁹⁶ kg/m³
Planck angular frequency	Frequency (T^-1)	$\omega _{p}=1/t_{p}=(c^{5}/\hbar G)^{\frac {1}{2}}$	1.855 × 10⁴³ rad/s
Planck pressure	Pressure (ML^-1T^-2)	$p_{p}=F_{p}/l_{p}^{2}=c^{7}/(\hbar G^{2})$	4.635 × 10¹¹³ Pa
Planck current	Electric current (QT^-1)	$I_{p}=q_{p}/t_{p}=(c^{6}4\pi \varepsilon _{0}/G)^{\frac {1}{2}}$	3.479 × 10²⁵ A
Planck voltage	Voltage (ML²T^-2Q^-1)	$V_{p}=E_{p}/q_{p}=(c^{4}/(G4\pi \varepsilon _{0}))^{\frac {1}{2}}$	1.0432 × 10²⁷ V
Planck impedance	Resistance (ML²T^&-2TQ^-2)	$Z_{p}=V_{p}/I_{p}=1/(4\pi \varepsilon _{0}c)=Z_{0}/4\pi$	2.9986 × 10¹ Ω

Other Natural Units

Although not part of the system of Planck units (nor are necessarily physical units requiring reference to any physical constants), the following SI derived units are defined naturally from a pure mathematical perspective.

Name	Dimensions	Expression	Approx. SI equivalent measure
Radian	Angle (dimensionless)		1 rad
Steradian	Solid angle (dimensionless)		1 sr

Discussion

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.

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

The Planck impedance comes out to be the Characteristic Impedance of Free Space $Z_{0}$ scaled down by $4\pi$ meaning that, in terms of Planck Units, that $Z_{0}=4\pi$ . This factor comes from the fact that the Coulomb Force Constant $k=1/(4\pi \varepsilon _{0})$ in Coulomb's law that is normallized to 1, rather than the permittivity of free space $\varepsilon _{0}$ . This, and that fact that the Gravitational constant $G$ is normalized rather than $4\pi G$ , could be considered to be an arbitrary definition and perhaps an erroneous one, from the perspective of defining the most natural physical units as the choice for Planck Units.

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'...

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