# Planck units

## Contents

In physics, Planck units are physical units of measurement originally proposed by Max Planck. They form a system of natural units because they are defined exclusively in terms of the following fundamental physical constants — the units are natural because the numerical values of these five universal constants become 1 when expressed in units of this system.

Constant Symbol Dimension
speed of light in vacuum ${ c } \$ L T-1
Gravitational constant ${ G } \$ M-1L3T-2
"reduced Planck's constant" or Dirac's constant $\hbar=\frac{h}{2 \pi}$ where ${h} \$ is Planck's constant ML2T-1
Coulomb force constant $\frac{1}{4 \pi \epsilon_0}$ where ${ \epsilon_0 } \$ is the permittivity in vacuum Q-2 M L3 T-2
Boltzmann constant ${ k } \$ ML2T-2Θ-1

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 :

...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)]...(June 2001 Physics Today)

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.

Natural units have the advantage of simplifying many equations in physics by removing conversion factors. For this reason, they are popular in quantum gravity research.

Newton's Law of universal gravitation

$F = G \frac{m_1 m_2}{r^2}$
becomes
$F = \frac{m_1 m_2}{r^2}$ using Planck units.
$- \frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \frac{\partial \psi}{\partial t} (\mathbf{r}, t)$
becomes
$- \frac{1}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \frac{\partial \psi}{\partial t} (\mathbf{r}, t)$

The energy of a particle or photon with radian frequency ${ \omega } \$ in its wave function

${ E = \hbar \omega } \$
becomes
${ E = \omega } \$ .

Einstein's famous mass-energy equation

${ E = m c^2} \$
becomes simply
${ E = m } \$
${ G_{\mu \nu} = 8 \pi {G \over c^4} T_{\mu \nu}} \$
becomes
${ G_{\mu \nu} = 8 \pi T_{\mu \nu} } \$ .

The unit of temperature is defined so that the mean amount of thermal kinetic energy carried per particle per degree of freedom of motion

${ E = \frac{1}{2} k T } \$
becomes
${ E = \frac{1}{2} T } \$
$F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2}$
becomes
$F = \frac{q_1 q_2}{r^2}$ .
$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0}\rho$
$\nabla \cdot \mathbf{B} = 0 \$
$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}$
become
$\nabla \cdot \mathbf{E} = 4 \pi \rho \$
$\nabla \cdot \mathbf{B} = 0 \$
$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
$\nabla \times \mathbf{B} = 4 \pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}$
when using Planck Units. (The $4 \pi \$ factors would have been eliminated if $\epsilon_0 \$ had been normalized instead of the Coulomb Force Constant $1/(4 \pi \epsilon_0) \$.)

## 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) $t_P = \sqrt{\frac{\hbar G}{c^5}}$ 5.39121 × 10-44 s
Planck length Length (L) $l_P = c \ t_P = \sqrt{\frac{\hbar G}{c^3}}$ 1.61624 × 10-35 m
Planck mass Mass (M) $m_P = \sqrt{\frac{\hbar c}{G}}$ 2.17645 × 10-8 kg
Planck charge Electric charge (Q) $q_P = \sqrt{\hbar c 4 \pi \epsilon_0}$ 1.8755459 × 10-18 C
Planck temperature Temperature (Θ) $T_P = \frac{m_P c^2}{k} = \sqrt{\frac{\hbar c^5}{G k^2}}$ 1.41679 × 1032 K

Accordingly these five dimensionful fundamental physical constants can be expressed in terms of the five base Planck units as:

$c = \frac{l_P}{t_P} \$
$\hbar = \frac{m_P l^2_P}{t_P} \$
$G = \frac{l^3_P}{m_P t^2_P} \$
$\epsilon_0 = \frac{q^2_P t^2_P}{4 \pi m_P l^3_P} \$
$k = \frac{m_P l^2_P}{t^2_P T_P} \$

