Planck units

In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants, shown in the table below, in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. Planck units elegantly simplify particular algebraic expressions appearing in physical law. Originally proposed by Max Planck, these units are also known as natural units because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are only one system of natural units among other systems, but might be considered unique in that these units are not based on properties of any prototype, object, or particle (that could be thought of as arbitrarily chosen) but are based only on properties of free space.

Constant	Symbol	Dimension
speed of light in vacuum	${c}\$	L T −1
Gravitational constant	${G}\$	L³ M−1 T −2
Dirac's constant or "reduced Planck's constant"	$\hbar ={\frac {h}{2\pi }}$ where ${h}\$ is Planck's constant	L² M T −1
Coulomb force constant	${\frac {1}{4\pi \epsilon _{0}}}$ where ${\epsilon _{0}}\$ is the permittivity of free space	L³ M T −2 Q−2
Boltzmann constant	${k}\$	L² M T −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 argue that communication with extra-terrestrial intelligence would have to use such a system of units.

Natural units can help physicists reframe questions. Frank Wilczek once described it as:

...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 Planck 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 is approximately the Planck unit of charge but the mass of the protons is far, far less than the Planck mass.

Base Planck units

Constraining the numerical values of the above five fundamental constants to be 1 defines units of length, mass, time, charge, and temperature, called "base units." As is the case with other systems of units (e.g., SI units and cgs units), a host of other "derived" units can be defined in terms of these base units.

Name	Quantity	Expressions	Approximate SI equivalent	Other equivalent
Planck length	Length (L)	$l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}$	1.616252 × 10−35 m
Planck mass	Mass (M)	$m_{P}={\sqrt {\frac {\hbar c}{G}}}$	2.17645 × 10−8 kg	1.311 × 1019 u
Planck time	Time (T)	$t_{P}={\frac {l_{P}}{c}}={\frac {\hbar }{m_{P}c^{2}}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	5.39121 × 10−44 s
Planck charge	Electric charge (Q)	$q_{P}={\sqrt {\hbar c4\pi \epsilon _{0}}}$	1.8755459 × 10−18 C	11.70624 e
Planck temperature	Temperature (Θ)	$T_{P}={\frac {m_{P}c^{2}}{k}}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}$	1.41679 × 1032 K

Derived Planck units

As in other systems of units, the following physical units are derived from the base units.

Name	Quantity	Expression	Approximate SI equivalent
Planck area	Area (L²)	$l_{P}^{2}={\frac {\hbar G}{c^{3}}}$	2.61223 × 10-70 m2
Planck momentum	Momentum (MLT−1)	$m_{P}c={\frac {\hbar }{l_{P}}}={\sqrt {\frac {\hbar c^{3}}{G}}}$	6.52485 kg m/s
Planck energy	Energy (ML²T−2)	$E_{P}=m_{P}c^{2}={\frac {\hbar }{t_{P}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$	1.9561 × 109 J
Planck force	Force (MLT−2)	$F_{P}={\frac {E_{P}}{l_{P}}}={\frac {\hbar }{l_{P}t_{P}}}={\frac {c^{4}}{G}}$	1.21027 × 1044 N
Planck power	Power (ML²T−3)	$P_{P}={\frac {E_{P}}{t_{P}}}={\frac {\hbar }{t_{P}^{2}}}={\frac {c^{5}}{G}}$	3.62831 × 1052 W
Planck density	Density (ML−3)	$\rho _{P}={\frac {m_{P}}{l_{P}^{3}}}={\frac {\hbar t_{P}}{l_{P}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}$	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)	$p_{P}={\frac {F_{P}}{l_{P}^{2}}}={\frac {\hbar }{l_{P}^{3}t_{P}}}={\frac {c^{7}}{\hbar G^{2}}}$	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)	$V_{P}={\frac {E_{P}}{q_{P}}}={\frac {\hbar }{t_{P}q_{P}}}={\sqrt {\frac {c^{4}}{G4\pi \epsilon _{0}}}}$	1.04295 × 1027 V
Planck impedance	Resistance (ML²T−1Q−2)	$Z_{P}={\frac {V_{P}}{I_{P}}}={\frac {\hbar }{q_{P}^{2}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}$	29.9792458 Ω

Nondimensionalization of fundamental physical equations

By setting the numerical values of the five fundamental constants to unity, natural units simplify many equations of physical law, making them nondimensional; examples are shown below. The simplifications Planck units afford make them common in quantum gravity research.

	Common form with dimensional conversion factor	Nondimensionalized form
Newton's Law of universal gravitation	$F=-G{\frac {m_{1}m_{2}}{r^{2}}}$	$F=-{\frac {m_{1}m_{2}}{r^{2}}}$
Schrödinger's equation	$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)$ $=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)$	$-{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)$ $=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)$
Equation relating particle energy to the radian frequency ${\omega }\$ of the wave function	${E=\hbar \omega }\$	${E=\omega }\$
Einstein's mass/energy equation of special relativity	${E=mc^{2}}\$	${E=m}\$
Einstein's field equation for general relativity	${G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\$	${G_{\mu \nu }=8\pi T_{\mu \nu }}\$
Thermal energy per particle per degree of freedom	${E={\frac {1}{2}}kT}\$	${E={\frac {1}{2}}T}\$
Coulomb's law	$F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}$	$F={\frac {q_{1}q_{2}}{r^{2}}}$
Maxwell's equations	$\nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$	$\nabla \cdot \mathbf {E} =4\pi \rho \$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$

Discussion

With some exceptions (the Planck momentum and impedance, possibly the Planck mass and energy as well), base and derived Planck units are impractical for empirical science, engineering, and everyday use, unless rescaled by many orders of magnitude. In fact, 1 Planck unit often represents the largest or smallest value of a physical quantity that makes sense given the current understanding of physical theory. For instance:

• A Planck velocity of 1 equals the speed of light in a vacuum, a maximum;
• At lengths and times of less than about 1 Planck unit, quantum theory as presently understood no longer applies;
• At a Planck temperature of 1, the four fundamental forces unify and all symmetries broken since the start of the Big Bang are restored.