## 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) $E_P = m_P c^2 = \sqrt{\frac{\hbar c^5}{G}}$ 1.9561 × 109 J
Planck force Force (MLT-2) $F_P = \frac{E_P}{l_P} = \frac{c^4}{G}$ 1.21027 × 1044 N
Planck power Power (ML2T-3) $P_P = \frac{E_P}{t_P} = \frac{c^5}{G}$ 3.62831 × 1052 W
Planck density Density (ML-3) $\rho_P = \frac{m_P}{l_P^3} = \frac{c^5}{\hbar G^2}$ 5.15500 × 1096 kg/m3
Planck angular frequency Frequency (T-1) $\omega_P = \frac{1}{t_P} = \sqrt{\frac{c^5}{\hbar G}}$ 1.85487 × 1043 s-1
Planck pressure Pressure (ML-1T-2) $p_P = \frac{F_P}{l_P^2} =\frac{c^7}{\hbar G^2}$ 4.63309 × 10113 Pa
Planck current Electric current (QT-1) $I_P = \frac{q_P}{t_P} = \sqrt{\frac{c^6 4 \pi \epsilon_0}{G}}$ 3.4789 × 1025 A
Planck voltage Voltage (ML2T-2Q-1) $V_P = \frac{E_P}{q_P} = \sqrt{\frac{c^4}{G 4 \pi \epsilon_0} }$ 1.04295 × 1027 V
Planck impedance Resistance (ML2T-1Q-2) $Z_P = \frac{V_P}{I_P} = \frac{1}{4 \pi \epsilon_0 c} = \frac{Z_0}{4 \pi}$ 2.99792458 × 101 Ω

## 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.

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 ${G} \$ (which has an uncertainty of 1 in 7000).

The Planck charge was not originally defined or proposed by Planck. It is a definition of unit charge that is a natural extension of how the other Planck units were defined and is referred to by physicists in some publications. It may be of interest to note that the elementary charge, measured in terms of the Planck charge, comes out to be

$e = \sqrt{\alpha} \ q_P = 0.085424543 \ q_P \$

where ${\alpha} \$ is the fine-structure constant

$\alpha =\left ( \frac{e}{q_P} \right )^2 = \frac{e^2}{\hbar c 4 \pi \epsilon_0} = \frac{1}{137.03599911}$ .

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. 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 $\sqrt{\alpha} \$), the strength of the electromagnetic force relative to other fundamental forces is proportional to ${\alpha} \$.

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, $Z_0 = 4 \pi Z_P \$. This factor comes from the fact that it is the Coulomb Force Constant $1/(4 \pi \epsilon_0) \$ 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 $\epsilon_0 \$. This, and the fact that the gravitational constant $G \$ is normalized (rather than $4 \pi G \$ or $8 \pi G \$ or $16 \pi G \$) 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.

An increasingly common convention in the literature of particle physics and cosmology is to use reduced Planck units in which $8 \pi G=1 \$ (so called because the Planck mass is reduced by $\sqrt{8 \pi} \$ in these units). These units have the advantage of removing a factor of $8 \pi \$ from the Einstein equation, Einstein-Hilbert action, Friedmann equation, and the Poisson equation for gravitation, at the expense of introducing one into Newton's law of universal gravitation. Another convention which is occasionally seen is to set $16 \pi G=1 \$, which sets the coefficient of R in the Einstein-Hilbert action to unity. Still, another convention sets $4 \pi G = 1 \$ so that the dimensionful constants in the gravitoelectromagnetic (GEM) counterparts to Maxwell's equations are eliminated. The GEM equations are of the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetic interaction with mass (or mass density) replacing charge (or charge density) and $1/(4 \pi G) \$ replacing the permittivity $\epsilon_0 \$ and are applicable in weak gravitational fields or reasonably flat space-time. Like electromagnetic radiation, gravitational radiation propagates at the speed of $c \$ and has characteristic impedance of free space $Z_0 = (4 \pi G)/c \$ which becomes unity if units are judiciously defined so that $c=1 \$ and $4 \pi G = 1 \$.

## Planck units and the invariant scaling of nature

Referring to Duff Comment on time-variation of fundamental constants and 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, as all physical quantities are measured relative to some other like dimensioned values.) We can notice a difference if some dimensionless physical quantity such as $\alpha \$ 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 √8 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:

$a_0 = {{4\pi\epsilon_0\hbar^2}\over{m_e e^2}}= {{m_P}\over{m_e \alpha}} l_P$

Then atoms would be bigger (in one dimension) by √8, each of us would be taller by √8, and so would our meter sticks be taller (and wider and thicker) by a factor of √8 and we would not know the difference.

Our clocks would tick slower by a factor of √32 (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by √32, 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 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 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"...