In fact, we presently have no understanding of the Big Bang before the age and size of the universe exceeded approximately one Planck time and one Planck length, and its temperature fell below approximately one Planck temperature. Physical theory applicable on the scale of approximately 1 Planck unit of distance, time, density, or temperature, requires taking account of both quantum effects and general relativity. Doing so would require a theory of quantum gravity which does not yet exist.

At present, the numerical value of the gravitational constant G cannot be determined experimentally to better accuracy than about 1 part in 7000. The resulting uncertainty in G (in SI, cgs, or similar systems of units) is far greater than that of any of the four other fundamental empirical constants. This uncertainty carries over to all Planck units that depend on G. By contrast, the speed of light in SI units is no longer subject to measurement error, because the SI unit of length, the metre, is now defined as some chosen fraction of the distance light travels in 1 second. Hence the speed of light is an exact defined quantity.

Planck neither defined nor proposed the Planck charge. Rather, its definition is a natural extension of the definitions of the other Planck units [1]. Note that the elementary charge e, measured in terms of the Planck charge, is

$e={\sqrt {\alpha }}\ q_{P}=0.085424543\ q_{P}\$

where α is the dimensionless fine-structure constant

$\alpha =\left({\frac {e}{q_{P}}}\right)^{2}={\frac {e^{2}}{\hbar c4\pi \epsilon _{0}}}={\frac {1}{137.03599907}}.$

The numerical value of the fine-structure constant can be seen as resulting from the amount of charge, measured in Planck units, that nature has assigned to electrons, protons, and other charged particles. Because the electromagnetic force between two charged particles is proportional to the product of the charges on each particle (which, given Planck units, are proportional to α^1/2 ), the strength of the electromagnetic force relative to other fundamental forces is proportional to α.

Planck units normalize the Coulomb force constant (4πε₀)⁻¹ to 1, as does the cgs system of units. Consequently, the Planck impedance, Z_P, equals Z₀/4π, where Z₀ is the characteristic impedance of free space. If Planck units normalized the permittivity of free space ε₀ instead, the 4π factors in Maxwell's equations would vanish and the Planck impedance, Z_P, would be identical to Z₀.

Planck units normalize the gravitational constant G in Newton's law of universal gravitation to 1. However Gauss's Law and the concept of flux apply to Newtonian gravity, and the resulting equations are simplified if 4πG rather than G is normalized to 1 (the factor 4π results from the surface area of a sphere of radius r being 4πr ² ). In general relativity and cosmology, G is nearly always preceded by 4π or an integer multiple thereof. These facts suggest other normalizations for G such as:

• (4π)⁻¹. This eliminates the factor of 4π from the gravitoelectromagnetic (GEM) counterparts to Maxwell's equations. (This normalization also sets the ubiquitous 8π of general relativity to 2.) The GEM equations hold in weak gravitational fields or reasonably flat space-time. They have 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 (4πG)⁻¹ replacing the permittivity ε₀. Normalizing G to (4π)⁻¹ also sets to 1 the characteristic impedance of gravitational radiation in free space, Z₀ = 4πG/c. This last normalization is possible because by general relativity, gravitational radiation propagates at the same velocity c as electromagnetic radiation.
• (8π)⁻¹. This removes the ubiquitous factor 8π from the equations of general relativity and cosmology, e.g., the Einstein equation, Einstein-Hilbert action, Friedmann equations, and the Poisson equation for gravitation. Planck units modified so that 8πG is set to 1 are known as reduced Planck units because the Planck mass is divided by a factor of (8π)^1/2.
• (16π)⁻¹. This sets the coefficient of R in the Einstein-Hilbert action to 1.

Planck units and the invariant scaling of nature

Some theoreticians and experimentalists have conjectured that some physical "constants" might actually change over time, a proposition that introduces many difficult questions. A few such questions that are relevant here might be: How would such a change make a noticeable operational difference in physical measurement or, more basically, our perception of reality? If some physical constant had changed, would we even notice it? How would physical reality be different? Which changed constants would result in a meaningful and measurable difference?

[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass m_P] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.

— Barrow 2002

Referring to Michael 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 α or the proton/electron mass ratio changes (atomic structures would change) but if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantity), 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 dimensional 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 perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.

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 (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). 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 (c⁄2 √(32)⁄√(8) continues to equal 299792458 m/s). We would not notice any difference.

In one sense this contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow suggests that if a dimension-dependent universal constant such as c changed, we would easily notice the difference. The disagreement is better thought of as the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether (1) all other dimensionless constants were kept the same, or whether (2) all other dimension-dependent constants are kept the same. The second choice is a somewhat confusing possibility, since most of our units of measurement are defined in relation to the outcomes of physical experiments, and the experimental results depend on the constants. (The only exception is the kilogram.) Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the second choice for "changing a physical constant".

Planck's discovery of 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' (On irreversible radiative processes). 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. The following quote from the 1899 paper 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"...

George Johnstone Stoney introduced a different set of natural units (Stoney units) in 1881, based on G, c, and the electron charge e.

See also

• Dimensional analysis
• Physical constants
• Natural units

References

• John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.

External links

• 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 Planck, M., "Ueber irreversible Strahlungsvorgaenge", Sitzungsbericht Deutsche Akad. Wiss. Berlin, Math-Phys Tech. Kl., 440 (1899)
• K.A. Tomilin: NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System An overview/tutorial of various systems of natural units with some emphasis on Planck units and comparing all systems.